Notation by work

Hypergraph Lumpability and Bayesian Refinement

All notation · Glossary · Source coverage

647 symbol spellings in 850 mathematical passages across 6 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.

The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.

All 647 entries are available below.

*Notation table · 15 passages

Search aliases: *

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3059
z_*,k · positive evidence floor
Exact mathematical passage
```
z_{*,k}
```

+Notation table · 63 passages

Search aliases: +

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3058
e_k^-,e_k^+ · aggregate errors before and after an update
Exact mathematical passage
```
e_k^-,e_k^+
```

=Notation table · 315 passages

Search aliases: =

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3046
P_C=EC · representative projection
Exact mathematical passage
```
P_C=EC
```

ANotation table · 110 passages

Search aliases: A

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3048
A_m^ f,A_m^ c · fine and coarse interaction tensors of order m
Exact mathematical passage
```
A_m^{\fine},A_m^{\coarse}
```

aNotation table · 50 passages

Search aliases: a

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3042
P_i,overline P_a · fine and coarse packet projectors
Exact mathematical passage
```
P_i,\overline P_a
```

A_m^{\coarse}Notation table · 47 passages

Search aliases: A_m^coarse A_m^ c coarse mathrm

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3048
A_m^ f,A_m^ c · fine and coarse interaction tensors of order m
Exact mathematical passage
```
A_m^{\fine},A_m^{\coarse}
```

A_m^{\fine}Notation table · 67 passages

Search aliases: A_m^fine A_m^ f fine mathrm

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3048
A_m^ f,A_m^ c · fine and coarse interaction tensors of order m
Exact mathematical passage
```
A_m^{\fine},A_m^{\coarse}
```

\BupNotation table · 35 passages

Search aliases: Bup B mathsf

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3053
B_λ · Bayesian normalized likelihood update
Exact mathematical passage
```
\Bup_{\lambda}
```

\Bup_{\lambda}Notation table · 5 passages

Search aliases: Bup_λ B_λ Bup lambda mathsf lambda

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3053
B_λ · Bayesian normalized likelihood update
Exact mathematical passage
```
\Bup_{\lambda}
```

CNotation table · 263 passages

Search aliases: C

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3044
C · surjective aggregation or quotient map
Exact mathematical passage
```
C
```

\coarseNotation table · 216 passages

Search aliases: coarse c mathrm

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3041
X_ f,X_ c · fine and coarse state spaces
Exact mathematical passage
```
X_{\fine},X_{\coarse}
```

dNotation table · 17 passages

Search aliases: d

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3056
d_Φ,k · direct prediction quotient defect at step k
Exact mathematical passage
```
d_{\Phi,k}
```

d_{\Bup,k}Notation table · 8 passages

Search aliases: d_Bup,k d_ B,k Bup mathsf

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3057
d_ B,k · direct Bayesian refinement defect at step k
Exact mathematical passage
```
d_{\Bup,k}
```

d_{\Phi,k}Notation table · 8 passages

Search aliases: d_Φ,k Phi

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3056
d_Φ,k · direct prediction quotient defect at step k
Exact mathematical passage
```
d_{\Phi,k}
```

ENotation table · 39 passages

Search aliases: E

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3045
E · right inverse selecting a fine representative of each coarse state
Exact mathematical passage
```
E
```

eNotation table · 39 passages

Search aliases: e

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3058
e_k^-,e_k^+ · aggregate errors before and after an update
Exact mathematical passage
```
e_k^-,e_k^+
```

e_k^+Notation table · 6 passages

Search aliases: e_k^+

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3058
e_k^-,e_k^+ · aggregate errors before and after an update
Exact mathematical passage
```
e_k^-,e_k^+
```

e_k^-Notation table · 10 passages

Search aliases: e_k^-

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3058
e_k^-,e_k^+ · aggregate errors before and after an update
Exact mathematical passage
```
e_k^-,e_k^+
```

FNotation table · 58 passages

Search aliases: F

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3050
F_ f,F_ c · fine and coarse vector fields
Exact mathematical passage
```
F_{\fine},F_{\coarse}
```

F_{\coarse}Notation table · 29 passages

Search aliases: F_coarse F_ c coarse mathrm

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3050
F_ f,F_ c · fine and coarse vector fields
Exact mathematical passage
```
F_{\fine},F_{\coarse}
```

F_{\fine}Notation table · 26 passages

Search aliases: F_fine F_ f fine mathrm

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3050
F_ f,F_ c · fine and coarse vector fields
Exact mathematical passage
```
F_{\fine},F_{\coarse}
```

\fineNotation table · 245 passages

Search aliases: fine f mathrm

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3041
X_ f,X_ c · fine and coarse state spaces
Exact mathematical passage
```
X_{\fine},X_{\coarse}
```

HNotation table · 3 passages

Search aliases: H

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3061
H_m · entropy of the normalized hyperedge energy distribution at order m
Exact mathematical passage
```
H_m
```

H_mNotation table · 3 passages

Search aliases: H_m

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3061
H_m · entropy of the normalized hyperedge energy distribution at order m
Exact mathematical passage
```
H_m
```

INotation table · 24 passages

Search aliases: I

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3047
Q_C=I-EC · projection onto the hidden directions, with range Ker C
Exact mathematical passage
```
Q_C=I-EC
```

iNotation table · 40 passages

Search aliases: i

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3042
P_i,overline P_a · fine and coarse packet projectors
Exact mathematical passage
```
P_i,\overline P_a
```

kNotation table · 94 passages

Search aliases: k

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3054
Γ_k · single step prediction and update map
Exact mathematical passage
```
\Gamma_k
```

\KerNotation table · 19 passages

Search aliases: Ker operatorname

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3047
Q_C=I-EC · projection onto the hidden directions, with range Ker C
Exact mathematical passage
```
\Ker C
```

LNotation table · 51 passages

Search aliases: L

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3052
L_ f,L_ c · positive observation transfer operators
Exact mathematical passage
```
L_{\fine},L_{\coarse}
```

L_{\coarse}Notation table · 11 passages

Search aliases: L_coarse L_ c coarse mathrm

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3052
L_ f,L_ c · positive observation transfer operators
Exact mathematical passage
```
L_{\fine},L_{\coarse}
```

L_{\fine}Notation table · 10 passages

Search aliases: L_fine L_ f fine mathrm

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3052
L_ f,L_ c · positive observation transfer operators
Exact mathematical passage
```
L_{\fine},L_{\coarse}
```

mNotation table · 204 passages

Search aliases: m

Link to this entry ·

Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3048
A_m^ f,A_m^ c · fine and coarse interaction tensors of order m
Exact mathematical passage
```
A_m^{\fine},A_m^{\coarse}
```

NNotation table · 27 passages

Search aliases: N

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3055
Ψ_1:N · cocycle for a finite observation sequence
Exact mathematical passage
```
\Psi_{1:N}
```

\overline PNotation table · 3 passages

Search aliases: overline P overline

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3042
P_i,overline P_a · fine and coarse packet projectors
Exact mathematical passage
```
P_i,\overline P_a
```

\overline P_aNotation table · 3 passages

Search aliases: overline P_a overline

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3042
P_i,overline P_a · fine and coarse packet projectors
Exact mathematical passage
```
P_i,\overline P_a
```

PNotation table · 23 passages

Search aliases: P

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3042
P_i,overline P_a · fine and coarse packet projectors
Exact mathematical passage
```
P_i,\overline P_a
```

P_CNotation table · 7 passages

Search aliases: P_C

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3046
P_C=EC · representative projection
Exact mathematical passage
```
P_C=EC
```

P_iNotation table · 8 passages

Search aliases: P_i

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3042
P_i,overline P_a · fine and coarse packet projectors
Exact mathematical passage
```
P_i,\overline P_a
```

QNotation table · 19 passages

Search aliases: Q

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3047
Q_C=I-EC · projection onto the hidden directions, with range Ker C
Exact mathematical passage
```
Q_C=I-EC
```

Q_CNotation table · 10 passages

Search aliases: Q_C

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3047
Q_C=I-EC · projection onto the hidden directions, with range Ker C
Exact mathematical passage
```
Q_C=I-EC
```

RNotation table · 28 passages

Search aliases: R

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3060
R_m · omitted hyperedge residual tensor
Exact mathematical passage
```
R_m
```

R_mNotation table · 5 passages

Search aliases: R_m

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3060
R_m · omitted hyperedge residual tensor
Exact mathematical passage
```
R_m
```

tNotation table · 34 passages

Search aliases: t

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3051
Φ_t^ f,Φ_t^ c · fine and coarse flows or prediction maps
Exact mathematical passage
```
\Phi_t^{\fine},\Phi_t^{\coarse}
```

XNotation table · 56 passages

Search aliases: X

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3041
X_ f,X_ c · fine and coarse state spaces
Exact mathematical passage
```
X_{\fine},X_{\coarse}
```

X_{\coarse}Notation table · 23 passages

Search aliases: X_coarse X_ c coarse mathrm

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3041
X_ f,X_ c · fine and coarse state spaces
Exact mathematical passage
```
X_{\fine},X_{\coarse}
```

X_{\fine}Notation table · 23 passages

Search aliases: X_fine X_ f fine mathrm

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3041
X_ f,X_ c · fine and coarse state spaces
Exact mathematical passage
```
X_{\fine},X_{\coarse}
```

zNotation table · 54 passages

Search aliases: z

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3059
z_*,k · positive evidence floor
Exact mathematical passage
```
z_{*,k}
```

z_{*,k}Notation table · 6 passages

Search aliases: z_*,k

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3059
z_*,k · positive evidence floor
Exact mathematical passage
```
z_{*,k}
```

\GammaNotation table · 4 passages

Search aliases: Γ Gamma

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3054
Γ_k · single step prediction and update map
Exact mathematical passage
```
\Gamma_k
```

\Gamma_kNotation table · 1 passage

Search aliases: Γ_k Gamma

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3054
Γ_k · single step prediction and update map
Exact mathematical passage
```
\Gamma_k
```

\DeltaNotation table · 33 passages

Search aliases: Δ Delta

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3049
Δ_m · quotient defect at interaction order m
Exact mathematical passage
```
\Delta_m
```

\Delta_mNotation table · 7 passages

Search aliases: Δ_m Delta

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3049
Δ_m · quotient defect at interaction order m
Exact mathematical passage
```
\Delta_m
```

\lambdaNotation table · 75 passages

Search aliases: λ lambda

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3053
B_λ · Bayesian normalized likelihood update
Exact mathematical passage
```
\Bup_{\lambda}
```

\piNotation table · 23 passages

Search aliases: π pi

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3043
π · map from fine packet labels to coarse labels
Exact mathematical passage
```
\pi
```

\PhiNotation table · 33 passages

Search aliases: Φ Phi

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3051
Φ_t^ f,Φ_t^ c · fine and coarse flows or prediction maps
Exact mathematical passage
```
\Phi_t^{\fine},\Phi_t^{\coarse}
```

\Phi_t^{\coarse}Notation table · 7 passages

Search aliases: Φ_t^coarse Φ_t^ c Phi coarse Phi mathrm

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3051
Φ_t^ f,Φ_t^ c · fine and coarse flows or prediction maps
Exact mathematical passage
```
\Phi_t^{\fine},\Phi_t^{\coarse}
```

\Phi_t^{\fine}Notation table · 7 passages

Search aliases: Φ_t^fine Φ_t^ f Phi fine Phi mathrm

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3051
Φ_t^ f,Φ_t^ c · fine and coarse flows or prediction maps
Exact mathematical passage
```
\Phi_t^{\fine},\Phi_t^{\coarse}
```

\PsiNotation table · 6 passages

Search aliases: Ψ Psi

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3055
Ψ_1:N · cocycle for a finite observation sequence
Exact mathematical passage
```
\Psi_{1:N}
```

\Psi_{1:N}Notation table · 1 passage

Search aliases: Ψ_1:N Psi

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Notation table · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 3055
Ψ_1:N · cocycle for a finite observation sequence
Exact mathematical passage
```
\Psi_{1:N}
```

A_0^{\coarse}Definition passage · 2 passages

Search aliases: A_0^coarse A_0^ c coarse mathrm

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Definition passage: Exact interaction quotient · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 381
m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.
Exact mathematical passage
```
CA_0^{\fine}=A_0^{\coarse}
```

A_0^{\fine}Definition passage · 3 passages

Search aliases: A_0^fine A_0^ f fine mathrm

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Definition passage: Exact interaction quotient · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 381
m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.
Exact mathematical passage
```
CA_0^{\fine}=A_0^{\coarse}
```

A_{i\mid j_1,\ldots,j_m}Definition passage · 2 passages

Search aliases: A_imid j_1,…,j_m mid ldots

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Definition passage: Resolved multilinear block and hyperedge · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 293
[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.
Exact mathematical passage
```
\label{eq:resolved-block}
  A_{i\mid j_1,\ldots,j_m}
  =
  Q_iA(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
```

A_mDefinition passage · 4 passages

Search aliases: A_m

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 912
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\cA=(A_m)_{m\geq1}
```

A_m^{\coarse,E}Definition passage · 2 passages

Search aliases: A_m^coarse,E A_m^ c,E coarse mathrm

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Definition passage: Coarse tensor induced by a section · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1546
[Coarse tensor induced by a section] Fix E with CE=I. Define A_m^ c,E = CA_m^ fE^× m.
Exact mathematical passage
```
\label{eq:section-coarse-tensor}
  A_m^{\coarse,E}
  =
  CA_m^{\fine}E^{\times m}.
```

BDefinition passage · 9 passages

Search aliases: B

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Definition passage: Interaction defect · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 392
[Interaction defect] For a proposed coarse tensor B_m, define Δ_m[B_m] = CA_m^ f-B_mC^× m. Exact descent is the condition Δ_m[B_m]=0.
Exact mathematical passage
```
B_m
```

B_+Definition passage · 3 passages

Search aliases: B_+

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 917
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
  \tau=B_+(\tau_1,\ldots,\tau_m)
```

B_mDefinition passage · 6 passages

Search aliases: B_m

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Definition passage: Interaction defect · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 392
[Interaction defect] For a proposed coarse tensor B_m, define Δ_m[B_m] = CA_m^ f-B_mC^× m. Exact descent is the condition Δ_m[B_m]=0.
Exact mathematical passage
```
B_m
```

\bulletDefinition passage · 2 passages

Search aliases: bullet

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 912
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\bullet
```

C^{\times m}Definition passage · 26 passages

Search aliases: C^× m times

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Definition passage: Exact interaction quotient · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 376
[Exact interaction quotient] Let A_m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.
Exact mathematical passage
```
\label{eq:exact-interaction-quotient}
  C A_m^{\fine}
  =
  A_m^{\coarse}C^{\times m}.
```

\cADefinition passage · 1 passage

Search aliases: cA A mathcal

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 912
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\cA=(A_m)_{m\geq1}
```

\cLDefinition passage · 5 passages

Search aliases: cL L mathcal

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Definition passage: Best quotient error · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1519
[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.
Exact mathematical passage
```
\label{eq:best-quotient-error}
  \varepsilon_m^\star
  =
  \inf_{B_m\in\cL^m(X_{\coarse};X_{\coarse})}
  \norm{CA_m^{\fine}-B_mC^{\times m}}.
```

\cL^mDefinition passage · 3 passages

Search aliases: cL^m L^m cL mathcal

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Definition passage: Best quotient error · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1519
[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.
Exact mathematical passage
```
\label{eq:best-quotient-error}
  \varepsilon_m^\star
  =
  \inf_{B_m\in\cL^m(X_{\coarse};X_{\coarse})}
  \norm{CA_m^{\fine}-B_mC^{\times m}}.
```

\cVDefinition passage · 4 passages

Search aliases: cV V mathcal

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 913
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
  \cV_{\bullet}=I_X.
```

\cV_{\bullet}Definition passage · 1 passage

Search aliases: cV_bullet V_bullet cV bullet mathcal bullet

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 913
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
  \cV_{\bullet}=I_X.
```

\cV_{\tau}Definition passage · 1 passage

Search aliases: cV_τ V_τ cV tau mathcal tau

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 921
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\label{eq:tree-evaluation}
  \cV_{\tau}(x_1,\ldots,x_{|\tau|})
  =
  A_m\bigl(
  \cV_{\tau_1}(\mathbf x_1),\ldots,
  \cV_{\tau_m}(\mathbf x_m)
  \bigr),
```

\cV_{\tau_1}Definition passage · 1 passage

Search aliases: cV_τ_1 V_τ_1 cV tau mathcal tau

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 921
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\label{eq:tree-evaluation}
  \cV_{\tau}(x_1,\ldots,x_{|\tau|})
  =
  A_m\bigl(
  \cV_{\tau_1}(\mathbf x_1),\ldots,
  \cV_{\tau_m}(\mathbf x_m)
  \bigr),
```

\cV_{\tau_m}Definition passage · 1 passage

Search aliases: cV_τ_m V_τ_m cV tau mathcal tau

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 921
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\label{eq:tree-evaluation}
  \cV_{\tau}(x_1,\ldots,x_{|\tau|})
  =
  A_m\bigl(
  \cV_{\tau_1}(\mathbf x_1),\ldots,
  \cV_{\tau_m}(\mathbf x_m)
  \bigr),
```

\DownarrowDefinition passage · 3 passages

Search aliases: Downarrow

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Definition passage: Exact interaction quotient · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 372
[Exact interaction quotient] Let A_m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.
Exact mathematical passage
```
  A_m^{\fine}\Downarrow_C A_m^{\coarse},
```

\Downarrow_CDefinition passage · 1 passage

Search aliases: Downarrow_C Downarrow

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Definition passage: Exact interaction quotient · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 372
[Exact interaction quotient] Let A_m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.
Exact mathematical passage
```
  A_m^{\fine}\Downarrow_C A_m^{\coarse},
```

E^{\times m}Definition passage · 1 passage

Search aliases: E^× m times

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Definition passage: Coarse tensor induced by a section · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1546
[Coarse tensor induced by a section] Fix E with CE=I. Define A_m^ c,E = CA_m^ fE^× m.
Exact mathematical passage
```
\label{eq:section-coarse-tensor}
  A_m^{\coarse,E}
  =
  CA_m^{\fine}E^{\times m}.
```

I_{\coarse}Definition passage · 6 passages

Search aliases: I_coarse I_ c coarse mathrm

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Definition passage: Compatible packet aggregation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 353
[Compatible packet aggregation] Suppose (P_i)_i∈ I_ f and (overline P_a)_a∈ I_ c resolve the fine and coarse spaces. A surjection π:I_ f↠ I_ c is compatible with C when overline P_aC = C∑_π(i)=aP_i (a∈ I_ c).
Exact mathematical passage
```
(\overline P_a)_{a\in I_{\coarse}}
```

I_{\fine}Definition passage · 4 passages

Search aliases: I_fine I_ f fine mathrm

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Definition passage: Compatible packet aggregation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 353
[Compatible packet aggregation] Suppose (P_i)_i∈ I_ f and (overline P_a)_a∈ I_ c resolve the fine and coarse spaces. A surjection π:I_ f↠ I_ c is compatible with C when overline P_aC = C∑_π(i)=aP_i (a∈ I_ c).
Exact mathematical passage
```
(P_i)_{i\in I_{\fine}}
```

I_XDefinition passage · 3 passages

Search aliases: I_X

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Definition passage: Packet resolution · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 276
[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.
Exact mathematical passage
```
  P_iP_j=\delta_{ij}P_i,
  \qquad
  \sum_{i\in I}P_i=I_X.
```

\infDefinition passage · 1 passage

Search aliases: inf

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Definition passage: Best quotient error · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1519
[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.
Exact mathematical passage
```
\label{eq:best-quotient-error}
  \varepsilon_m^\star
  =
  \inf_{B_m\in\cL^m(X_{\coarse};X_{\coarse})}
  \norm{CA_m^{\fine}-B_mC^{\times m}}.
```

\inf_{B_m\in\cL^m(X_{\coarse};X_{\coarse})}Definition passage · 1 passage

Search aliases: inf_B_m∈cL^m(X_coarse;X_coarse) inf_B_m∈ L^m(X_ c;X_ c) inf in cL coarse coarse inf in mathcal mathrm mathrm

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Definition passage: Best quotient error · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1519
[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.
Exact mathematical passage
```
\label{eq:best-quotient-error}
  \varepsilon_m^\star
  =
  \inf_{B_m\in\cL^m(X_{\coarse};X_{\coarse})}
  \norm{CA_m^{\fine}-B_mC^{\times m}}.
```

jDefinition passage · 33 passages

Search aliases: j

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Definition passage: Packet resolution · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 276
[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.
Exact mathematical passage
```
  P_iP_j=\delta_{ij}P_i,
  \qquad
  \sum_{i\in I}P_i=I_X.
```

j_1Definition passage · 15 passages

Search aliases: j_1

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Definition passage: Resolved multilinear block and hyperedge · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 293
[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.
Exact mathematical passage
```
\label{eq:resolved-block}
  A_{i\mid j_1,\ldots,j_m}
  =
  Q_iA(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
```

j_mDefinition passage · 15 passages

Search aliases: j_m

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Definition passage: Resolved multilinear block and hyperedge · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 293
[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.
Exact mathematical passage
```
\label{eq:resolved-block}
  A_{i\mid j_1,\ldots,j_m}
  =
  Q_iA(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
```

\maxDefinition passage · 8 passages

Search aliases: max

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Definition passage: Fiber oscillation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1887
[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).
Exact mathematical passage
```
  \osc_a(\lambda^{\fine})
  =
  \max_{i:\pi(i)=a}\lambda_i^{\fine}
  -
  \min_{i:\pi(i)=a}\lambda_i^{\fine},
```

\max_aDefinition passage · 1 passage

Search aliases: max_a max

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Definition passage: Fiber oscillation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1895
[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).
Exact mathematical passage
```
  \osc_C(\lambda^{\fine})
  =\max_a\osc_a(\lambda^{\fine}).
```

\max_{i:\pi(i)=a}Definition passage · 1 passage

Search aliases: max_i:π(i)=a max pi

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Definition passage: Fiber oscillation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1887
[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).
Exact mathematical passage
```
  \osc_a(\lambda^{\fine})
  =
  \max_{i:\pi(i)=a}\lambda_i^{\fine}
  -
  \min_{i:\pi(i)=a}\lambda_i^{\fine},
```

\midDefinition passage · 6 passages

Search aliases: mid

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Definition passage: Resolved multilinear block and hyperedge · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 293
[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.
Exact mathematical passage
```
\label{eq:resolved-block}
  A_{i\mid j_1,\ldots,j_m}
  =
  Q_iA(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
```

\minDefinition passage · 8 passages

Search aliases: min

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Definition passage: Fiber oscillation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1887
[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).
Exact mathematical passage
```
  \osc_a(\lambda^{\fine})
  =
  \max_{i:\pi(i)=a}\lambda_i^{\fine}
  -
  \min_{i:\pi(i)=a}\lambda_i^{\fine},
```

\min_{i:\pi(i)=a}Definition passage · 1 passage

Search aliases: min_i:π(i)=a min pi

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Definition passage: Fiber oscillation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1887
[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).
Exact mathematical passage
```
  \osc_a(\lambda^{\fine})
  =
  \max_{i:\pi(i)=a}\lambda_i^{\fine}
  -
  \min_{i:\pi(i)=a}\lambda_i^{\fine},
```

\normDefinition passage · 54 passages

Search aliases: norm

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Definition passage: Best quotient error · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1519
[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.
Exact mathematical passage
```
\label{eq:best-quotient-error}
  \varepsilon_m^\star
  =
  \inf_{B_m\in\cL^m(X_{\coarse};X_{\coarse})}
  \norm{CA_m^{\fine}-B_mC^{\times m}}.
```

\NrmDefinition passage · 9 passages

Search aliases: Nrm N mathsf

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Definition passage: Normalized positive transfer · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1109
[Normalized positive transfer] For a positive linear operator L:X→ X, define N_ℓ(L)(x) = fracLxℓ(Lx) whenever ℓ(Lx)>0.
Exact mathematical passage
```
\label{eq:normalized-transfer}
  \Nrm_{\ell}(L)(x)
  =
  \frac{Lx}{\ell(Lx)}
```

\Nrm_{\ell}Definition passage · 3 passages

Search aliases: Nrm_ℓ N_ℓ Nrm ell mathsf ell

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Definition passage: Normalized positive transfer · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1109
[Normalized positive transfer] For a positive linear operator L:X→ X, define N_ℓ(L)(x) = fracLxℓ(Lx) whenever ℓ(Lx)>0.
Exact mathematical passage
```
\label{eq:normalized-transfer}
  \Nrm_{\ell}(L)(x)
  =
  \frac{Lx}{\ell(Lx)}
```

\oscDefinition passage · 4 passages

Search aliases: osc operatorname

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Definition passage: Fiber oscillation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1887
[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).
Exact mathematical passage
```
  \osc_a(\lambda^{\fine})
  =
  \max_{i:\pi(i)=a}\lambda_i^{\fine}
  -
  \min_{i:\pi(i)=a}\lambda_i^{\fine},
```

\osc_aDefinition passage · 2 passages

Search aliases: osc_a osc operatorname

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Definition passage: Fiber oscillation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1887
[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).
Exact mathematical passage
```
  \osc_a(\lambda^{\fine})
  =
  \max_{i:\pi(i)=a}\lambda_i^{\fine}
  -
  \min_{i:\pi(i)=a}\lambda_i^{\fine},
```

\osc_CDefinition passage · 3 passages

Search aliases: osc_C osc operatorname

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Definition passage: Fiber oscillation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1895
[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).
Exact mathematical passage
```
  \osc_C(\lambda^{\fine})
  =\max_a\osc_a(\lambda^{\fine}).
```

P_jDefinition passage · 4 passages

Search aliases: P_j

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Definition passage: Packet resolution · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 276
[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.
Exact mathematical passage
```
  P_iP_j=\delta_{ij}P_i,
  \qquad
  \sum_{i\in I}P_i=I_X.
```

P_{j_1}Definition passage · 4 passages

Search aliases: P_j_1

Link to this entry ·

Definition passage: Resolved multilinear block and hyperedge · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 293
[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.
Exact mathematical passage
```
\label{eq:resolved-block}
  A_{i\mid j_1,\ldots,j_m}
  =
  Q_iA(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
```

P_{j_m}Definition passage · 4 passages

Search aliases: P_j_m

Link to this entry ·

Definition passage: Resolved multilinear block and hyperedge · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 293
[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.
Exact mathematical passage
```
\label{eq:resolved-block}
  A_{i\mid j_1,\ldots,j_m}
  =
  Q_iA(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
```

Q_iDefinition passage · 6 passages

Search aliases: Q_i

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Definition passage: Resolved multilinear block and hyperedge · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 293
[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.
Exact mathematical passage
```
\label{eq:resolved-block}
  A_{i\mid j_1,\ldots,j_m}
  =
  Q_iA(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
```

\mathbf xDefinition passage · 1 passage

Search aliases: x mathbf

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 921
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\label{eq:tree-evaluation}
  \cV_{\tau}(x_1,\ldots,x_{|\tau|})
  =
  A_m\bigl(
  \cV_{\tau_1}(\mathbf x_1),\ldots,
  \cV_{\tau_m}(\mathbf x_m)
  \bigr),
```

xDefinition passage · 89 passages

Search aliases: x

Link to this entry ·

Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 921
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\label{eq:tree-evaluation}
  \cV_{\tau}(x_1,\ldots,x_{|\tau|})
  =
  A_m\bigl(
  \cV_{\tau_1}(\mathbf x_1),\ldots,
  \cV_{\tau_m}(\mathbf x_m)
  \bigr),
```

X^mDefinition passage · 4 passages

Search aliases: X^m

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Definition passage: Resolved multilinear block and hyperedge · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 292
[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.
Exact mathematical passage
```
A:X^m\to Y
```

\mathbf x_1Definition passage · 1 passage

Search aliases: x_1 mathbf

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 921
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\label{eq:tree-evaluation}
  \cV_{\tau}(x_1,\ldots,x_{|\tau|})
  =
  A_m\bigl(
  \cV_{\tau_1}(\mathbf x_1),\ldots,
  \cV_{\tau_m}(\mathbf x_m)
  \bigr),
```

x_1Definition passage · 25 passages

Search aliases: x_1

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 921
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\label{eq:tree-evaluation}
  \cV_{\tau}(x_1,\ldots,x_{|\tau|})
  =
  A_m\bigl(
  \cV_{\tau_1}(\mathbf x_1),\ldots,
  \cV_{\tau_m}(\mathbf x_m)
  \bigr),
```

X_{\coarse}^mDefinition passage · 2 passages

Search aliases: X_coarse^m X_ c^m coarse mathrm

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Definition passage: Exact interaction quotient · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 371
[Exact interaction quotient] Let A_m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.
Exact mathematical passage
```
A_m^{\coarse}:X_{\coarse}^m\to X_{\coarse}
```

X_{\fine}^mDefinition passage · 4 passages

Search aliases: X_fine^m X_ f^m fine mathrm

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Definition passage: Exact interaction quotient · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 371
[Exact interaction quotient] Let A_m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.
Exact mathematical passage
```
A_m^{\fine}:X_{\fine}^m\to X_{\fine}
```

X_iDefinition passage · 1 passage

Search aliases: X_i

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Definition passage: Packet resolution · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 282
[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.
Exact mathematical passage
```
  X=\bigoplus_{i\in I}X_i,
  \qquad X_i=P_iX.
```

\mathbf x_mDefinition passage · 1 passage

Search aliases: x_m mathbf

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 921
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\label{eq:tree-evaluation}
  \cV_{\tau}(x_1,\ldots,x_{|\tau|})
  =
  A_m\bigl(
  \cV_{\tau_1}(\mathbf x_1),\ldots,
  \cV_{\tau_m}(\mathbf x_m)
  \bigr),
```

x_{|\tau|}Definition passage · 1 passage

Search aliases: x_|τ| tau

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 921
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\label{eq:tree-evaluation}
  \cV_{\tau}(x_1,\ldots,x_{|\tau|})
  =
  A_m\bigl(
  \cV_{\tau_1}(\mathbf x_1),\ldots,
  \cV_{\tau_m}(\mathbf x_m)
  \bigr),
```

YDefinition passage · 7 passages

Search aliases: Y

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Definition passage: Resolved multilinear block and hyperedge · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 292
[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.
Exact mathematical passage
```
A:X^m\to Y
```

|Definition passage · 6 passages

Search aliases: |

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 921
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\label{eq:tree-evaluation}
  \cV_{\tau}(x_1,\ldots,x_{|\tau|})
  =
  A_m\bigl(
  \cV_{\tau_1}(\mathbf x_1),\ldots,
  \cV_{\tau_m}(\mathbf x_m)
  \bigr),
```

\cdotDefinition passage · 1 passage

Search aliases: · cdot

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Definition passage: Resolved multilinear block and hyperedge · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 293
[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.
Exact mathematical passage
```
\label{eq:resolved-block}
  A_{i\mid j_1,\ldots,j_m}
  =
  Q_iA(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
```

\timesDefinition passage · 66 passages

Search aliases: × times

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Definition passage: Exact interaction quotient · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 376
[Exact interaction quotient] Let A_m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.
Exact mathematical passage
```
\label{eq:exact-interaction-quotient}
  C A_m^{\fine}
  =
  A_m^{\coarse}C^{\times m}.
```

\deltaDefinition passage · 11 passages

Search aliases: δ delta

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Definition passage: Packet resolution · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 276
[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.
Exact mathematical passage
```
  P_iP_j=\delta_{ij}P_i,
  \qquad
  \sum_{i\in I}P_i=I_X.
```

\delta_{ij}Definition passage · 1 passage

Search aliases: δ_ij delta

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Definition passage: Packet resolution · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 276
[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.
Exact mathematical passage
```
  P_iP_j=\delta_{ij}P_i,
  \qquad
  \sum_{i\in I}P_i=I_X.
```

\varepsilonDefinition passage · 16 passages

Search aliases: ε varepsilon

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Definition passage: Best quotient error · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1519
[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.
Exact mathematical passage
```
\label{eq:best-quotient-error}
  \varepsilon_m^\star
  =
  \inf_{B_m\in\cL^m(X_{\coarse};X_{\coarse})}
  \norm{CA_m^{\fine}-B_mC^{\times m}}.
```

\varepsilon_m^\starDefinition passage · 2 passages

Search aliases: ε_m^⋆ varepsilon star

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Definition passage: Best quotient error · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1519
[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.
Exact mathematical passage
```
\label{eq:best-quotient-error}
  \varepsilon_m^\star
  =
  \inf_{B_m\in\cL^m(X_{\coarse};X_{\coarse})}
  \norm{CA_m^{\fine}-B_mC^{\times m}}.
```

\lambda^{\fine}Definition passage · 19 passages

Search aliases: λ^fine λ^ f lambda fine lambda mathrm

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Definition passage: Fiber oscillation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1887
[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).
Exact mathematical passage
```
  \osc_a(\lambda^{\fine})
  =
  \max_{i:\pi(i)=a}\lambda_i^{\fine}
  -
  \min_{i:\pi(i)=a}\lambda_i^{\fine},
```

\lambda_i^{\fine}Definition passage · 4 passages

Search aliases: λ_i^fine λ_i^ f lambda fine lambda mathrm

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Definition passage: Fiber oscillation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1887
[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).
Exact mathematical passage
```
  \osc_a(\lambda^{\fine})
  =
  \max_{i:\pi(i)=a}\lambda_i^{\fine}
  -
  \min_{i:\pi(i)=a}\lambda_i^{\fine},
```

\tauDefinition passage · 7 passages

Search aliases: τ tau

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 917
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
  \tau=B_+(\tau_1,\ldots,\tau_m)
```

\tau_1Definition passage · 4 passages

Search aliases: τ_1 tau

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 917
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
  \tau=B_+(\tau_1,\ldots,\tau_m)
```

\tau_mDefinition passage · 4 passages

Search aliases: τ_m tau

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 917
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
  \tau=B_+(\tau_1,\ldots,\tau_m)
```

\ldotsDefinition passage · 45 passages

Search aliases: … ldots

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Definition passage: Resolved multilinear block and hyperedge · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 293
[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.
Exact mathematical passage
```
\label{eq:resolved-block}
  A_{i\mid j_1,\ldots,j_m}
  =
  Q_iA(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
```

\ellDefinition passage · 17 passages

Search aliases: ℓ ell

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Definition passage: Normalized positive transfer · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1109
[Normalized positive transfer] For a positive linear operator L:X→ X, define N_ℓ(L)(x) = fracLxℓ(Lx) whenever ℓ(Lx)>0.
Exact mathematical passage
```
\label{eq:normalized-transfer}
  \Nrm_{\ell}(L)(x)
  =
  \frac{Lx}{\ell(Lx)}
```

\toDefinition passage · 22 passages

Search aliases: → to

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Definition passage: Resolved multilinear block and hyperedge · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 292
[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.
Exact mathematical passage
```
A:X^m\to Y
```

\twoheadrightarrowDefinition passage · 7 passages

Search aliases: ↠ twoheadrightarrow

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Definition passage: Compatible packet aggregation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 354
[Compatible packet aggregation] Suppose (P_i)_i∈ I_ f and (overline P_a)_a∈ I_ c resolve the fine and coarse spaces. A surjection π:I_ f↠ I_ c is compatible with C when overline P_aC = C∑_π(i)=aP_i (a∈ I_ c).
Exact mathematical passage
```
  \pi:I_{\fine}\twoheadrightarrow I_{\coarse}
```

\inDefinition passage · 40 passages

Search aliases: ∈ in

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Definition passage: Packet resolution · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 275
[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.
Exact mathematical passage
```
(P_i)_{i\in I}
```

\sumDefinition passage · 42 passages

Search aliases: ∑ sum

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Definition passage: Packet resolution · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 276
[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.
Exact mathematical passage
```
  P_iP_j=\delta_{ij}P_i,
  \qquad
  \sum_{i\in I}P_i=I_X.
```

\sum_{i\in I}Definition passage · 1 passage

Search aliases: ∑_i∈ I sum in

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Definition passage: Packet resolution · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 276
[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.
Exact mathematical passage
```
  P_iP_j=\delta_{ij}P_i,
  \qquad
  \sum_{i\in I}P_i=I_X.
```

\sum_{\pi(i)=a}Definition passage · 2 passages

Search aliases: ∑_π(i)=a sum pi

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Definition passage: Compatible packet aggregation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 358
[Compatible packet aggregation] Suppose (P_i)_i∈ I_ f and (overline P_a)_a∈ I_ c resolve the fine and coarse spaces. A surjection π:I_ f↠ I_ c is compatible with C when overline P_aC = C∑_π(i)=aP_i (a∈ I_ c).
Exact mathematical passage
```
\label{eq:packet-aggregation-compatibility}
  \overline P_aC
  =
  C\sum_{\pi(i)=a}P_i
  \qquad(a\in I_{\coarse}).
```

\geqDefinition passage · 21 passages

Search aliases: ≥ geq

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Definition passage: Interaction tree evaluation · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 912
[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
Exact mathematical passage
```
\cA=(A_m)_{m\geq1}
```

\bigoplusDefinition passage · 1 passage

Search aliases: ⊕ bigoplus

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Definition passage: Packet resolution · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 282
[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.
Exact mathematical passage
```
  X=\bigoplus_{i\in I}X_i,
  \qquad X_i=P_iX.
```

\bigoplus_{i\in I}Definition passage · 1 passage

Search aliases: ⊕_i∈ I bigoplus in

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Definition passage: Packet resolution · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 282
[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.
Exact mathematical passage
```
  X=\bigoplus_{i\in I}X_i,
  \qquad X_i=P_iX.
```

\starDefinition passage · 2 passages

Search aliases: ⋆ star

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Definition passage: Best quotient error · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1519
[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.
Exact mathematical passage
```
\label{eq:best-quotient-error}
  \varepsilon_m^\star
  =
  \inf_{B_m\in\cL^m(X_{\coarse};X_{\coarse})}
  \norm{CA_m^{\fine}-B_mC^{\times m}}.
```

\longrightarrowDefinition passage · 5 passages

Search aliases: ⟶ longrightarrow

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Definition passage: Resolved multilinear block and hyperedge · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 299
[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.
Exact mathematical passage
```
  (j_1,\ldots,j_m)\longrightarrow i
```

a^{\fine,(m)}_{i;j_1,\ldots,j_m}Introduction passage · 3 passages

Search aliases: a^fine,(m)_i;j_1,…,j_m a^ f,(m)_i;j_1,…,j_m fine ldots mathrm ldots

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 544
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
\label{eq:fine-coordinates}
  A_m^{\fine}(e_{j_1},\ldots,e_{j_m})
  =
  \sum_{i=1}^n
  a^{\fine,(m)}_{i;j_1,\ldots,j_m}e_i.
```

bIntroduction passage · 25 passages

Search aliases: b

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1719
Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.
Exact mathematical passage
```
a>b
```

C^{\otimes m}Introduction passage · 4 passages

Search aliases: C^⊗ m otimes

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 389
Here C^× m(x_1,…,x_m) =(Cx_1,…,Cx_m). We write C^⊗ m for the induced linear map between algebraic tensor powers.
Exact mathematical passage
```
C^{\otimes m}
```

C_{10}Introduction passage · 9 passages

Search aliases: C_10

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1714
Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.
Exact mathematical passage
```
  X_L\xrightarrow{C_{L,L-1}}X_{L-1}
  \longrightarrow\cdots\longrightarrow
  X_1\xrightarrow{C_{10}}X_0.
```

C_{a,a-1}Introduction passage · 1 passage

Search aliases: C_a,a-1

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1720
Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.
Exact mathematical passage
```
  C_{ab}=C_{b+1,b}\cdots C_{a,a-1}.
```

C_{ab}Introduction passage · 2 passages

Search aliases: C_ab

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1720
Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.
Exact mathematical passage
```
  C_{ab}=C_{b+1,b}\cdots C_{a,a-1}.
```

C_{b+1,b}Introduction passage · 1 passage

Search aliases: C_b+1,b

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1720
Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.
Exact mathematical passage
```
  C_{ab}=C_{b+1,b}\cdots C_{a,a-1}.
```

C_{L,L-1}Introduction passage · 1 passage

Search aliases: C_L,L-1

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1714
Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.
Exact mathematical passage
```
  X_L\xrightarrow{C_{L,L-1}}X_{L-1}
  \longrightarrow\cdots\longrightarrow
  X_1\xrightarrow{C_{10}}X_0.
```

\cCIntroduction passage · 2 passages

Search aliases: cC C mathcal

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2857
Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of…
Exact mathematical passage
```
\cC
```

\cL_{\coarse}Introduction passage · 1 passage

Search aliases: cL_coarse L_ c cL coarse mathcal mathrm

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2858
d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of the drift and covariance, prove positivity of the…
Exact mathematical passage
```
  \cC\cL_{\fine}=\cL_{\coarse}\cC
```

\cL_{\fine}Introduction passage · 1 passage

Search aliases: cL_fine L_ f cL fine mathcal mathrm

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2858
d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of the drift and covariance, prove positivity of the…
Exact mathematical passage
```
  \cC\cL_{\fine}=\cL_{\coarse}\cC
```

DIntroduction passage · 6 passages

Search aliases: D

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2852
Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of…
Exact mathematical passage
```
  \cL\varphi
  =D\varphi[b]
  +\frac12\tr\bigl(\sigma\sigma^TD^2\varphi\bigr).
```

D^2Introduction passage · 1 passage

Search aliases: D^2

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2852
Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of…
Exact mathematical passage
```
  \cL\varphi
  =D\varphi[b]
  +\frac12\tr\bigl(\sigma\sigma^TD^2\varphi\bigr).
```

\ddIntroduction passage · 4 passages

Search aliases: dd d mathrm

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2848
Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of…
Exact mathematical passage
```
  \dd X_t=b(X_t)\dd t+\sigma(X_t)\dd W_t,
```

\dot xIntroduction passage · 6 passages

Search aliases: dot x dot

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 149
The quotient problem. Let X_ f be a fine state space, let X_ c be a coarse state space, and let C:X_ f↠ X_ c be a surjective linear map. A fine trajectory satisfies dot x(t)=F_ f(x(t)), x(0)=x_0, and the coarse observable of interest is z(t)=Cx(t). The first question is whether z(t) obeys a closed equation of its own. Equivalently, can we define a vector field F_ c such that C F_ f=F_ cC? When this identity holds and both initial value problems have unique solutions, aggregation commutes with ev…
Exact mathematical passage
```
\label{eq:fine-flow-intro}
  \dot x(t)=F_{\fine}(x(t)),
  \qquad x(0)=x_0,
```

e_iIntroduction passage · 3 passages

Search aliases: e_i

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 537
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
(e_i)
```

e_{j_1}Introduction passage · 2 passages

Search aliases: e_j_1

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 544
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
\label{eq:fine-coordinates}
  A_m^{\fine}(e_{j_1},\ldots,e_{j_m})
  =
  \sum_{i=1}^n
  a^{\fine,(m)}_{i;j_1,\ldots,j_m}e_i.
```

e_{j_m}Introduction passage · 2 passages

Search aliases: e_j_m

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 544
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
\label{eq:fine-coordinates}
  A_m^{\fine}(e_{j_1},\ldots,e_{j_m})
  =
  \sum_{i=1}^n
  a^{\fine,(m)}_{i;j_1,\ldots,j_m}e_i.
```

fIntroduction passage · 10 passages

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 537
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
(f_a)
```

f_aIntroduction passage · 2 passages

Search aliases: f_a

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 537
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
(f_a)
```

f_{\pi(i)}Introduction passage · 1 passage

Search aliases: f_π(i) pi

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 538
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
\label{eq:block-sum-C}
  Ce_i=f_{\pi(i)},
  \qquad
  (Cx)_a=\sum_{\pi(i)=a}x_i.
```

nIntroduction passage · 13 passages

Search aliases: n

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 528
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
  I_{\fine}=\{1,\ldots,n\},
  \qquad
  I_{\coarse}=\{1,\ldots,q\},
```

\oneIntroduction passage · 11 passages

Search aliases: one 1 mathbf

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1792
Defect in the observation intertwining law. Let L_ f and L_ c be positive linear observation operators, and choose a scale s>0. On a finite simplex, we write N(L) for normalization with respect to the total mass 1^T. Define R_L = CL_ f-sL_ cC.
Exact mathematical passage
```
\one^T
```

\one^TIntroduction passage · 10 passages

Search aliases: one^T 1^T one mathbf

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1792
Defect in the observation intertwining law. Let L_ f and L_ c be positive linear observation operators, and choose a scale s>0. On a finite simplex, we write N(L) for normalization with respect to the total mass 1^T. Define R_L = CL_ f-sL_ cC.
Exact mathematical passage
```
\one^T
```

qIntroduction passage · 17 passages

Search aliases: q

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 528
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
  I_{\fine}=\{1,\ldots,n\},
  \qquad
  I_{\coarse}=\{1,\ldots,q\},
```

\RIntroduction passage · 12 passages

Search aliases: R mathbb

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 537
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
\R^n
```

\R^nIntroduction passage · 1 passage

Search aliases: R^n R mathbb

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 537
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
\R^n
```

\R^qIntroduction passage · 3 passages

Search aliases: R^q R mathbb

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 537
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
\R^q
```

R_LIntroduction passage · 5 passages

Search aliases: R_L

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1793
Defect in the observation intertwining law. Let L_ f and L_ c be positive linear observation operators, and choose a scale s>0. On a finite simplex, we write N(L) for normalization with respect to the total mass 1^T. Define R_L = CL_ f-sL_ cC.
Exact mathematical passage
```
\label{eq:observation-linear-defect}
  R_L
  =
  CL_{\fine}-sL_{\coarse}C.
```

sIntroduction passage · 46 passages

Search aliases: s

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1792
Defect in the observation intertwining law. Let L_ f and L_ c be positive linear observation operators, and choose a scale s>0. On a finite simplex, we write N(L) for normalization with respect to the total mass 1^T. Define R_L = CL_ f-sL_ cC.
Exact mathematical passage
```
s>0
```

TIntroduction passage · 43 passages

Search aliases: T

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1792
Defect in the observation intertwining law. Let L_ f and L_ c be positive linear observation operators, and choose a scale s>0. On a finite simplex, we write N(L) for normalization with respect to the total mass 1^T. Define R_L = CL_ f-sL_ cC.
Exact mathematical passage
```
\one^T
```

\trIntroduction passage · 1 passage

Search aliases: tr operatorname

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2852
Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of…
Exact mathematical passage
```
  \cL\varphi
  =D\varphi[b]
  +\frac12\tr\bigl(\sigma\sigma^TD^2\varphi\bigr).
```

UIntroduction passage · 4 passages

Search aliases: U

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 995
[Exact quotient flow] Let U_ f⊆ X_ f and U_ c⊆ X_ c be open, with C(U_ f)⊆ U_ c. Suppose F_ f and F_ c are locally Lipschitz and satisfy CF_ f=F_ cC on U_ f. Let Φ_t^ f(x) and Φ_t^ c(z) denote the unique maximal local flows. Whenever both sides are defined, CΦ_t^ f(x) = Φ_t^ c(Cx).
Exact mathematical passage
```
U_{\fine}\subseteq X_{\fine}
```

U_{\coarse}Introduction passage · 2 passages

Search aliases: U_coarse U_ c coarse mathrm

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 995
[Exact quotient flow] Let U_ f⊆ X_ f and U_ c⊆ X_ c be open, with C(U_ f)⊆ U_ c. Suppose F_ f and F_ c are locally Lipschitz and satisfy CF_ f=F_ cC on U_ f. Let Φ_t^ f(x) and Φ_t^ c(z) denote the unique maximal local flows. Whenever both sides are defined, CΦ_t^ f(x) = Φ_t^ c(Cx).
Exact mathematical passage
```
U_{\coarse}\subseteq X_{\coarse}
```

U_{\fine}Introduction passage · 3 passages

Search aliases: U_fine U_ f fine mathrm

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 995
[Exact quotient flow] Let U_ f⊆ X_ f and U_ c⊆ X_ c be open, with C(U_ f)⊆ U_ c. Suppose F_ f and F_ c are locally Lipschitz and satisfy CF_ f=F_ cC on U_ f. Let Φ_t^ f(x) and Φ_t^ c(z) denote the unique maximal local flows. Whenever both sides are defined, CΦ_t^ f(x) = Φ_t^ c(Cx).
Exact mathematical passage
```
U_{\fine}\subseteq X_{\fine}
```

WIntroduction passage · 1 passage

Search aliases: W

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2848
Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of…
Exact mathematical passage
```
  \dd X_t=b(X_t)\dd t+\sigma(X_t)\dd W_t,
```

W_tIntroduction passage · 1 passage

Search aliases: W_t

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2848
Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of…
Exact mathematical passage
```
  \dd X_t=b(X_t)\dd t+\sigma(X_t)\dd W_t,
```

X_0Introduction passage · 2 passages

Search aliases: X_0

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1714
Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.
Exact mathematical passage
```
  X_L\xrightarrow{C_{L,L-1}}X_{L-1}
  \longrightarrow\cdots\longrightarrow
  X_1\xrightarrow{C_{10}}X_0.
```

x_0Introduction passage · 5 passages

Search aliases: x_0

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 149
The quotient problem. Let X_ f be a fine state space, let X_ c be a coarse state space, and let C:X_ f↠ X_ c be a surjective linear map. A fine trajectory satisfies dot x(t)=F_ f(x(t)), x(0)=x_0, and the coarse observable of interest is z(t)=Cx(t). The first question is whether z(t) obeys a closed equation of its own. Equivalently, can we define a vector field F_ c such that C F_ f=F_ cC? When this identity holds and both initial value problems have unique solutions, aggregation commutes with ev…
Exact mathematical passage
```
\label{eq:fine-flow-intro}
  \dot x(t)=F_{\fine}(x(t)),
  \qquad x(0)=x_0,
```

X_1Introduction passage · 2 passages

Search aliases: X_1

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1714
Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.
Exact mathematical passage
```
  X_L\xrightarrow{C_{L,L-1}}X_{L-1}
  \longrightarrow\cdots\longrightarrow
  X_1\xrightarrow{C_{10}}X_0.
```

x_iIntroduction passage · 1 passage

Search aliases: x_i

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 538
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
\label{eq:block-sum-C}
  Ce_i=f_{\pi(i)},
  \qquad
  (Cx)_a=\sum_{\pi(i)=a}x_i.
```

X_LIntroduction passage · 1 passage

Search aliases: X_L

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1714
Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.
Exact mathematical passage
```
  X_L\xrightarrow{C_{L,L-1}}X_{L-1}
  \longrightarrow\cdots\longrightarrow
  X_1\xrightarrow{C_{10}}X_0.
```

X_{L-1}Introduction passage · 1 passage

Search aliases: X_L-1

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1714
Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.
Exact mathematical passage
```
  X_L\xrightarrow{C_{L,L-1}}X_{L-1}
  \longrightarrow\cdots\longrightarrow
  X_1\xrightarrow{C_{10}}X_0.
```

x_mIntroduction passage · 12 passages

Search aliases: x_m

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 385
Here C^× m(x_1,…,x_m) =(Cx_1,…,Cx_m). We write C^⊗ m for the induced linear map between algebraic tensor powers.
Exact mathematical passage
```
  C^{\times m}(x_1,\ldots,x_m)
  =(Cx_1,\ldots,Cx_m).
```

X_tIntroduction passage · 1 passage

Search aliases: X_t

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2848
Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of…
Exact mathematical passage
```
  \dd X_t=b(X_t)\dd t+\sigma(X_t)\dd W_t,
```

\xrightarrowIntroduction passage · 2 passages

Search aliases: xrightarrow

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1714
Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.
Exact mathematical passage
```
  X_L\xrightarrow{C_{L,L-1}}X_{L-1}
  \longrightarrow\cdots\longrightarrow
  X_1\xrightarrow{C_{10}}X_0.
```

\Delta_m^{ab}Introduction passage · 1 passage

Search aliases: Δ_m^ab Delta

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1723
Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.
Exact mathematical passage
```
\Delta_m^{ab}
```

\sigmaIntroduction passage · 2 passages

Search aliases: σ sigma

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2848
Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of…
Exact mathematical passage
```
  \dd X_t=b(X_t)\dd t+\sigma(X_t)\dd W_t,
```

\sigma^TIntroduction passage · 1 passage

Search aliases: σ^T sigma

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2852
Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of…
Exact mathematical passage
```
  \cL\varphi
  =D\varphi[b]
  +\frac12\tr\bigl(\sigma\sigma^TD^2\varphi\bigr).
```

\varphiIntroduction passage · 1 passage

Search aliases: φ varphi

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2852
Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pullback or averaging map appropriate to observables. A complete theorem would characterize closure of…
Exact mathematical passage
```
  \cL\varphi
  =D\varphi[b]
  +\frac12\tr\bigl(\sigma\sigma^TD^2\varphi\bigr).
```

\sum_{i=1}^nIntroduction passage · 1 passage

Search aliases: ∑_i=1^n sum

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 544
Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
\label{eq:fine-coordinates}
  A_m^{\fine}(e_{j_1},\ldots,e_{j_m})
  =
  \sum_{i=1}^n
  a^{\fine,(m)}_{i;j_1,\ldots,j_m}e_i.
```

\subseteqIntroduction passage · 7 passages

Search aliases: ⊆ subseteq

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 995
[Exact quotient flow] Let U_ f⊆ X_ f and U_ c⊆ X_ c be open, with C(U_ f)⊆ U_ c. Suppose F_ f and F_ c are locally Lipschitz and satisfy CF_ f=F_ cC on U_ f. Let Φ_t^ f(x) and Φ_t^ c(z) denote the unique maximal local flows. Whenever both sides are defined, CΦ_t^ f(x) = Φ_t^ c(Cx).
Exact mathematical passage
```
U_{\fine}\subseteq X_{\fine}
```

\otimesIntroduction passage · 17 passages

Search aliases: ⊗ otimes

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 389
Here C^× m(x_1,…,x_m) =(Cx_1,…,Cx_m). We write C^⊗ m for the induced linear map between algebraic tensor powers.
Exact mathematical passage
```
C^{\otimes m}
```

\cdotsIntroduction passage · 8 passages

Search aliases: ⋯ cdots

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Introduction passage · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1714
Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.
Exact mathematical passage
```
  X_L\xrightarrow{C_{L,L-1}}X_{L-1}
  \longrightarrow\cdots\longrightarrow
  X_1\xrightarrow{C_{10}}X_0.
```

a^{\coarse,(m)}_{a;\mathbf b}Use in source · 1 passage

Search aliases: a^coarse,(m)_a; b a^ c,(m)_a; b coarse mathbf mathrm mathbf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2394
put a and coarse input tuple b: [label=arabic*.,leftmargin=2.3em] item Evaluate g^(m)_a; j for all j∈ G_ b. item In exact arithmetic, test whether all values are equal. With floating point input, compare their spread with a declared tolerance and record that tolerance. item When the exact test passes, assign the common value to a^ c,(m)_a; b. When only the numerical tolerance is met, assign a declared representative such as the mean or the midrange. item Otherwise, return a witness (a, j, j') wi…
Exact mathematical passage
```
a^{\coarse,(m)}_{a;\mathbf b}
```

a^{\coarse,(m)}_{a;b_1,\ldots,b_m}Use in source · 1 passage

Search aliases: a^coarse,(m)_a;b_1,…,b_m a^ c,(m)_a;b_1,…,b_m coarse ldots mathrm ldots

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 571
The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m, π(j_r)=b_r.
Exact mathematical passage
```
\label{eq:coarse-coordinate-formula}
  a^{\coarse,(m)}_{a;b_1,\ldots,b_m}
  =
  g^{(m)}_{a;j_1,\ldots,j_m},
  \qquad
  \pi(j_r)=b_r.
```

a^{\coarse,\mathrm{LS}}_{a;\mathbf b}Use in source · 1 passage

Search aliases: a^coarse,LS_a; b a^ c,LS_a; b coarse mathrm mathbf mathrm mathrm mathbf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2411
Approximate coarse coefficients. When exact fiber constancy fails, the approximation depends on a chosen loss and, when relevant, a task distribution. Under scalar squared loss with nonnegative weights w_ j on a fiber product, assume their total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ G_ bw_ j . For scalar worst case loss, a minimax representative is the midrange a^ c,∞_a; b = frac12 ( max_ j∈ G_ bg_a; j + min_ j∈ G_ bg_a; j ), with…
Exact mathematical passage
```
\label{eq:least-squares-coarse-coeff}
  a^{\coarse,\mathrm{LS}}_{a;\mathbf b}
  =
  \frac{
  \sum_{\mathbf j\in\cG_{\mathbf b}}
  w_{\mathbf j}g_{a;\mathbf j}
  }{
  \sum_{\mathbf j\in\cG_{\mathbf b}}w_{\mathbf j}
  }.
```

a^{\coarse,\infty}_{a;\mathbf b}Use in source · 1 passage

Search aliases: a^coarse,∞_a; b a^ c,∞_a; b coarse infty mathbf mathrm infty mathbf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2422
r total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ G_ bw_ j . For scalar worst case loss, a minimax representative is the midrange a^ c,∞_a; b = frac12 ( max_ j∈ G_ bg_a; j + min_ j∈ G_ bg_a; j ), with optimal scalar fiber error ε^∞_a; b = frac12 ( max_ j∈ G_ bg_a; j - min_ j∈ G_ bg_a; j ). These formulas are local coefficient summaries. A global operator norm can couple several fibers and may require a joint optimization.
Exact mathematical passage
```
\label{eq:minimax-coarse-coeff}
  a^{\coarse,\infty}_{a;\mathbf b}
  =
  \frac12
  \left(
  \max_{\mathbf j\in\cG_{\mathbf b}}g_{a;\mathbf j}
  +
  \min_{\mathbf j\in\cG_{\mathbf b}}g_{a;\mathbf j}
  \right),
```

a^{\fine,(1)}_{i;j}Use in source · 1 passage

Search aliases: a^fine,(1)_i;j a^ f,(1)_i;j fine mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 624
[Unary aggregate rate criterion] For m=1, exact descent is equivalent to ∑_i:π(i)=aa^ f,(1)_i;j = ∑_i:π(i)=aa^ f,(1)_i;j' for all coarse outputs a whenever π(j)=π(j').
Exact mathematical passage
```
\label{eq:unary-aggregate-rate}
  \sum_{i:\pi(i)=a}a^{\fine,(1)}_{i;j}
  =
  \sum_{i:\pi(i)=a}a^{\fine,(1)}_{i;j'}
```

a^{\fine,(1)}_{i;j'}Use in source · 1 passage

Search aliases: a^fine,(1)_i;j' a^ f,(1)_i;j' fine mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 624
[Unary aggregate rate criterion] For m=1, exact descent is equivalent to ∑_i:π(i)=aa^ f,(1)_i;j = ∑_i:π(i)=aa^ f,(1)_i;j' for all coarse outputs a whenever π(j)=π(j').
Exact mathematical passage
```
\label{eq:unary-aggregate-rate}
  \sum_{i:\pi(i)=a}a^{\fine,(1)}_{i;j}
  =
  \sum_{i:\pi(i)=a}a^{\fine,(1)}_{i;j'}
```

a^{\fine,(2)}_{i;j',k'}Use in source · 1 passage

Search aliases: a^fine,(2)_i;j',k' a^ f,(2)_i;j',k' fine mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 640
[Quadratic hypergraph criterion] For m=2, exact descent is equivalent to ∑_i:π(i)=aa^ f,(2)_i;j,k = ∑_i:π(i)=aa^ f,(2)_i;j',k' whenever π(j)=π(j'), π(k)=π(k'). The resulting coarse interaction is the directed quadratic hyperedge (π(j),π(k))⟶ a.
Exact mathematical passage
```
  \sum_{i:\pi(i)=a}a^{\fine,(2)}_{i;j,k}
  =
  \sum_{i:\pi(i)=a}a^{\fine,(2)}_{i;j',k'}
```

a^{\fine,(2)}_{i;j,k}Use in source · 1 passage

Search aliases: a^fine,(2)_i;j,k a^ f,(2)_i;j,k fine mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 640
[Quadratic hypergraph criterion] For m=2, exact descent is equivalent to ∑_i:π(i)=aa^ f,(2)_i;j,k = ∑_i:π(i)=aa^ f,(2)_i;j',k' whenever π(j)=π(j'), π(k)=π(k'). The resulting coarse interaction is the directed quadratic hyperedge (π(j),π(k))⟶ a.
Exact mathematical passage
```
  \sum_{i:\pi(i)=a}a^{\fine,(2)}_{i;j,k}
  =
  \sum_{i:\pi(i)=a}a^{\fine,(2)}_{i;j',k'}
```

A_1^{\fine}Use in source · 1 passage

Search aliases: A_1^fine A_1^ f fine mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 174
Why hypergraphs appear. Suppose a local analytic vector field has the expansion F_ f(x)=∑_m≥0A_m^ f[x,…,x]. The linear coefficient A_1^ f moves one input packet into one output packet. The quadratic coefficient A_2^ f combines two input packets before producing an output. In general, A_m^ f combines m inputs. We record this incidence by the directed hyperedge (j_1,…,j_m)⟶ i.
Exact mathematical passage
```
A_1^{\fine}
```

A_2^{\fine}Use in source · 1 passage

Search aliases: A_2^fine A_2^ f fine mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 174
Why hypergraphs appear. Suppose a local analytic vector field has the expansion F_ f(x)=∑_m≥0A_m^ f[x,…,x]. The linear coefficient A_1^ f moves one input packet into one output packet. The quadratic coefficient A_2^ f combines two input packets before producing an output. In general, A_m^ f combines m inputs. We record this incidence by the directed hyperedge (j_1,…,j_m)⟶ i.
Exact mathematical passage
```
A_2^{\fine}
```

A_{\coarse}Use in source · 2 passages

Search aliases: A_coarse A_ c coarse mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1021
[Linear semigroup naturality] If CA_ f=A_ cC, then Ce^tA_ f = e^tA_ cC for every t.
Exact mathematical passage
```
CA_{\fine}=A_{\coarse}C
```

A_{\coarse}^kUse in source · 1 passage

Search aliases: A_coarse^k A_ c^k coarse mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1031
Either apply [source reference] to the linear systems or sum the exponential series using CA_ f^k=A_ c^kC.
Exact mathematical passage
```
CA_{\fine}^k=A_{\coarse}^kC
```

A_{\fine}Use in source · 2 passages

Search aliases: A_fine A_ f fine mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1021
[Linear semigroup naturality] If CA_ f=A_ cC, then Ce^tA_ f = e^tA_ cC for every t.
Exact mathematical passage
```
CA_{\fine}=A_{\coarse}C
```

A_{\fine}^kUse in source · 1 passage

Search aliases: A_fine^k A_ f^k fine mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1031
Either apply [source reference] to the linear systems or sum the exponential series using CA_ f^k=A_ c^kC.
Exact mathematical passage
```
CA_{\fine}^k=A_{\coarse}^kC
```

A_m^{(0)}Use in source · 6 passages

Search aliases: A_m^(0)

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 717
[Transitivity of exact interaction quotients] If A_m^(2)Downarrow_C_21A_m^(1) and A_m^(1)Downarrow_C_10A_m^(0), then A_m^(2)Downarrow_C_20A_m^(0).
Exact mathematical passage
```
  A_m^{(2)}\Downarrow_{C_{21}}A_m^{(1)}
  \qquad\text{and}\qquad
  A_m^{(1)}\Downarrow_{C_{10}}A_m^{(0)},
```

A_m^{(1)}Use in source · 6 passages

Search aliases: A_m^(1)

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 717
[Transitivity of exact interaction quotients] If A_m^(2)Downarrow_C_21A_m^(1) and A_m^(1)Downarrow_C_10A_m^(0), then A_m^(2)Downarrow_C_20A_m^(0).
Exact mathematical passage
```
  A_m^{(2)}\Downarrow_{C_{21}}A_m^{(1)}
  \qquad\text{and}\qquad
  A_m^{(1)}\Downarrow_{C_{10}}A_m^{(0)},
```

A_m^{(2)}Use in source · 6 passages

Search aliases: A_m^(2)

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 717
[Transitivity of exact interaction quotients] If A_m^(2)Downarrow_C_21A_m^(1) and A_m^(1)Downarrow_C_10A_m^(0), then A_m^(2)Downarrow_C_20A_m^(0).
Exact mathematical passage
```
  A_m^{(2)}\Downarrow_{C_{21}}A_m^{(1)}
  \qquad\text{and}\qquad
  A_m^{(1)}\Downarrow_{C_{10}}A_m^{(0)},
```

A_m^cUse in source · 2 passages

Search aliases: A_m^c

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Use in source · MANUSCRIPT_STATUS.md, line 56
CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.
Exact mathematical passage
```
CA_m^f=A_m^cC^{\times m}
\Rightarrow
CF_f=F_cC
\Rightarrow
C\Phi_t^f=\Phi_t^cC.
```

A_m^fUse in source · 2 passages

Search aliases: A_m^f

Link to this entry ·

Use in source · MANUSCRIPT_STATUS.md, line 56
CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.
Exact mathematical passage
```
CA_m^f=A_m^cC^{\times m}
\Rightarrow
CF_f=F_cC
\Rightarrow
C\Phi_t^f=\Phi_t^cC.
```

A_m^{\mathrm{keep}}Use in source · 2 passages

Search aliases: A_m^keep mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2318
Combined quotient and pruning defect. Let the full fine coefficient split as A_m^ f =A_m^keep+R_m^ f. For a proposed coarse coefficient A_m^ c, define Δ_m^total =CA_m^ f-A_m^ cC^× m. Then Δ_m^total · = underbraceCA_m^keep-A_m^ cC^× m _Δ_m^keep +CR_m^ f. Therefore normΔ_m^total ≤ normΔ_m^keep +normCnormR_m^ f.
Exact mathematical passage
```
  A_m^{\fine}
  =A_m^{\mathrm{keep}}+R_m^{\fine}.
```

A_{PP}Use in source · 1 passage

Search aliases: A_PP

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2865
Coarse dynamics with memory. Eliminating hidden packets generally produces memory. The target form is dot z(t) =A_PPz(t) +∫_0^tK(t-s)z(s) d s +η(t). The quotient theory should allow a fine system without memory to produce coarse dynamics with memory. Volterra hyperedges and fractional kernels would then be treated as native coarse interactions rather than as evidence that quotienting failed.
Exact mathematical passage
```
  \dot z(t)
  =A_{PP}z(t)
  +\int_0^tK(t-s)z(s)\dd s
  +\eta(t).
```

\absUse in source · 12 passages

Search aliases: abs

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1773
Write a= 1^Tu and b= 1^Tv. Then normfracua-fracvb_1 · ≤ fracnormu-v_1a + normv_1absfrac1a-frac1b · = fracnormu-v_1a + fracabsa-ba · ≤ frac2normu-v_1a. Interchanging u and v gives the same inequality with b in the denominator. Taking the smaller of the two upper bounds yields the denominator max{a,b}.
Exact mathematical passage
```
\norm{\frac{u}{a}-\frac{v}{b}}_1
&\leq
\frac{\norm{u-v}_1}{a}
+
\norm{v}_1\abs{\frac1a-\frac1b}\\
&=
\frac{\norm{u-v}_1}{a}
+
\frac{\abs{a-b}}{a}\\
&\leq
\frac{2\norm{u-v}_1}{a}.
```

\mathbf bUse in source · 8 passages

Search aliases: b mathbf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2382
Exact test for every interaction order. For a tensor of order m in scalar packet coordinates, compute the aggregated coefficient array g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m. For every coarse input tuple b=(b_1,…,b_m), define the fine fiber product G_ b = {(j_1,…,j_m):π(j_r)=b_r for all r}.
Exact mathematical passage
```
\mathbf b=(b_1,\ldots,b_m)
```

\mathsf BUse in source · 3 passages

Search aliases: B mathsf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2966
Certificate schemas and reproducibility. A minimal interaction certificate stores L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite ledger as JSON files and CSV tables. A certificate records a computation under specified inputs. The ledger provides cryptographic authenticity only …
Exact mathematical passage
```
\label{eq:bayes-certificate-schema}
  \mathsf B_y
  =
  (\lambda_y^{\fine},\lambda_y^{\coarse},s_y,r_y,
  z_{*,y},d_{\Bup,y}).
```

b_1Use in source · 3 passages

Search aliases: b_1

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 571
The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m, π(j_r)=b_r.
Exact mathematical passage
```
\label{eq:coarse-coordinate-formula}
  a^{\coarse,(m)}_{a;b_1,\ldots,b_m}
  =
  g^{(m)}_{a;j_1,\ldots,j_m},
  \qquad
  \pi(j_r)=b_r.
```

\mathsf B_cUse in source · 1 passage

Search aliases: B_c mathsf

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Use in source · README.md, line 31
CL_f=sL_cC Longrightarrow C B_f= B_cC.
Exact mathematical passage
```
CL_f=sL_cC
\Longrightarrow
C\mathsf B_f=\mathsf B_cC.
```

\mathsf B_fUse in source · 1 passage

Search aliases: B_f mathsf

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Use in source · README.md, line 31
CL_f=sL_cC Longrightarrow C B_f= B_cC.
Exact mathematical passage
```
CL_f=sL_cC
\Longrightarrow
C\mathsf B_f=\mathsf B_cC.
```

b_mUse in source · 3 passages

Search aliases: b_m

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 571
The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m, π(j_r)=b_r.
Exact mathematical passage
```
\label{eq:coarse-coordinate-formula}
  a^{\coarse,(m)}_{a;b_1,\ldots,b_m}
  =
  g^{(m)}_{a;j_1,\ldots,j_m},
  \qquad
  \pi(j_r)=b_r.
```

b_rUse in source · 2 passages

Search aliases: b_r

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 571
The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m, π(j_r)=b_r.
Exact mathematical passage
```
\label{eq:coarse-coordinate-formula}
  a^{\coarse,(m)}_{a;b_1,\ldots,b_m}
  =
  g^{(m)}_{a;j_1,\ldots,j_m},
  \qquad
  \pi(j_r)=b_r.
```

\mathsf B_yUse in source · 1 passage

Search aliases: B_y mathsf

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2966
Certificate schemas and reproducibility. A minimal interaction certificate stores L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite ledger as JSON files and CSV tables. A certificate records a computation under specified inputs. The ledger provides cryptographic authenticity only …
Exact mathematical passage
```
\label{eq:bayes-certificate-schema}
  \mathsf B_y
  =
  (\lambda_y^{\fine},\lambda_y^{\coarse},s_y,r_y,
  z_{*,y},d_{\Bup,y}).
```

\mathsf B_y^cUse in source · 1 passage

Search aliases: B_y^c mathsf

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Use in source · MANUSCRIPT_STATUS.md, line 66
CL_y^f=s_yL_y^cC ⇒ C B_y^f= B_y^cC ⇒ CΨ_1:N^f=Ψ_1:N^cC.
Exact mathematical passage
```
CL_y^f=s_yL_y^cC
\Rightarrow
C\mathsf B_y^f=\mathsf B_y^cC
\Rightarrow
C\Psi_{1:N}^f=\Psi_{1:N}^cC.
```

\mathsf B_y^fUse in source · 1 passage

Search aliases: B_y^f mathsf

Link to this entry ·

Use in source · MANUSCRIPT_STATUS.md, line 66
CL_y^f=s_yL_y^cC ⇒ C B_y^f= B_y^cC ⇒ CΨ_1:N^f=Ψ_1:N^cC.
Exact mathematical passage
```
CL_y^f=s_yL_y^cC
\Rightarrow
C\mathsf B_y^f=\mathsf B_y^cC
\Rightarrow
C\Psi_{1:N}^f=\Psi_{1:N}^cC.
```

\Bup_k^{\coarse}Use in source · 8 passages

Search aliases: Bup_k^coarse B_k^ c Bup coarse mathsf mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1317
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Bup_k^{\fine}=\Bup_{\lambda_{y_k}^{\fine}},
  \qquad
  \Bup_k^{\coarse}=\Bup_{\lambda_{y_k}^{\coarse}}.
```

\Bup_k^{\fine}Use in source · 7 passages

Search aliases: Bup_k^fine B_k^ f Bup fine mathsf mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1317
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Bup_k^{\fine}=\Bup_{\lambda_{y_k}^{\fine}},
  \qquad
  \Bup_k^{\coarse}=\Bup_{\lambda_{y_k}^{\coarse}}.
```

\Bup_{\lambda^{\coarse}}Use in source · 5 passages

Search aliases: Bup_λ^coarse B_λ^ c Bup lambda coarse mathsf lambda mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 199
Classical sufficiency theory asks whether a statistic preserves all information relevant to a chosen observation family [source reference]. Here the operator question is C B_λ^ f = B_λ^ cC. When both the flow identity [source reference] and the Bayesian identity [source reference] hold at each observation time, the whole filtering sequence commutes with aggregation. Hidden Markov filtering is the linear unary case [source reference]. The same argument applies when the prediction step comes from …
Exact mathematical passage
```
\label{eq:bayes-question}
  C\Bup_{\lambda^{\fine}}
  =
  \Bup_{\lambda^{\coarse}}C.
```

\Bup_{\lambda^{\fine}}Use in source · 5 passages

Search aliases: Bup_λ^fine B_λ^ f Bup lambda fine mathsf lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 199
Classical sufficiency theory asks whether a statistic preserves all information relevant to a chosen observation family [source reference]. Here the operator question is C B_λ^ f = B_λ^ cC. When both the flow identity [source reference] and the Bayesian identity [source reference] hold at each observation time, the whole filtering sequence commutes with aggregation. Hidden Markov filtering is the linear unary case [source reference]. The same argument applies when the prediction step comes from …
Exact mathematical passage
```
\label{eq:bayes-question}
  C\Bup_{\lambda^{\fine}}
  =
  \Bup_{\lambda^{\coarse}}C.
```

\Bup_{\lambda_1}Use in source · 1 passage

Search aliases: Bup_λ_1 B_λ_1 Bup lambda mathsf lambda

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1187
Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.
Exact mathematical passage
```
\label{eq:bayes-composition}
  \Bup_{\lambda_2}\circ\Bup_{\lambda_1}
  =
  \Bup_{\lambda_1\odot\lambda_2}.
```

\Bup_{\lambda_1\odot\lambda_2}Use in source · 1 passage

Search aliases: Bup_λ_1odotλ_2 B_λ_1odotλ_2 Bup lambda odot lambda mathsf lambda odot lambda

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1187
Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.
Exact mathematical passage
```
\label{eq:bayes-composition}
  \Bup_{\lambda_2}\circ\Bup_{\lambda_1}
  =
  \Bup_{\lambda_1\odot\lambda_2}.
```

\Bup_{\lambda_2}Use in source · 1 passage

Search aliases: Bup_λ_2 B_λ_2 Bup lambda mathsf lambda

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1187
Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.
Exact mathematical passage
```
\label{eq:bayes-composition}
  \Bup_{\lambda_2}\circ\Bup_{\lambda_1}
  =
  \Bup_{\lambda_1\odot\lambda_2}.
```

\Bup_{\lambda_{y_1}^{\coarse}}Use in source · 1 passage

Search aliases: Bup_λ_y_1^coarse B_λ_y_1^ c Bup lambda coarse mathsf lambda mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1462
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.
Exact mathematical passage
```
&C\bigl(
\Bup_{\lambda_{y_N}^{\fine}}\circ M_{\fine}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\fine}}\circ M_{\fine}
\bigr)\\
&\qquad=
\bigl(
\Bup_{\lambda_{y_N}^{\coarse}}\circ M_{\coarse}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\coarse}}\circ M_{\coarse}
\bigr)C.
```

\Bup_{\lambda_{y_1}^{\fine}}Use in source · 1 passage

Search aliases: Bup_λ_y_1^fine B_λ_y_1^ f Bup lambda fine mathsf lambda mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1462
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.
Exact mathematical passage
```
&C\bigl(
\Bup_{\lambda_{y_N}^{\fine}}\circ M_{\fine}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\fine}}\circ M_{\fine}
\bigr)\\
&\qquad=
\bigl(
\Bup_{\lambda_{y_N}^{\coarse}}\circ M_{\coarse}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\coarse}}\circ M_{\coarse}
\bigr)C.
```

\Bup_{\lambda_{y_k}^{\coarse}}Use in source · 1 passage

Search aliases: Bup_λ_y_k^coarse B_λ_y_k^ c Bup lambda coarse mathsf lambda mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1317
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Bup_k^{\fine}=\Bup_{\lambda_{y_k}^{\fine}},
  \qquad
  \Bup_k^{\coarse}=\Bup_{\lambda_{y_k}^{\coarse}}.
```

\Bup_{\lambda_{y_k}^{\fine}}Use in source · 1 passage

Search aliases: Bup_λ_y_k^fine B_λ_y_k^ f Bup lambda fine mathsf lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1317
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Bup_k^{\fine}=\Bup_{\lambda_{y_k}^{\fine}},
  \qquad
  \Bup_k^{\coarse}=\Bup_{\lambda_{y_k}^{\coarse}}.
```

\Bup_{\lambda_{y_N}^{\coarse}}Use in source · 1 passage

Search aliases: Bup_λ_y_N^coarse B_λ_y_N^ c Bup lambda coarse mathsf lambda mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1462
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.
Exact mathematical passage
```
&C\bigl(
\Bup_{\lambda_{y_N}^{\fine}}\circ M_{\fine}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\fine}}\circ M_{\fine}
\bigr)\\
&\qquad=
\bigl(
\Bup_{\lambda_{y_N}^{\coarse}}\circ M_{\coarse}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\coarse}}\circ M_{\coarse}
\bigr)C.
```

\Bup_{\lambda_{y_N}^{\fine}}Use in source · 1 passage

Search aliases: Bup_λ_y_N^fine B_λ_y_N^ f Bup lambda fine mathsf lambda mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1462
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.
Exact mathematical passage
```
&C\bigl(
\Bup_{\lambda_{y_N}^{\fine}}\circ M_{\fine}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\fine}}\circ M_{\fine}
\bigr)\\
&\qquad=
\bigl(
\Bup_{\lambda_{y_N}^{\coarse}}\circ M_{\coarse}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\coarse}}\circ M_{\coarse}
\bigr)C.
```

\CUse in source · 1 passage

Search aliases: C mathbb

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 272
Finite packet resolutions. The main algebraic results are stated for vector spaces of finite dimension over R or C. The quotient and tensor factorizations are then purely algebraic. The later analytic results add the stated convergence conditions and the usual assumptions for local uniqueness.
Exact mathematical passage
```
\C
```

cUse in source · 7 passages

Search aliases: c

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2447
Certificate pipeline. The reference implementation follows the dependency chain c interaction tensors downarrow fiber constancy and kernel tests downarrow coarse tensor and Δ_m downarrow d_F(R) and d_Φ,k downarrow r_λ,k,z_*,k,d_ B,k downarrow e_k^-,e_k^+, E_N. Each numerical record stores the theorem identifier, the inputs, the residual, the tolerance, and the outcome. The suite includes examples built to satisfy the exact identities and perturbations built to violate them.
Exact mathematical passage
```
\begin{array}{c}
\text{interaction tensors}\\
\downarrow\\
\text{fiber constancy and kernel tests}\\
\downarrow\\
\text{coarse tensor and }\Delta_m\\
\downarrow\\
\mathfrak d_F(R)\text{ and }d_{\Phi,k}\\
\downarrow\\
 r_{\lambda,k},z_{*,k},d_{\Bup,k}\\
\downarrow\\
 e_k^-,e_k^+,\Err_N.
\end{array}
```

C^rUse in source · 1 passage

Search aliases: C^r

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 889
[Jet naturality] Let F_ f and F_ c be C^r maps satisfying CF_ f=F_ cC. Then, for every 1≤ k≤ r, C D^kF_ f(x)[h_1,…,h_k] = D^kF_ c(Cx)[Ch_1,…,Ch_k].
Exact mathematical passage
```
C^r
```

C^TUse in source · 10 passages

Search aliases: C^T

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1198
[Sufficient condition for Bayesian refinement] Let λ^ f∈ R_>0^n and λ^ c∈ R_>0^q. If λ^ f=sC^Tλ^ c for some s>0, then C B_λ^ f = B_λ^ cC. Moreover, Z_λ^ f(μ) = sZ_λ^ c(Cμ).
Exact mathematical passage
```
\label{eq:fiber-likelihood-sufficient}
  \lambda^{\fine}=sC^T\lambda^{\coarse}
```

C^{\times |\tau|}Use in source · 1 passage

Search aliases: C^× |τ| times tau

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 941
[Treewise quotient naturality] Suppose every operation in a fine family descends: CA_m^ f=A_m^ cC^× m (m≥1). Then, for every rooted interaction tree τ, C V_τ^ f = V_τ^ cC^× |τ|.
Exact mathematical passage
```
\label{eq:treewise-quotient}
  \boxed{
  C\cV_{\tau}^{\fine}
  =
  \cV_{\tau}^{\coarse}C^{\times |\tau|}.}
```

C^{\times|\tau_1|}Use in source · 1 passage

Search aliases: C^×|τ_1| times tau

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 955
Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.
Exact mathematical passage
```
C\cV_{\tau}^{\fine}
&=
CA_m^{\fine}
\bigl(
\cV_{\tau_1}^{\fine},\ldots,
\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
C\cV_{\tau_1}^{\fine},\ldots,
C\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
\cV_{\tau_1}^{\coarse}C^{\times|\tau_1|},\ldots,
\cV_{\tau_m}^{\coarse}C^{\times|\tau_m|}
\bigr)\\
&=
\cV_{\tau}^{\coarse}C^{\times|\tau|}.
```

C^{\times|\tau_m|}Use in source · 1 passage

Search aliases: C^×|τ_m| times tau

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 955
Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.
Exact mathematical passage
```
C\cV_{\tau}^{\fine}
&=
CA_m^{\fine}
\bigl(
\cV_{\tau_1}^{\fine},\ldots,
\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
C\cV_{\tau_1}^{\fine},\ldots,
C\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
\cV_{\tau_1}^{\coarse}C^{\times|\tau_1|},\ldots,
\cV_{\tau_m}^{\coarse}C^{\times|\tau_m|}
\bigr)\\
&=
\cV_{\tau}^{\coarse}C^{\times|\tau|}.
```

C^{\times|\tau|}Use in source · 1 passage

Search aliases: C^×|τ| times tau

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 955
Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.
Exact mathematical passage
```
C\cV_{\tau}^{\fine}
&=
CA_m^{\fine}
\bigl(
\cV_{\tau_1}^{\fine},\ldots,
\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
C\cV_{\tau_1}^{\fine},\ldots,
C\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
\cV_{\tau_1}^{\coarse}C^{\times|\tau_1|},\ldots,
\cV_{\tau_m}^{\coarse}C^{\times|\tau_m|}
\bigr)\\
&=
\cV_{\tau}^{\coarse}C^{\times|\tau|}.
```

C_{00}Use in source · 1 passage

Search aliases: C_00

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1735
[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.
Exact mathematical passage
```
C_{00}=I
```

C_{10}^{\times m}Use in source · 3 passages

Search aliases: C_10^× m times

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 730
Compose the two identities: C_20A_m^(2) =C_10A_m^(1)C_21^× m =A_m^(0)C_10^× mC_21^× m =A_m^(0)C_20^× m.
Exact mathematical passage
```
  C_{20}A_m^{(2)}
  =C_{10}A_m^{(1)}C_{21}^{\times m}
  =A_m^{(0)}C_{10}^{\times m}C_{21}^{\times m}
  =A_m^{(0)}C_{20}^{\times m}.
```

C_{20}Use in source · 4 passages

Search aliases: C_20

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 709
Composition across refinement levels. Let X_2xrightarrowC_21X_1xrightarrowC_10X_0, C_20=C_10C_21.
Exact mathematical passage
```
  X_2\xrightarrow{C_{21}}X_1\xrightarrow{C_{10}}X_0,
  \qquad
  C_{20}=C_{10}C_{21}.
```

C_{20}^{\times m}Use in source · 2 passages

Search aliases: C_20^× m times

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 730
Compose the two identities: C_20A_m^(2) =C_10A_m^(1)C_21^× m =A_m^(0)C_10^× mC_21^× m =A_m^(0)C_20^× m.
Exact mathematical passage
```
  C_{20}A_m^{(2)}
  =C_{10}A_m^{(1)}C_{21}^{\times m}
  =A_m^{(0)}C_{10}^{\times m}C_{21}^{\times m}
  =A_m^{(0)}C_{20}^{\times m}.
```

C_{21}Use in source · 5 passages

Search aliases: C_21

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 709
Composition across refinement levels. Let X_2xrightarrowC_21X_1xrightarrowC_10X_0, C_20=C_10C_21.
Exact mathematical passage
```
  X_2\xrightarrow{C_{21}}X_1\xrightarrow{C_{10}}X_0,
  \qquad
  C_{20}=C_{10}C_{21}.
```

C_{21}^{\times m}Use in source · 5 passages

Search aliases: C_21^× m times

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 730
Compose the two identities: C_20A_m^(2) =C_10A_m^(1)C_21^× m =A_m^(0)C_10^× mC_21^× m =A_m^(0)C_20^× m.
Exact mathematical passage
```
  C_{20}A_m^{(2)}
  =C_{10}A_m^{(1)}C_{21}^{\times m}
  =A_m^{(0)}C_{10}^{\times m}C_{21}^{\times m}
  =A_m^{(0)}C_{20}^{\times m}.
```

C_{L,\ell+1}Use in source · 1 passage

Search aliases: C_L,ℓ+1 ell

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1736
[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.
Exact mathematical passage
```
\label{eq:tower-defect-bound}
  \norm{\Delta_m^{L0}}
  \leq
  \sum_{\ell=0}^{L-1}
  \norm{C_{\ell0}}
  \norm{\Delta_m^{\ell+1,\ell}}
  \norm{C_{L,\ell+1}}^m.
```

C_{L,\ell+1}^{\times m}Use in source · 1 passage

Search aliases: C_L,ℓ+1^× m ell times

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1727
[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.
Exact mathematical passage
```
\label{eq:tower-defect}
  \Delta_m^{L0}
  =
  \sum_{\ell=0}^{L-1}
  C_{\ell0}\,
  \Delta_m^{\ell+1,\ell}
  C_{L,\ell+1}^{\times m},
```

C_{LL}Use in source · 1 passage

Search aliases: C_LL

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1735
[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.
Exact mathematical passage
```
C_{LL}=I
```

C_{\ell0}Use in source · 2 passages

Search aliases: C_ℓ0 ell

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1727
[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.
Exact mathematical passage
```
\label{eq:tower-defect}
  \Delta_m^{L0}
  =
  \sum_{\ell=0}^{L-1}
  C_{\ell0}\,
  \Delta_m^{\ell+1,\ell}
  C_{L,\ell+1}^{\times m},
```

\cEUse in source · 5 passages

Search aliases: cE E mathcal

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2241
Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.
Exact mathematical passage
```
\cE_m^{\mathrm{keep}}
```

\cE_m^{\mathrm{keep}}Use in source · 4 passages

Search aliases: cE_m^keep E_m^keep cE mathrm mathcal mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2241
Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.
Exact mathematical passage
```
\cE_m^{\mathrm{keep}}
```

\cGUse in source · 5 passages

Search aliases: cG G mathcal

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2383
Exact test for every interaction order. For a tensor of order m in scalar packet coordinates, compute the aggregated coefficient array g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m. For every coarse input tuple b=(b_1,…,b_m), define the fine fiber product G_ b = {(j_1,…,j_m):π(j_r)=b_r for all r}.
Exact mathematical passage
```
  \cG_{\mathbf b}
  =
  \{(j_1,\ldots,j_m):\pi(j_r)=b_r\text{ for all }r\}.
```

\cG_{\mathbf b}Use in source · 5 passages

Search aliases: cG_ b G_ b cG mathbf mathcal mathbf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2383
Exact test for every interaction order. For a tensor of order m in scalar packet coordinates, compute the aggregated coefficient array g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m. For every coarse input tuple b=(b_1,…,b_m), define the fine fiber product G_ b = {(j_1,…,j_m):π(j_r)=b_r for all r}.
Exact mathematical passage
```
  \cG_{\mathbf b}
  =
  \{(j_1,\ldots,j_m):\pi(j_r)=b_r\text{ for all }r\}.
```

\operatorname{Complexity}Use in source · 1 passage

Search aliases: Complexity operatorname

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2821
Approximate joint design. A finite approximate problem is min_C,{A_m^ c},{λ_y^ c} [ ∑_mw_mnormCA_m^ f-A_m^ cC^× m +∑_yv_yd_ B,y +γComplexity(C) ]. The main questions concern existence, tractable relaxations, and stability of the minimizers. Another question is whether the task weights can be learned without invalidating the certificate after a distribution shift.
Exact mathematical passage
```
  \min_{C,\{A_m^{\coarse}\},\{\lambda_y^{\coarse}\}}
  \left[
  \sum_mw_m\norm{CA_m^{\fine}-A_m^{\coarse}C^{\times m}}
  +\sum_yv_yd_{\Bup,y}
  +\gamma\operatorname{Complexity}(C)
  \right].
```

\operatorname{Cost}Use in source · 1 passage

Search aliases: Cost operatorname

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2358
A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability conditions on C. The three terms measure failure of dynamical closure, computational cost, and loss of observational information. Their weights and th…
Exact mathematical passage
```
\label{eq:budgeted-design}
  \min_{C,\{A_m^{\coarse}\},\{\cE_m^{\mathrm{keep}}\}}
  \left[
  \sum_mw_m\norm{\Delta_m^{\mathrm{total}}}
  +\lambda\sum_m\operatorname{Cost}(\cE_m^{\mathrm{keep}})
  +\gamma\sum_y d_{\Bup,y}
  \right].
```

\cSUse in source · 1 passage

Search aliases: cS S mathcal

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1095
Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.
Exact mathematical passage
```
  \cS_{\fine}
  =\{x\in K_{\fine}:\ell_{\fine}(x)=1\},
  \qquad
  \cS_{\coarse}
  =\{z\in K_{\coarse}:\ell_{\coarse}(z)=1\}.
```

\cS_{\coarse}Use in source · 1 passage

Search aliases: cS_coarse S_ c cS coarse mathcal mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1095
Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.
Exact mathematical passage
```
  \cS_{\fine}
  =\{x\in K_{\fine}:\ell_{\fine}(x)=1\},
  \qquad
  \cS_{\coarse}
  =\{z\in K_{\coarse}:\ell_{\coarse}(z)=1\}.
```

\cS_{\fine}Use in source · 1 passage

Search aliases: cS_fine S_ f cS fine mathcal mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1095
Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.
Exact mathematical passage
```
  \cS_{\fine}
  =\{x\in K_{\fine}:\ell_{\fine}(x)=1\},
  \qquad
  \cS_{\coarse}
  =\{z\in K_{\coarse}:\ell_{\coarse}(z)=1\}.
```

\cV_{\tau}^{\coarse}Use in source · 2 passages

Search aliases: cV_τ^coarse V_τ^ c cV tau coarse mathcal tau mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 941
[Treewise quotient naturality] Suppose every operation in a fine family descends: CA_m^ f=A_m^ cC^× m (m≥1). Then, for every rooted interaction tree τ, C V_τ^ f = V_τ^ cC^× |τ|.
Exact mathematical passage
```
\label{eq:treewise-quotient}
  \boxed{
  C\cV_{\tau}^{\fine}
  =
  \cV_{\tau}^{\coarse}C^{\times |\tau|}.}
```

\cV_{\tau}^{\fine}Use in source · 2 passages

Search aliases: cV_τ^fine V_τ^ f cV tau fine mathcal tau mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 941
[Treewise quotient naturality] Suppose every operation in a fine family descends: CA_m^ f=A_m^ cC^× m (m≥1). Then, for every rooted interaction tree τ, C V_τ^ f = V_τ^ cC^× |τ|.
Exact mathematical passage
```
\label{eq:treewise-quotient}
  \boxed{
  C\cV_{\tau}^{\fine}
  =
  \cV_{\tau}^{\coarse}C^{\times |\tau|}.}
```

\cV_{\tau_1}^{\coarse}Use in source · 1 passage

Search aliases: cV_τ_1^coarse V_τ_1^ c cV tau coarse mathcal tau mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 955
Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.
Exact mathematical passage
```
C\cV_{\tau}^{\fine}
&=
CA_m^{\fine}
\bigl(
\cV_{\tau_1}^{\fine},\ldots,
\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
C\cV_{\tau_1}^{\fine},\ldots,
C\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
\cV_{\tau_1}^{\coarse}C^{\times|\tau_1|},\ldots,
\cV_{\tau_m}^{\coarse}C^{\times|\tau_m|}
\bigr)\\
&=
\cV_{\tau}^{\coarse}C^{\times|\tau|}.
```

\cV_{\tau_1}^{\fine}Use in source · 1 passage

Search aliases: cV_τ_1^fine V_τ_1^ f cV tau fine mathcal tau mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 955
Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.
Exact mathematical passage
```
C\cV_{\tau}^{\fine}
&=
CA_m^{\fine}
\bigl(
\cV_{\tau_1}^{\fine},\ldots,
\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
C\cV_{\tau_1}^{\fine},\ldots,
C\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
\cV_{\tau_1}^{\coarse}C^{\times|\tau_1|},\ldots,
\cV_{\tau_m}^{\coarse}C^{\times|\tau_m|}
\bigr)\\
&=
\cV_{\tau}^{\coarse}C^{\times|\tau|}.
```

\cV_{\tau_m}^{\coarse}Use in source · 1 passage

Search aliases: cV_τ_m^coarse V_τ_m^ c cV tau coarse mathcal tau mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 955
Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.
Exact mathematical passage
```
C\cV_{\tau}^{\fine}
&=
CA_m^{\fine}
\bigl(
\cV_{\tau_1}^{\fine},\ldots,
\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
C\cV_{\tau_1}^{\fine},\ldots,
C\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
\cV_{\tau_1}^{\coarse}C^{\times|\tau_1|},\ldots,
\cV_{\tau_m}^{\coarse}C^{\times|\tau_m|}
\bigr)\\
&=
\cV_{\tau}^{\coarse}C^{\times|\tau|}.
```

\cV_{\tau_m}^{\fine}Use in source · 1 passage

Search aliases: cV_τ_m^fine V_τ_m^ f cV tau fine mathcal tau mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 955
Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.
Exact mathematical passage
```
C\cV_{\tau}^{\fine}
&=
CA_m^{\fine}
\bigl(
\cV_{\tau_1}^{\fine},\ldots,
\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
C\cV_{\tau_1}^{\fine},\ldots,
C\cV_{\tau_m}^{\fine}
\bigr)\\
&=
A_m^{\coarse}
\bigl(
\cV_{\tau_1}^{\coarse}C^{\times|\tau_1|},\ldots,
\cV_{\tau_m}^{\coarse}C^{\times|\tau_m|}
\bigr)\\
&=
\cV_{\tau}^{\coarse}C^{\times|\tau|}.
```

\cYUse in source · 5 passages

Search aliases: cY Y mathcal

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2803
Coarsest exact joint partition. For a finite interaction family (A_m^ f)_m≤ M and likelihood family (λ_y^ f)_y∈ Y, seek min_π|I_ c| subject to CA_m^ f=A_m^ cC^× m (m≤ M), and λ_y^ f=s_yC^Tλ_y^ c (y∈ Y). The partition determined by the observations alone groups states with identical likelihood profiles. The dynamical condition also depends on the partition because changing the partition changes the output aggregation. A terminating refinement algorithm, together with a useful complexity bound, re…
Exact mathematical passage
```
(\lambda_y^{\fine})_{y\in\cY}
```

\mathfrak dUse in source · 9 passages

Search aliases: d mathfrak

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1625
[Analytic defect majorant] If normΔ_m≤ε_m and the scalar series d_F(R)=∑_m≥0ε_mR^m converges, then sup_normx≤ Rnormδ_F(x) ≤ d_F(R).
Exact mathematical passage
```
  \mathfrak d_F(R)=\sum_{m\geq0}\varepsilon_mR^m
```

D^kUse in source · 1 passage

Search aliases: D^k

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 894
[Jet naturality] Let F_ f and F_ c be C^r maps satisfying CF_ f=F_ cC. Then, for every 1≤ k≤ r, C D^kF_ f(x)[h_1,…,h_k] = D^kF_ c(Cx)[Ch_1,…,Ch_k].
Exact mathematical passage
```
\label{eq:jet-naturality}
  \boxed{
  C D^kF_{\fine}(x)[h_1,\ldots,h_k]
  =
  D^kF_{\coarse}(Cx)[Ch_1,\ldots,Ch_k].}
```

D^rUse in source · 1 passage

Search aliases: D^r

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1292
[Bayesian jet naturality] Under the hypotheses of [source reference], for every r≥1 and every interior fine prior, C D^r B_λ^ f(μ) [h_1,…,h_r] = D^r B_λ^ c(Cμ) [Ch_1,…,Ch_r].
Exact mathematical passage
```
\label{eq:bayes-jet-naturality}
  C D^r\Bup_{\lambda^{\fine}}(\mu)
  [h_1,\ldots,h_r]
  =
  D^r\Bup_{\lambda^{\coarse}}(C\mu)
  [Ch_1,\ldots,Ch_r].
```

d_{\Bup}Use in source · 2 passages

Search aliases: d_Bup d_ B Bup mathsf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1874
_ f=Z_λ^ f(μ), z_ c=sZ_λ^ c(Cμ). Then norm C B_λ^ f(μ) - B_λ^ c(Cμ) _1 ≤ frac2∑_iμ_iabsr_λ,i max{z_ f,z_ c}. If both evidences are at least z_*>0, then d_ B ≤ frac2normr_λ_∞z_*.
Exact mathematical passage
```
\label{eq:bayes-refinement-uniform}
  d_{\Bup}
  \leq
  \frac{2\norm{r_{\lambda}}_{\infty}}{z_*}.
```

d_{\Bup,y}Use in source · 3 passages

Search aliases: d_Bup,y d_ B,y Bup mathsf

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2358
A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability conditions on C. The three terms measure failure of dynamical closure, computational cost, and loss of observational information. Their weights and th…
Exact mathematical passage
```
\label{eq:budgeted-design}
  \min_{C,\{A_m^{\coarse}\},\{\cE_m^{\mathrm{keep}}\}}
  \left[
  \sum_mw_m\norm{\Delta_m^{\mathrm{total}}}
  +\lambda\sum_m\operatorname{Cost}(\cE_m^{\mathrm{keep}})
  +\gamma\sum_y d_{\Bup,y}
  \right].
```

\mathfrak d_FUse in source · 7 passages

Search aliases: d_F mathfrak

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1625
[Analytic defect majorant] If normΔ_m≤ε_m and the scalar series d_F(R)=∑_m≥0ε_mR^m converges, then sup_normx≤ Rnormδ_F(x) ≤ d_F(R).
Exact mathematical passage
```
  \mathfrak d_F(R)=\sum_{m\geq0}\varepsilon_mR^m
```

\mathfrak d_{F,k}Use in source · 2 passages

Search aliases: d_F,k mathfrak

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2102
From interaction coefficients to posterior error. For a flow interval of length T_k, suppose the fine trajectory remains in a ball of radius R_k and the coarse field has Lipschitz constant L_F,k. By Corollary~[source reference], one may take d_Φ,k = d_F,k(R_k) frace^L_F,kT_k-1L_F,k, with the continuous value T_k d_F,k(R_k) at L_F,k=0. For the observation, Corollary~[source reference] gives d_ B,k ≤ frac2normr_λ,k_∞z_*,k. Substituting these bounds into [source reference] gives a certificate expre…
Exact mathematical passage
```
\label{eq:dphi-coefficient}
  d_{\Phi,k}
  =
  \mathfrak d_{F,k}(R_k)
  \frac{e^{L_{F,k}T_k}-1}{L_{F,k}},
```

\diagUse in source · 1 passage

Search aliases: diag operatorname

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1170
Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.
Exact mathematical passage
```
  L_{\lambda}=\diag(\lambda_1,\ldots,\lambda_n),
  \qquad
  \lambda_i>0.
```

\dot x_1Use in source · 2 passages

Search aliases: dot x_1 dot

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2558
h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x), dot x_2 · =-ax_2+bx_4-h_12(x), dot x_3 · = ax_1-bx_3+h_34(x), dot x_4 · = ax_2-bx_4-h_34(x).
Exact mathematical passage
```
\label{eq:fine-four-state}
\begin{aligned}
  \dot x_1&=-ax_1+bx_3+h_{12}(x),\\
  \dot x_2&=-ax_2+bx_4-h_{12}(x),\\
  \dot x_3&= ax_1-bx_3+h_{34}(x),\\
  \dot x_4&= ax_2-bx_4-h_{34}(x).
\end{aligned}
```

\dot x_2Use in source · 2 passages

Search aliases: dot x_2 dot

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2558
h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x), dot x_2 · =-ax_2+bx_4-h_12(x), dot x_3 · = ax_1-bx_3+h_34(x), dot x_4 · = ax_2-bx_4-h_34(x).
Exact mathematical passage
```
\label{eq:fine-four-state}
\begin{aligned}
  \dot x_1&=-ax_1+bx_3+h_{12}(x),\\
  \dot x_2&=-ax_2+bx_4-h_{12}(x),\\
  \dot x_3&= ax_1-bx_3+h_{34}(x),\\
  \dot x_4&= ax_2-bx_4-h_{34}(x).
\end{aligned}
```

\dot x_3Use in source · 2 passages

Search aliases: dot x_3 dot

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2558
h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x), dot x_2 · =-ax_2+bx_4-h_12(x), dot x_3 · = ax_1-bx_3+h_34(x), dot x_4 · = ax_2-bx_4-h_34(x).
Exact mathematical passage
```
\label{eq:fine-four-state}
\begin{aligned}
  \dot x_1&=-ax_1+bx_3+h_{12}(x),\\
  \dot x_2&=-ax_2+bx_4-h_{12}(x),\\
  \dot x_3&= ax_1-bx_3+h_{34}(x),\\
  \dot x_4&= ax_2-bx_4-h_{34}(x).
\end{aligned}
```

\dot x_4Use in source · 2 passages

Search aliases: dot x_4 dot

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2558
h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x), dot x_2 · =-ax_2+bx_4-h_12(x), dot x_3 · = ax_1-bx_3+h_34(x), dot x_4 · = ax_2-bx_4-h_34(x).
Exact mathematical passage
```
\label{eq:fine-four-state}
\begin{aligned}
  \dot x_1&=-ax_1+bx_3+h_{12}(x),\\
  \dot x_2&=-ax_2+bx_4-h_{12}(x),\\
  \dot x_3&= ax_1-bx_3+h_{34}(x),\\
  \dot x_4&= ax_2-bx_4-h_{34}(x).
\end{aligned}
```

\dot yUse in source · 1 passage

Search aliases: dot y dot

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1684
Let y(t)=Cx(t). From [source reference], dot y(t)=F_ c(y(t))+δ_F(x(t)). Subtract the integral equations for y and z: y(t)-z(t) =y(0)-z(0) +∫_0^t (F_ c(y(s))-F_ c(z(s))) d s +∫_0^tδ_F(x(s)) d s. Taking norms and applying the Grönwall inequality yields [source reference]. The uniform cases follow by evaluating the exponential integral.
Exact mathematical passage
```
  \dot y(t)=F_{\coarse}(y(t))+\delta_F(x(t)).
```

\dot zUse in source · 4 passages

Search aliases: dot z dot

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1010
Let x(t)=Φ_t^ f(x_0) and set z(t)=Cx(t). Then dot z(t) =Cdot x(t) =CF_ f(x(t)) =F_ c(Cx(t)) =F_ c(z(t)), with z(0)=Cx_0. Uniqueness for the coarse initial value problem gives z(t)=Φ_t^ c(Cx_0).
Exact mathematical passage
```
  \dot z(t)
  =C\dot x(t)
  =CF_{\fine}(x(t))
  =F_{\coarse}(Cx(t))
  =F_{\coarse}(z(t)),
```

\dot z_1Use in source · 1 passage

Search aliases: dot z_1 dot

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2540
0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
Exact mathematical passage
```
\label{eq:coarse-two-state}
  \dot z_1=-az_1+bz_2,
  \qquad
  \dot z_2=az_1-bz_2,
  \qquad a,b>0.
```

\dot z_2Use in source · 1 passage

Search aliases: dot z_2 dot

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2540
0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
Exact mathematical passage
```
\label{eq:coarse-two-state}
  \dot z_1=-az_1+bz_2,
  \qquad
  \dot z_2=az_1-bz_2,
  \qquad a,b>0.
```

\downarrowUse in source · 1 passage

Search aliases: downarrow

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2447
Certificate pipeline. The reference implementation follows the dependency chain c interaction tensors downarrow fiber constancy and kernel tests downarrow coarse tensor and Δ_m downarrow d_F(R) and d_Φ,k downarrow r_λ,k,z_*,k,d_ B,k downarrow e_k^-,e_k^+, E_N. Each numerical record stores the theorem identifier, the inputs, the residual, the tolerance, and the outcome. The suite includes examples built to satisfy the exact identities and perturbations built to violate them.
Exact mathematical passage
```
\begin{array}{c}
\text{interaction tensors}\\
\downarrow\\
\text{fiber constancy and kernel tests}\\
\downarrow\\
\text{coarse tensor and }\Delta_m\\
\downarrow\\
\mathfrak d_F(R)\text{ and }d_{\Phi,k}\\
\downarrow\\
 r_{\lambda,k},z_{*,k},d_{\Bup,k}\\
\downarrow\\
 e_k^-,e_k^+,\Err_N.
\end{array}
```

\Downarrow_{C_{10}}Use in source · 1 passage

Search aliases: Downarrow_C_10 Downarrow

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 717
[Transitivity of exact interaction quotients] If A_m^(2)Downarrow_C_21A_m^(1) and A_m^(1)Downarrow_C_10A_m^(0), then A_m^(2)Downarrow_C_20A_m^(0).
Exact mathematical passage
```
  A_m^{(2)}\Downarrow_{C_{21}}A_m^{(1)}
  \qquad\text{and}\qquad
  A_m^{(1)}\Downarrow_{C_{10}}A_m^{(0)},
```

\Downarrow_{C_{20}}Use in source · 1 passage

Search aliases: Downarrow_C_20 Downarrow

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 723
[Transitivity of exact interaction quotients] If A_m^(2)Downarrow_C_21A_m^(1) and A_m^(1)Downarrow_C_10A_m^(0), then A_m^(2)Downarrow_C_20A_m^(0).
Exact mathematical passage
```
  A_m^{(2)}\Downarrow_{C_{20}}A_m^{(0)}.
```

\Downarrow_{C_{21}}Use in source · 1 passage

Search aliases: Downarrow_C_21 Downarrow

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 717
[Transitivity of exact interaction quotients] If A_m^(2)Downarrow_C_21A_m^(1) and A_m^(1)Downarrow_C_10A_m^(0), then A_m^(2)Downarrow_C_20A_m^(0).
Exact mathematical passage
```
  A_m^{(2)}\Downarrow_{C_{21}}A_m^{(1)}
  \qquad\text{and}\qquad
  A_m^{(1)}\Downarrow_{C_{10}}A_m^{(0)},
```

e^{H_m}Use in source · 1 passage

Search aliases: e^H_m

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2311
Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The quantity deliberately depends on the resolution. It evaluates a representation rather than an invariant of the unresolved tensor alone.
Exact mathematical passage
```
  N_{\mathrm{eff},m}=e^{H_m}.
```

e^{L(t-s)}Use in source · 1 passage

Search aliases: e^L(t-s)

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1662
[Approximate quotient flow bound] Let dot x=F_ f(x), dot z=F_ c(z), and suppose both solutions remain in domains on which normF_ c(u)-F_ c(v) ≤ Lnormu-v. Set e(t)=normCx(t)-z(t). Then e(t) ≤ e^Lte(0) + ∫_0^t e^L(t-s)normδ_F(x(s)) d s. If e(0)=0 and normδ_F(x(s))≤ε_F on [0,t], then e(t) ≤ displaystyle ε_Ffrace^Lt-1L, · L>0,
Exact mathematical passage
```
\label{eq:approx-flow-general}
  \boxed{
  e(t)
  \leq
  e^{Lt}e(0)
  +
  \int_0^t e^{L(t-s)}\norm{\delta_F(x(s))}\dd s.}
```

e^{L_{F,k}T_k}Use in source · 1 passage

Search aliases: e^L_F,kT_k

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2102
From interaction coefficients to posterior error. For a flow interval of length T_k, suppose the fine trajectory remains in a ball of radius R_k and the coarse field has Lipschitz constant L_F,k. By Corollary~[source reference], one may take d_Φ,k = d_F,k(R_k) frace^L_F,kT_k-1L_F,k, with the continuous value T_k d_F,k(R_k) at L_F,k=0. For the observation, Corollary~[source reference] gives d_ B,k ≤ frac2normr_λ,k_∞z_*,k. Substituting these bounds into [source reference] gives a certificate expre…
Exact mathematical passage
```
\label{eq:dphi-coefficient}
  d_{\Phi,k}
  =
  \mathfrak d_{F,k}(R_k)
  \frac{e^{L_{F,k}T_k}-1}{L_{F,k}},
```

e^{LT}Use in source · 1 passage

Search aliases: e^LT

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1700
[Coefficient certificate for a flow segment] If normx(s)≤ R and Proposition~[source reference] applies, then a flow interval of length T satisfies normCΦ_T^ f(x_0)-Φ_T^ c(Cx_0) ≤ d_F(R)frace^LT-1L for L>0, with T d_F(R) at L=0.
Exact mathematical passage
```
  \norm{C\Phi_T^{\fine}(x_0)-\Phi_T^{\coarse}(Cx_0)}
  \leq
  \mathfrak d_F(R)\frac{e^{LT}-1}{L}
```

e^{Lt}Use in source · 2 passages

Search aliases: e^Lt

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1662
[Approximate quotient flow bound] Let dot x=F_ f(x), dot z=F_ c(z), and suppose both solutions remain in domains on which normF_ c(u)-F_ c(v) ≤ Lnormu-v. Set e(t)=normCx(t)-z(t). Then e(t) ≤ e^Lte(0) + ∫_0^t e^L(t-s)normδ_F(x(s)) d s. If e(0)=0 and normδ_F(x(s))≤ε_F on [0,t], then e(t) ≤ displaystyle ε_Ffrace^Lt-1L, · L>0,
Exact mathematical passage
```
\label{eq:approx-flow-general}
  \boxed{
  e(t)
  \leq
  e^{Lt}e(0)
  +
  \int_0^t e^{L(t-s)}\norm{\delta_F(x(s))}\dd s.}
```

e^{tA_{\coarse}}Use in source · 1 passage

Search aliases: e^tA_coarse e^tA_ c coarse mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1022
[Linear semigroup naturality] If CA_ f=A_ cC, then Ce^tA_ f = e^tA_ cC for every t.
Exact mathematical passage
```
\label{eq:linear-semigroup-naturality}
  Ce^{tA_{\fine}}
  =
  e^{tA_{\coarse}}C
```

e^{tA_{\fine}}Use in source · 1 passage

Search aliases: e^tA_fine e^tA_ f fine mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1022
[Linear semigroup naturality] If CA_ f=A_ cC, then Ce^tA_ f = e^tA_ cC for every t.
Exact mathematical passage
```
\label{eq:linear-semigroup-naturality}
  Ce^{tA_{\fine}}
  =
  e^{tA_{\coarse}}C
```

e^{tQ_{\coarse}}Use in source · 1 passage

Search aliases: e^tQ_coarse e^tQ_ c coarse mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2753
Markov lumpability and Markov functions. For unary stochastic dynamics, the identity CQ_ f=Q_ cC is a standard exact lumping condition. It implies Ce^tQ_ f=e^tQ_ cC and means that the coarse process is Markov for every fine initial distribution compatible with the quotient. Classical treatments distinguish several forms of strong, ordinary, exact, and weak lumpability, depending on the admissible initial laws and the direction of the stochastic convention [source reference]. Here we fix one colu…
Exact mathematical passage
```
  Ce^{tQ_{\fine}}=e^{tQ_{\coarse}}C
```

e^{tQ_{\fine}}Use in source · 1 passage

Search aliases: e^tQ_fine e^tQ_ f fine mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2753
Markov lumpability and Markov functions. For unary stochastic dynamics, the identity CQ_ f=Q_ cC is a standard exact lumping condition. It implies Ce^tQ_ f=e^tQ_ cC and means that the coarse process is Markov for every fine initial distribution compatible with the quotient. Classical treatments distinguish several forms of strong, ordinary, exact, and weak lumpability, depending on the admissible initial laws and the direction of the stochastic convention [source reference]. Here we fix one colu…
Exact mathematical passage
```
  Ce^{tQ_{\fine}}=e^{tQ_{\coarse}}C
```

e_0^+Use in source · 7 passages

Search aliases: e_0^+

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1975
=Φ_k^ c(μ_k-1^ c), μ_k^ c = B_k^ c(widehatμ_k^ c). Define the discrepancies before and after each update by e_k^- =normCwidehatμ_k^ f-widehatμ_k^ c_1, e_k^+ =normCμ_k^ f-μ_k^ c_1. At the initial time, set e_0^+=normCμ_0^ f-μ_0^ c_1.
Exact mathematical passage
```
  e_0^+=\norm{C\mu_0^{\fine}-\mu_0^{\coarse}}_1.
```

e_hUse in source · 2 passages

Search aliases: e_h

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2272
texorpdfstringRetaining the k largest energy hyperedgesRetaining the k largest energy hyperedges. Let e_h=normwidehat A_m,h_HS^2 and order the energies e_h_1≥ e_h_2≥⋯≥ e_h_M.
Exact mathematical passage
```
  e_h=\norm{\widehat A_{m,h}}_{\mathrm{HS}}^2
```

e_{h_1}Use in source · 1 passage

Search aliases: e_h_1

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2276
texorpdfstringRetaining the k largest energy hyperedgesRetaining the k largest energy hyperedges. Let e_h=normwidehat A_m,h_HS^2 and order the energies e_h_1≥ e_h_2≥⋯≥ e_h_M.
Exact mathematical passage
```
  e_{h_1}\geq e_{h_2}\geq\cdots\geq e_{h_M}.
```

e_{h_2}Use in source · 1 passage

Search aliases: e_h_2

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2276
texorpdfstringRetaining the k largest energy hyperedgesRetaining the k largest energy hyperedges. Let e_h=normwidehat A_m,h_HS^2 and order the energies e_h_1≥ e_h_2≥⋯≥ e_h_M.
Exact mathematical passage
```
  e_{h_1}\geq e_{h_2}\geq\cdots\geq e_{h_M}.
```

e_{h_M}Use in source · 1 passage

Search aliases: e_h_M

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2276
texorpdfstringRetaining the k largest energy hyperedgesRetaining the k largest energy hyperedges. Let e_h=normwidehat A_m,h_HS^2 and order the energies e_h_1≥ e_h_2≥⋯≥ e_h_M.
Exact mathematical passage
```
  e_{h_1}\geq e_{h_2}\geq\cdots\geq e_{h_M}.
```

e_{h_r}Use in source · 1 passage

Search aliases: e_h_r

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2286
[Optimal retention of k orthogonal hyperedges] Among all retained sets of cardinality k, the set {h_1,…,h_k} minimizes the Hilbert Schmidt residual. The minimum is min_| E|=k normR_m( E)_HS^2 = ∑_r>ke_h_r.
Exact mathematical passage
```
\label{eq:best-k-residual}
  \boxed{
  \min_{|\cE|=k}
  \norm{R_m(\cE)}_{\mathrm{HS}}^2
  =
  \sum_{r>k}e_{h_r}.}
```

e_{k-1}^+Use in source · 3 passages

Search aliases: e_k-1^+

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1995
[Prediction bound for one step] Under [source reference], e_k^- ≤ L_Φ,ke_k-1^+ +d_Φ,k.
Exact mathematical passage
```
\label{eq:prediction-step}
  e_k^-
  \leq
  L_{\Phi,k}e_{k-1}^+
  +d_{\Phi,k}.
```

E_mUse in source · 2 passages

Search aliases: E_m

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2399
A dense implementation requires O(n^m+1) arithmetic to aggregate a tensor of order m on n states. If the fine tensor is stored as E_m active scalar hyperedges, the aggregation pass is O(E_m), followed by grouping and consistency checks over the represented coarse incidences.
Exact mathematical passage
```
E_m
```

e_N^+Use in source · 7 passages

Search aliases: e_N^+

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2038
[Certified approximate filtering] Under the assumptions above, e_N^+ ≤ (∏_j=1^Nα_j)e_0^+ + ∑_k=1^N (∏_j=k+1^Nα_j) β_k. Here and below, an empty product equals one. With exact initial aggregation e_0^+=0, e_N^+ ≤ ∑_k=1^N (∏_j=k+1^Nα_j) (L_ B,kd_Φ,k+d_ B,k).
Exact mathematical passage
```
\label{eq:posterior-certificate}
  \boxed{
  e_N^+
  \leq
  \left(\prod_{j=1}^N\alpha_j\right)e_0^+
  +
  \sum_{k=1}^N
  \left(\prod_{j=k+1}^N\alpha_j\right)
  \beta_k.}
```

e_{\eta}Use in source · 1 passage

Search aliases: e_η eta

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2305
Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The quantity deliberately depends on the resolution. It evaluates a representation rather than an invariant of the unresolved tensor alone.
Exact mathematical passage
```
  p_h=\frac{e_h}{\sum_{\eta}e_{\eta}},
  \qquad
  H_m=-\sum_hp_h\log p_h,
```

\mathrm{eff}Use in source · 1 passage

Search aliases: eff mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2311
Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The quantity deliberately depends on the resolution. It evaluates a representation rather than an invariant of the unresolved tensor alone.
Exact mathematical passage
```
  N_{\mathrm{eff},m}=e^{H_m}.
```

\EffectiveHyperedgeCountUse in source · 1 passage

Search aliases: EffectiveHyperedgeCount

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2709
Concentrated hyperedge field. A synthetic interaction field with decreasing orthogonal block energies illustrates [source reference]. The computed hyperedge entropy is HyperedgeEntropyValue, corresponding to an effective count of approximately EffectiveHyperedgeCount hyperedges despite a larger ambient support. The ranked spectrum in [source reference] shows why the field is compressible in this representation.
Exact mathematical passage
```
\EffectiveHyperedgeCount
```

\ErrUse in source · 1 passage

Search aliases: Err E mathfrak

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2447
Certificate pipeline. The reference implementation follows the dependency chain c interaction tensors downarrow fiber constancy and kernel tests downarrow coarse tensor and Δ_m downarrow d_F(R) and d_Φ,k downarrow r_λ,k,z_*,k,d_ B,k downarrow e_k^-,e_k^+, E_N. Each numerical record stores the theorem identifier, the inputs, the residual, the tolerance, and the outcome. The suite includes examples built to satisfy the exact identities and perturbations built to violate them.
Exact mathematical passage
```
\begin{array}{c}
\text{interaction tensors}\\
\downarrow\\
\text{fiber constancy and kernel tests}\\
\downarrow\\
\text{coarse tensor and }\Delta_m\\
\downarrow\\
\mathfrak d_F(R)\text{ and }d_{\Phi,k}\\
\downarrow\\
 r_{\lambda,k},z_{*,k},d_{\Bup,k}\\
\downarrow\\
 e_k^-,e_k^+,\Err_N.
\end{array}
```

\Err_NUse in source · 1 passage

Search aliases: Err_N E_N Err mathfrak

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2447
Certificate pipeline. The reference implementation follows the dependency chain c interaction tensors downarrow fiber constancy and kernel tests downarrow coarse tensor and Δ_m downarrow d_F(R) and d_Φ,k downarrow r_λ,k,z_*,k,d_ B,k downarrow e_k^-,e_k^+, E_N. Each numerical record stores the theorem identifier, the inputs, the residual, the tolerance, and the outcome. The suite includes examples built to satisfy the exact identities and perturbations built to violate them.
Exact mathematical passage
```
\begin{array}{c}
\text{interaction tensors}\\
\downarrow\\
\text{fiber constancy and kernel tests}\\
\downarrow\\
\text{coarse tensor and }\Delta_m\\
\downarrow\\
\mathfrak d_F(R)\text{ and }d_{\Phi,k}\\
\downarrow\\
 r_{\lambda,k},z_{*,k},d_{\Bup,k}\\
\downarrow\\
 e_k^-,e_k^+,\Err_N.
\end{array}
```

\ExactEvidenceResidualUse in source · 1 passage

Search aliases: ExactEvidenceResidual

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2621
[Exact nonlinear filtering residuals] For the deterministic run with four states, the largest aggregate prediction residual was ExactPredictionResidual. The largest posterior residual after an update was ExactPosteriorResidual, and the largest residual in a single evidence factor was ExactEvidenceResidual. The small nonzero values come from floating point integration and arithmetic in the finite example.
Exact mathematical passage
```
\ExactEvidenceResidual
```

\ExactPosteriorResidualUse in source · 1 passage

Search aliases: ExactPosteriorResidual

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2621
[Exact nonlinear filtering residuals] For the deterministic run with four states, the largest aggregate prediction residual was ExactPredictionResidual. The largest posterior residual after an update was ExactPosteriorResidual, and the largest residual in a single evidence factor was ExactEvidenceResidual. The small nonzero values come from floating point integration and arithmetic in the finite example.
Exact mathematical passage
```
\ExactPosteriorResidual
```

\ExactPredictionResidualUse in source · 1 passage

Search aliases: ExactPredictionResidual

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2621
[Exact nonlinear filtering residuals] For the deterministic run with four states, the largest aggregate prediction residual was ExactPredictionResidual. The largest posterior residual after an update was ExactPosteriorResidual, and the largest residual in a single evidence factor was ExactEvidenceResidual. The small nonzero values come from floating point integration and arithmetic in the finite example.
Exact mathematical passage
```
\ExactPredictionResidual
```

\mathsf FUse in source · 1 passage

Search aliases: F mathsf

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2973
abeleq:lumpability-certificate-schema L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite ledger as JSON files and CSV tables. A certificate records a computation under specified inputs. The ledger provides cryptographic authenticity only when it is paired with a separate signed dig…
Exact mathematical passage
```
\label{eq:filter-certificate-schema}
  \mathsf F_N
  =
  (e_0^+,\{L_{\Phi,k}\},\{L_{\Bup,k}\},
  \{d_{\Phi,k}\},\{d_{\Bup,k}\},
  \{e_k^-,e_k^+\}).
```

F_cUse in source · 2 passages

Search aliases: F_c

Link to this entry ·

Use in source · MANUSCRIPT_STATUS.md, line 56
CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.
Exact mathematical passage
```
CA_m^f=A_m^cC^{\times m}
\Rightarrow
CF_f=F_cC
\Rightarrow
C\Phi_t^f=\Phi_t^cC.
```

F_{\coarse}^{\circ n}Use in source · 1 passage

Search aliases: F_coarse^∘ n F_ c^∘ n coarse circ mathrm circ

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1036
[Quotients of discrete dynamics] If CF_ f=F_ cC, then CF_ f^∘ n = F_ c^∘ nC for every integer n≥0.
Exact mathematical passage
```
  CF_{\fine}^{\circ n}
  =
  F_{\coarse}^{\circ n}C
```

F_fUse in source · 2 passages

Search aliases: F_f

Link to this entry ·

Use in source · MANUSCRIPT_STATUS.md, line 56
CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.
Exact mathematical passage
```
CA_m^f=A_m^cC^{\times m}
\Rightarrow
CF_f=F_cC
\Rightarrow
C\Phi_t^f=\Phi_t^cC.
```

F_{\fine}^{\circ n}Use in source · 1 passage

Search aliases: F_fine^∘ n F_ f^∘ n fine circ mathrm circ

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1036
[Quotients of discrete dynamics] If CF_ f=F_ cC, then CF_ f^∘ n = F_ c^∘ nC for every integer n≥0.
Exact mathematical passage
```
  CF_{\fine}^{\circ n}
  =
  F_{\coarse}^{\circ n}C
```

\mathsf F_NUse in source · 1 passage

Search aliases: F_N mathsf

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2973
abeleq:lumpability-certificate-schema L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite ledger as JSON files and CSV tables. A certificate records a computation under specified inputs. The ledger provides cryptographic authenticity only when it is paired with a separate signed dig…
Exact mathematical passage
```
\label{eq:filter-certificate-schema}
  \mathsf F_N
  =
  (e_0^+,\{L_{\Phi,k}\},\{L_{\Bup,k}\},
  \{d_{\Phi,k}\},\{d_{\Bup,k}\},
  \{e_k^-,e_k^+\}).
```

f_{\pi(j_1)}Use in source · 1 passage

Search aliases: f_π(j_1) pi

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 589
Applying C to [source reference] gives CA_m^ f(e_j_1,…,e_j_m) = ∑_a=1^q g^(m)_a;j_1,…,j_mf_a. If a coarse tensor exists, this vector equals A_m^ c(f_π(j_1),…,f_π(j_m)), which depends only on the coarse input labels. This proves necessity. Conversely, constancy on each fiber makes [source reference] well defined. Equality on every tuple of fine basis vectors then gives CA_m^ f=A_m^ cC^× m by multilinearity.
Exact mathematical passage
```
  A_m^{\coarse}(f_{\pi(j_1)},\ldots,f_{\pi(j_m)}),
```

f_{\pi(j_m)}Use in source · 1 passage

Search aliases: f_π(j_m) pi

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 589
Applying C to [source reference] gives CA_m^ f(e_j_1,…,e_j_m) = ∑_a=1^q g^(m)_a;j_1,…,j_mf_a. If a coarse tensor exists, this vector equals A_m^ c(f_π(j_1),…,f_π(j_m)), which depends only on the coarse input labels. This proves necessity. Conversely, constancy on each fiber makes [source reference] well defined. Equality on every tuple of fine basis vectors then gives CA_m^ f=A_m^ cC^× m by multilinearity.
Exact mathematical passage
```
  A_m^{\coarse}(f_{\pi(j_1)},\ldots,f_{\pi(j_m)}),
```

GUse in source · 3 passages

Search aliases: G

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 981
[Compatibility with Fa a di Bruno composition] Let F_ f and F_ c be smooth maps with CF_ f=F_ cC, and let G_ f and G_ c satisfy the same relation. Then each rooted tree in the Fa a di Bruno expansion, written in packet coordinates, descends to the corresponding coarse tree. Moreover, C(G_ f∘ F_ f) =(G_ c∘ F_ c)C.
Exact mathematical passage
```
G_{\fine}
```

gUse in source · 11 passages

Search aliases: g

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 551
R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
\label{eq:aggregated-coefficient}
  g^{(m)}_{a;j_1,\ldots,j_m}
  =
  \sum_{i:\pi(i)=a}
  a^{\fine,(m)}_{i;j_1,\ldots,j_m}.
```

g^{(m)}_{a;\mathbf j}Use in source · 1 passage

Search aliases: g^(m)_a; j mathbf

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2392
[Exact packet coordinate lumpability test] For every coarse output a and coarse input tuple b: [label=arabic*.,leftmargin=2.3em] item Evaluate g^(m)_a; j for all j∈ G_ b. item In exact arithmetic, test whether all values are equal. With floating point input, compare their spread with a declared tolerance and record that tolerance. item When the exact test passes, assign the common value to a^ c,(m)_a; b. When only the numerical tolerance is met, assign a declared representative such as the mean …
Exact mathematical passage
```
g^{(m)}_{a;\mathbf j}
```

g^{(m)}_{a;j_1',\ldots,j_m'}Use in source · 1 passage

Search aliases: g^(m)_a;j_1',…,j_m' ldots

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 560
[Coefficient criterion for every interaction order] The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m, π(j_r)=b_r.
Exact mathematical passage
```
\label{eq:packet-fiber-constancy}
  g^{(m)}_{a;j_1,\ldots,j_m}
  =
  g^{(m)}_{a;j_1',\ldots,j_m'}
```

g^{(m)}_{a;j_1,\ldots,j_m}Use in source · 5 passages

Search aliases: g^(m)_a;j_1,…,j_m ldots

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 551
R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
\label{eq:aggregated-coefficient}
  g^{(m)}_{a;j_1,\ldots,j_m}
  =
  \sum_{i:\pi(i)=a}
  a^{\fine,(m)}_{i;j_1,\ldots,j_m}.
```

g^TUse in source · 1 passage

Search aliases: g^T

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2121
[Coarse observable error] For any coarse observable g∈ R^q, absg^TCμ_N^ f-g^Tμ_N^ c ≤ normg_∞e_N^+. In particular, TV(Cμ_N^ f,μ_N^ c) =frac12e_N^+.
Exact mathematical passage
```
  \abs{g^TC\mu_N^{\fine}-g^T\mu_N^{\coarse}}
  \leq
  \norm{g}_{\infty}e_N^+.
```

g_{a;\mathbf j}Use in source · 3 passages

Search aliases: g_a; j mathbf

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2411
Approximate coarse coefficients. When exact fiber constancy fails, the approximation depends on a chosen loss and, when relevant, a task distribution. Under scalar squared loss with nonnegative weights w_ j on a fiber product, assume their total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ G_ bw_ j . For scalar worst case loss, a minimax representative is the midrange a^ c,∞_a; b = frac12 ( max_ j∈ G_ bg_a; j + min_ j∈ G_ bg_a; j ), with…
Exact mathematical passage
```
\label{eq:least-squares-coarse-coeff}
  a^{\coarse,\mathrm{LS}}_{a;\mathbf b}
  =
  \frac{
  \sum_{\mathbf j\in\cG_{\mathbf b}}
  w_{\mathbf j}g_{a;\mathbf j}
  }{
  \sum_{\mathbf j\in\cG_{\mathbf b}}w_{\mathbf j}
  }.
```

G_{\coarse}Use in source · 2 passages

Search aliases: G_coarse G_ c coarse mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 981
[Compatibility with Fa a di Bruno composition] Let F_ f and F_ c be smooth maps with CF_ f=F_ cC, and let G_ f and G_ c satisfy the same relation. Then each rooted tree in the Fa a di Bruno expansion, written in packet coordinates, descends to the corresponding coarse tree. Moreover, C(G_ f∘ F_ f) =(G_ c∘ F_ c)C.
Exact mathematical passage
```
G_{\coarse}
```

G_{\fine}Use in source · 2 passages

Search aliases: G_fine G_ f fine mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 981
[Compatibility with Fa a di Bruno composition] Let F_ f and F_ c be smooth maps with CF_ f=F_ cC, and let G_ f and G_ c satisfy the same relation. Then each rooted tree in the Fa a di Bruno expansion, written in packet coordinates, descends to the corresponding coarse tree. Moreover, C(G_ f∘ F_ f) =(G_ c∘ F_ c)C.
Exact mathematical passage
```
G_{\fine}
```

hUse in source · 12 passages

Search aliases: h

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 894
[Jet naturality] Let F_ f and F_ c be C^r maps satisfying CF_ f=F_ cC. Then, for every 1≤ k≤ r, C D^kF_ f(x)[h_1,…,h_k] = D^kF_ c(Cx)[Ch_1,…,Ch_k].
Exact mathematical passage
```
\label{eq:jet-naturality}
  \boxed{
  C D^kF_{\fine}(x)[h_1,\ldots,h_k]
  =
  D^kF_{\coarse}(Cx)[Ch_1,\ldots,Ch_k].}
```

h_1Use in source · 4 passages

Search aliases: h_1

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 894
[Jet naturality] Let F_ f and F_ c be C^r maps satisfying CF_ f=F_ cC. Then, for every 1≤ k≤ r, C D^kF_ f(x)[h_1,…,h_k] = D^kF_ c(Cx)[Ch_1,…,Ch_k].
Exact mathematical passage
```
\label{eq:jet-naturality}
  \boxed{
  C D^kF_{\fine}(x)[h_1,\ldots,h_k]
  =
  D^kF_{\coarse}(Cx)[Ch_1,\ldots,Ch_k].}
```

h_{12}Use in source · 2 passages

Search aliases: h_12

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2547
0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x),
Exact mathematical passage
```
  h_{12}(x)
  &=
  (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
  \label{eq:h12}\\
  h_{34}(x)
  &=
  (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3,
  \label{eq:h34}
```

h_2Use in source · 1 passage

Search aliases: h_2

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2276
texorpdfstringRetaining the k largest energy hyperedgesRetaining the k largest energy hyperedges. Let e_h=normwidehat A_m,h_HS^2 and order the energies e_h_1≥ e_h_2≥⋯≥ e_h_M.
Exact mathematical passage
```
  e_{h_1}\geq e_{h_2}\geq\cdots\geq e_{h_M}.
```

h_{34}Use in source · 2 passages

Search aliases: h_34

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2547
0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x),
Exact mathematical passage
```
  h_{12}(x)
  &=
  (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
  \label{eq:h12}\\
  h_{34}(x)
  &=
  (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3,
  \label{eq:h34}
```

h_kUse in source · 2 passages

Search aliases: h_k

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 894
[Jet naturality] Let F_ f and F_ c be C^r maps satisfying CF_ f=F_ cC. Then, for every 1≤ k≤ r, C D^kF_ f(x)[h_1,…,h_k] = D^kF_ c(Cx)[Ch_1,…,Ch_k].
Exact mathematical passage
```
\label{eq:jet-naturality}
  \boxed{
  C D^kF_{\fine}(x)[h_1,\ldots,h_k]
  =
  D^kF_{\coarse}(Cx)[Ch_1,\ldots,Ch_k].}
```

h_MUse in source · 1 passage

Search aliases: h_M

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2276
texorpdfstringRetaining the k largest energy hyperedgesRetaining the k largest energy hyperedges. Let e_h=normwidehat A_m,h_HS^2 and order the energies e_h_1≥ e_h_2≥⋯≥ e_h_M.
Exact mathematical passage
```
  e_{h_1}\geq e_{h_2}\geq\cdots\geq e_{h_M}.
```

h_rUse in source · 2 passages

Search aliases: h_r

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1292
[Bayesian jet naturality] Under the hypotheses of [source reference], for every r≥1 and every interior fine prior, C D^r B_λ^ f(μ) [h_1,…,h_r] = D^r B_λ^ c(Cμ) [Ch_1,…,Ch_r].
Exact mathematical passage
```
\label{eq:bayes-jet-naturality}
  C D^r\Bup_{\lambda^{\fine}}(\mu)
  [h_1,\ldots,h_r]
  =
  D^r\Bup_{\lambda^{\coarse}}(C\mu)
  [Ch_1,\ldots,Ch_r].
```

\mathrm{HS}Use in source · 4 passages

Search aliases: HS mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2223
[Orthogonal hyperedge Parseval identity] Distinct packet blocks are orthogonal in Hilbert Schmidt inner product, and normwidehat A_HS^2 = ∑_i,j_1,…,j_m normwidehat A_imid j_1,…,j_m_HS^2.
Exact mathematical passage
```
\label{eq:hyperedge-parseval}
  \boxed{
  \norm{\widehat A}_{\mathrm{HS}}^2
  =
  \sum_{i,j_1,\ldots,j_m}
  \norm{\widehat A_{i\mid j_1,\ldots,j_m}}_{\mathrm{HS}}^2.}
```

\HyperedgeEntropyValueUse in source · 1 passage

Search aliases: HyperedgeEntropyValue

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2709
Concentrated hyperedge field. A synthetic interaction field with decreasing orthogonal block energies illustrates [source reference]. The computed hyperedge entropy is HyperedgeEntropyValue, corresponding to an effective count of approximately EffectiveHyperedgeCount hyperedges despite a larger ambient support. The ranked spectrum in [source reference] shows why the field is compressible in this representation.
Exact mathematical passage
```
\HyperedgeEntropyValue
```

I_qUse in source · 1 passage

Search aliases: I_q

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 663
[A quotient with invisible internal interactions] Let X_ c= R^q, X_ f= R^2q, and C=I_q · I_q, E=frac12I_q I_q. Let A_m^ c be arbitrary and let K_m:X_ f^m→ X_ f satisfy CK_m=0. Then A_m^ f = EA_m^ cC^× m+K_m satisfies CA_m^ f=A_m^ cC^× m. The tensor K_m may contain many fine hyperedges, so the fine model can remain complicated. Exact quotienting asks only that every hidden contribution disappear after applying C.
Exact mathematical passage
```
  C=\begin{bmatrix}I_q&I_q\end{bmatrix},
  \qquad
  E=\frac12\begin{bmatrix}I_q\\I_q\end{bmatrix}.
```

I_{X_{\coarse}}Use in source · 1 passage

Search aliases: I_X_coarse I_X_ c coarse mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 328
Coarse maps, sections, and hidden directions. Let C:X_ f↠ X_ c be a surjective linear coarse map. Finite dimensionality guarantees a linear right inverse E:X_ c→ X_ f, CE=I_X_ c. Define P_C=EC, Q_C=I_X_ f-EC. Then P_C^2=P_C, Q_C^2=Q_C, Ran Q_C=Ker C. Every fine state has the exact decomposition x=ECx+Q_Cx. The first term is the chosen fine representative of the coarse state. The second lies in Ker C, the hidden subspace. Exact quotienting means that motion inside this subspace cannot change the …
Exact mathematical passage
```
  E:X_{\coarse}\to X_{\fine},
  \qquad CE=I_{X_{\coarse}}.
```

I_{X_{\fine}}Use in source · 1 passage

Search aliases: I_X_fine I_X_ f fine mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 333
Coarse maps, sections, and hidden directions. Let C:X_ f↠ X_ c be a surjective linear coarse map. Finite dimensionality guarantees a linear right inverse E:X_ c→ X_ f, CE=I_X_ c. Define P_C=EC, Q_C=I_X_ f-EC. Then P_C^2=P_C, Q_C^2=Q_C, Ran Q_C=Ker C. Every fine state has the exact decomposition x=ECx+Q_Cx. The first term is the chosen fine representative of the coarse state. The second lies in Ker C, the hidden subspace. Exact quotienting means that motion inside this subspace cannot change the …
Exact mathematical passage
```
\label{eq:representative-hidden-proj}
  P_C=EC,
  \qquad
  Q_C=I_{X_{\fine}}-EC.
```

I_YUse in source · 1 passage

Search aliases: I_Y

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 319
Insert I_X=∑_jP_j in every input slot and I_Y=∑_iQ_i after applying A. Multilinearity distributes all finite sums and gives [source reference].
Exact mathematical passage
```
I_Y=\sum_iQ_i
```

JUse in source · 1 passage

Search aliases: J

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 289
Let (P_j)_j∈ J resolve a source X and (Q_i)_i∈ I resolve a target Y.
Exact mathematical passage
```
(P_j)_{j\in J}
```

\mathbf jUse in source · 7 passages

Search aliases: j mathbf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2392
[Exact packet coordinate lumpability test] For every coarse output a and coarse input tuple b: [label=arabic*.,leftmargin=2.3em] item Evaluate g^(m)_a; j for all j∈ G_ b. item In exact arithmetic, test whether all values are equal. With floating point input, compare their spread with a declared tolerance and record that tolerance. item When the exact test passes, assign the common value to a^ c,(m)_a; b. When only the numerical tolerance is met, assign a declared representative such as the mean …
Exact mathematical passage
```
g^{(m)}_{a;\mathbf j}
```

\mathbf j'Use in source · 1 passage

Search aliases: j' mathbf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2395
em In exact arithmetic, test whether all values are equal. With floating point input, compare their spread with a declared tolerance and record that tolerance. item When the exact test passes, assign the common value to a^ c,(m)_a; b. When only the numerical tolerance is met, assign a declared representative such as the mean or the midrange. item Otherwise, return a witness (a, j, j') with unequal aggregate coefficients.
Exact mathematical passage
```
(a,\mathbf j,\mathbf j')
```

j'Use in source · 4 passages

Search aliases: j'

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 624
[Unary aggregate rate criterion] For m=1, exact descent is equivalent to ∑_i:π(i)=aa^ f,(1)_i;j = ∑_i:π(i)=aa^ f,(1)_i;j' for all coarse outputs a whenever π(j)=π(j').
Exact mathematical passage
```
\label{eq:unary-aggregate-rate}
  \sum_{i:\pi(i)=a}a^{\fine,(1)}_{i;j}
  =
  \sum_{i:\pi(i)=a}a^{\fine,(1)}_{i;j'}
```

j_1'Use in source · 1 passage

Search aliases: j_1'

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 560
[Coefficient criterion for every interaction order] The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m, π(j_r)=b_r.
Exact mathematical passage
```
\label{eq:packet-fiber-constancy}
  g^{(m)}_{a;j_1,\ldots,j_m}
  =
  g^{(m)}_{a;j_1',\ldots,j_m'}
```

j_m'Use in source · 1 passage

Search aliases: j_m'

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 560
[Coefficient criterion for every interaction order] The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m, π(j_r)=b_r.
Exact mathematical passage
```
\label{eq:packet-fiber-constancy}
  g^{(m)}_{a;j_1,\ldots,j_m}
  =
  g^{(m)}_{a;j_1',\ldots,j_m'}
```

j_rUse in source · 3 passages

Search aliases: j_r

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 566
[Coefficient criterion for every interaction order] The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m, π(j_r)=b_r.
Exact mathematical passage
```
  \pi(j_r)=\pi(j_r')
  \qquad(r=1,\ldots,m).
```

j_r'Use in source · 1 passage

Search aliases: j_r'

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 566
[Coefficient criterion for every interaction order] The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m, π(j_r)=b_r.
Exact mathematical passage
```
  \pi(j_r)=\pi(j_r')
  \qquad(r=1,\ldots,m).
```

KUse in source · 15 passages

Search aliases: K

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 668
I_q. Let A_m^ c be arbitrary and let K_m:X_ f^m→ X_ f satisfy CK_m=0. Then A_m^ f = EA_m^ cC^× m+K_m satisfies CA_m^ f=A_m^ cC^× m. The tensor K_m may contain many fine hyperedges, so the fine model can remain complicated. Exact quotienting asks only that every hidden contribution disappear after applying C.
Exact mathematical passage
```
K_m:X_{\fine}^m\to X_{\fine}
```

k'Use in source · 2 passages

Search aliases: k'

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 640
[Quadratic hypergraph criterion] For m=2, exact descent is equivalent to ∑_i:π(i)=aa^ f,(2)_i;j,k = ∑_i:π(i)=aa^ f,(2)_i;j',k' whenever π(j)=π(j'), π(k)=π(k'). The resulting coarse interaction is the directed quadratic hyperedge (π(j),π(k))⟶ a.
Exact mathematical passage
```
  \sum_{i:\pi(i)=a}a^{\fine,(2)}_{i;j,k}
  =
  \sum_{i:\pi(i)=a}a^{\fine,(2)}_{i;j',k'}
```

K_{\coarse}Use in source · 2 passages

Search aliases: K_coarse K_ c coarse mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1088
Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.
Exact mathematical passage
```
K_{\coarse}
```

K_{\fine}Use in source · 2 passages

Search aliases: K_fine K_ f fine mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1088
Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.
Exact mathematical passage
```
K_{\fine}
```

K_mUse in source · 11 passages

Search aliases: K_m

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 668
I_q. Let A_m^ c be arbitrary and let K_m:X_ f^m→ X_ f satisfy CK_m=0. Then A_m^ f = EA_m^ cC^× m+K_m satisfies CA_m^ f=A_m^ cC^× m. The tensor K_m may contain many fine hyperedges, so the fine model can remain complicated. Exact quotienting asks only that every hidden contribution disappear after applying C.
Exact mathematical passage
```
K_m:X_{\fine}^m\to X_{\fine}
```

k_rUse in source · 1 passage

Search aliases: k_r

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2732
Interpretation for reaction networks. Consider a polynomial reaction system dot x=S v(x), v_r(x)=k_r∏_i x_i^α_ir. Each reaction monomial contributes one interaction hyperedge to every species changed by the corresponding stoichiometric column. The inputs are the reactants. A block sum aggregation is exact precisely when the monomial coefficients, after summation over each output fiber, are constant on the product fibers specified by [source reference]. This gives a coefficient test at every inte…
Exact mathematical passage
```
  \dot x=S v(x),
  \qquad
  v_r(x)=k_r\prod_i x_i^{\alpha_{ir}}.
```

\mathrm{keep}Use in source · 8 passages

Search aliases: keep mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2241
Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.
Exact mathematical passage
```
\cE_m^{\mathrm{keep}}
```

\mathsf LUse in source · 1 passage

Search aliases: L mathsf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2959
Certificate schemas and reproducibility. A minimal interaction certificate stores L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite ledger as JSON files and CSV tables. A certificate records a computation under specified inputs. The ledger provides cryptographic authenticity only …
Exact mathematical passage
```
\label{eq:lumpability-certificate-schema}
  \mathsf L_m
  =
  (m,C,A_m^{\fine},A_m^{\coarse},\Delta_m,
  \norm{\Delta_m},\text{norm},\text{tolerance}).
```

L_1Use in source · 2 passages

Search aliases: L_1

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1153
[Composition law for normalized transfers] For positive operators L_1,L_2 on the same ordered state space, N_ℓ(L_2) ∘ N_ℓ(L_1) = N_ℓ(L_2L_1) wherever both sides are defined.
Exact mathematical passage
```
L_1,L_2
```

L_2Use in source · 2 passages

Search aliases: L_2

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1153
[Composition law for normalized transfers] For positive operators L_1,L_2 on the same ordered state space, N_ℓ(L_2) ∘ N_ℓ(L_1) = N_ℓ(L_2L_1) wherever both sides are defined.
Exact mathematical passage
```
L_1,L_2
```

L_{\Bup,k}Use in source · 6 passages

Search aliases: L_Bup,k L_ B,k Bup mathsf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1985
Assume the coarse prediction is Lipschitz with constant L_Φ,k and that its quotient defect is bounded by sup_μ normCΦ_k^ f(μ)-Φ_k^ c(Cμ)_1 ≤ d_Φ,k. Assume the coarse update is Lipschitz with constant L_ B,k and that its direct refinement defect is bounded by sup_μ normC B_k^ f(μ)- B_k^ c(Cμ)_1 ≤ d_ B,k. The suprema in [source reference] and [source reference] are taken over a declared admissible set that contains both recursions.
Exact mathematical passage
```
L_{\Bup,k}
```

L_cUse in source · 1 passage

Search aliases: L_c

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Use in source · README.md, line 31
CL_f=sL_cC Longrightarrow C B_f= B_cC.
Exact mathematical passage
```
CL_f=sL_cC
\Longrightarrow
C\mathsf B_f=\mathsf B_cC.
```

L_fUse in source · 1 passage

Search aliases: L_f

Link to this entry ·

Use in source · README.md, line 31
CL_f=sL_cC Longrightarrow C B_f= B_cC.
Exact mathematical passage
```
CL_f=sL_cC
\Longrightarrow
C\mathsf B_f=\mathsf B_cC.
```

L_{F,k}Use in source · 3 passages

Search aliases: L_F,k

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2101
From interaction coefficients to posterior error. For a flow interval of length T_k, suppose the fine trajectory remains in a ball of radius R_k and the coarse field has Lipschitz constant L_F,k. By Corollary~[source reference], one may take d_Φ,k = d_F,k(R_k) frace^L_F,kT_k-1L_F,k, with the continuous value T_k d_F,k(R_k) at L_F,k=0. For the observation, Corollary~[source reference] gives d_ B,k ≤ frac2normr_λ,k_∞z_*,k. Substituting these bounds into [source reference] gives a certificate expre…
Exact mathematical passage
```
L_{F,k}
```

\mathsf L_mUse in source · 1 passage

Search aliases: L_m mathsf

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2959
Certificate schemas and reproducibility. A minimal interaction certificate stores L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite ledger as JSON files and CSV tables. A certificate records a computation under specified inputs. The ledger provides cryptographic authenticity only …
Exact mathematical passage
```
\label{eq:lumpability-certificate-schema}
  \mathsf L_m
  =
  (m,C,A_m^{\fine},A_m^{\coarse},\Delta_m,
  \norm{\Delta_m},\text{norm},\text{tolerance}).
```

L_y^cUse in source · 1 passage

Search aliases: L_y^c

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Use in source · MANUSCRIPT_STATUS.md, line 66
CL_y^f=s_yL_y^cC ⇒ C B_y^f= B_y^cC ⇒ CΨ_1:N^f=Ψ_1:N^cC.
Exact mathematical passage
```
CL_y^f=s_yL_y^cC
\Rightarrow
C\mathsf B_y^f=\mathsf B_y^cC
\Rightarrow
C\Psi_{1:N}^f=\Psi_{1:N}^cC.
```

L_y^fUse in source · 1 passage

Search aliases: L_y^f

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Use in source · MANUSCRIPT_STATUS.md, line 66
CL_y^f=s_yL_y^cC ⇒ C B_y^f= B_y^cC ⇒ CΨ_1:N^f=Ψ_1:N^cC.
Exact mathematical passage
```
CL_y^f=s_yL_y^cC
\Rightarrow
C\mathsf B_y^f=\mathsf B_y^cC
\Rightarrow
C\Psi_{1:N}^f=\Psi_{1:N}^cC.
```

L_{\lambda}Use in source · 3 passages

Search aliases: L_λ lambda

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1170
Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.
Exact mathematical passage
```
  L_{\lambda}=\diag(\lambda_1,\ldots,\lambda_n),
  \qquad
  \lambda_i>0.
```

L_{\lambda^{\coarse}}Use in source · 1 passage

Search aliases: L_λ^coarse L_λ^ c lambda coarse lambda mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1218
Equation [source reference] is equivalent to CL_λ^ f =sL_λ^ cC. Apply [source reference]. The evidence identity follows from (λ^ f)^Tμ =s(λ^ c)^TCμ.
Exact mathematical passage
```
  CL_{\lambda^{\fine}}
  =sL_{\lambda^{\coarse}}C.
```

L_{\lambda^{\fine}}Use in source · 1 passage

Search aliases: L_λ^fine L_λ^ f lambda fine lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1218
Equation [source reference] is equivalent to CL_λ^ f =sL_λ^ cC. Apply [source reference]. The evidence identity follows from (λ^ f)^Tμ =s(λ^ c)^TCμ.
Exact mathematical passage
```
  CL_{\lambda^{\fine}}
  =sL_{\lambda^{\coarse}}C.
```

L_{\Phi,k}Use in source · 5 passages

Search aliases: L_Φ,k Phi

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1979
Assume the coarse prediction is Lipschitz with constant L_Φ,k and that its quotient defect is bounded by sup_μ normCΦ_k^ f(μ)-Φ_k^ c(Cμ)_1 ≤ d_Φ,k. Assume the coarse update is Lipschitz with constant L_ B,k and that its direct refinement defect is bounded by sup_μ normC B_k^ f(μ)- B_k^ c(Cμ)_1 ≤ d_ B,k. The suprema in [source reference] and [source reference] are taken over a declared admissible set that contains both recursions.
Exact mathematical passage
```
L_{\Phi,k}
```

\limsupUse in source · 1 passage

Search aliases: limsup

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2067
[label=(roman*),leftmargin=2.8em] item If 0≤α<1, then e_N^+ ≤ α^Ne_0^+ + βfrac1-α^N1-α, limsup_N→∞e_N^+ ≤fracβ1-α.
Exact mathematical passage
```
\label{eq:stable-filter-bound}
  e_N^+
  \leq
  \alpha^Ne_0^+
  +
  \beta\frac{1-\alpha^N}{1-\alpha},
  \qquad
  \limsup_{N\to\infty}e_N^+
  \leq\frac{\beta}{1-\alpha}.
```

\limsup_{N\to\infty}Use in source · 1 passage

Search aliases: limsup_N→∞ limsup to infty

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2067
[label=(roman*),leftmargin=2.8em] item If 0≤α<1, then e_N^+ ≤ α^Ne_0^+ + βfrac1-α^N1-α, limsup_N→∞e_N^+ ≤fracβ1-α.
Exact mathematical passage
```
\label{eq:stable-filter-bound}
  e_N^+
  \leq
  \alpha^Ne_0^+
  +
  \beta\frac{1-\alpha^N}{1-\alpha},
  \qquad
  \limsup_{N\to\infty}e_N^+
  \leq\frac{\beta}{1-\alpha}.
```

\logUse in source · 4 passages

Search aliases: log

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2163
for one step] Assume widehatμ_k^ f is a probability vector. Under [source reference], absz_k^ f-s_kz_k^ c ≤ s_knormλ_k^ c_∞e_k^- +normr_k_∞. If z_k^ f,s_kz_k^ c≥ z_*,k>0, then abs log z_k^ f -log(s_kz_k^ c) ≤ frac s_knormλ_k^ c_∞e_k^- +normr_k_∞ z_*,k.
Exact mathematical passage
```
\label{eq:log-evidence-step}
  \boxed{
  \abs{
  \log z_k^{\fine}
  -\log(s_kz_k^{\coarse})}
  \leq
  \frac{
  s_k\norm{\lambda_k^{\coarse}}_{\infty}e_k^-
  +\norm{r_k}_{\infty}
  }{z_{*,k}}.}
```

\LongleftrightarrowUse in source · 1 passage

Search aliases: Longleftrightarrow

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2943
A finite observational partition lemma. Let Y be a finite observation alphabet and let each state i have a strictly positive likelihood profile Λ(i) = (λ_y(i))_y∈ Y. Define i∼_obsj Longleftrightarrow Λ(i)=Λ(j).
Exact mathematical passage
```
  i\sim_{\mathrm{obs}}j
  \quad\Longleftrightarrow\quad
  \Lambda(i)=\Lambda(j).
```

\LongrightarrowUse in source · 4 passages

Search aliases: Longrightarrow

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 236
The proof strategy can be summarized by the chain multilinear quotient Longrightarrow interaction tree quotient Longrightarrow flow quotient Longrightarrow Bayesian quotient Longrightarrow filtering cocycle quotient Longrightarrow posterior certificate.
Exact mathematical passage
```
\boxed{
\begin{gathered}
  \text{multilinear quotient}
  \Longrightarrow
  \text{interaction tree quotient}
  \Longrightarrow
  \text{flow quotient}
  \\
  \Longrightarrow
  \text{Bayesian quotient}
  \Longrightarrow
  \text{filtering cocycle quotient}
  \Longrightarrow
  \text{posterior certificate}.
\end{gathered}}
```

\mathrm{LS}Use in source · 1 passage

Search aliases: LS mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2411
Approximate coarse coefficients. When exact fiber constancy fails, the approximation depends on a chosen loss and, when relevant, a task distribution. Under scalar squared loss with nonnegative weights w_ j on a fiber product, assume their total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ G_ bw_ j . For scalar worst case loss, a minimax representative is the midrange a^ c,∞_a; b = frac12 ( max_ j∈ G_ bg_a; j + min_ j∈ G_ bg_a; j ), with…
Exact mathematical passage
```
\label{eq:least-squares-coarse-coeff}
  a^{\coarse,\mathrm{LS}}_{a;\mathbf b}
  =
  \frac{
  \sum_{\mathbf j\in\cG_{\mathbf b}}
  w_{\mathbf j}g_{a;\mathbf j}
  }{
  \sum_{\mathbf j\in\cG_{\mathbf b}}w_{\mathbf j}
  }.
```

MUse in source · 12 passages

Search aliases: M

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 856
[Coefficientwise characterization of polynomial descent] Let F_ f(x)=∑_m=0^MA_m^ f[x^⊗ m], F_ c(z)=∑_m=0^MA_m^ c[z^⊗ m], where every A_m^ f and A_m^ c is symmetric. Then CF_ f=F_ cC if and only if CA_m^ f=A_m^ cC^× m (0≤ m≤ M).
Exact mathematical passage
```
  F_{\fine}(x)=\sum_{m=0}^{M}A_m^{\fine}[x^{\otimes m}],
  \qquad
  F_{\coarse}(z)=\sum_{m=0}^{M}A_m^{\coarse}[z^{\otimes m}],
```

M_{\coarse}Use in source · 4 passages

Search aliases: M_coarse M_ c coarse mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1453
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f )
Exact mathematical passage
```
M_{\coarse}
```

M_{\fine}Use in source · 4 passages

Search aliases: M_fine M_ f fine mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1453
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f )
Exact mathematical passage
```
M_{\fine}
```

\max_{\mathbf j\in\cG_{\mathbf b}}Use in source · 2 passages

Search aliases: max_ j∈cG_ b max_ j∈ G_ b max mathbf in cG mathbf max mathbf in mathcal mathbf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2422
r total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ G_ bw_ j . For scalar worst case loss, a minimax representative is the midrange a^ c,∞_a; b = frac12 ( max_ j∈ G_ bg_a; j + min_ j∈ G_ bg_a; j ), with optimal scalar fiber error ε^∞_a; b = frac12 ( max_ j∈ G_ bg_a; j - min_ j∈ G_ bg_a; j ). These formulas are local coefficient summaries. A global operator norm can couple several fibers and may require a joint optimization.
Exact mathematical passage
```
\label{eq:minimax-coarse-coeff}
  a^{\coarse,\infty}_{a;\mathbf b}
  =
  \frac12
  \left(
  \max_{\mathbf j\in\cG_{\mathbf b}}g_{a;\mathbf j}
  +
  \min_{\mathbf j\in\cG_{\mathbf b}}g_{a;\mathbf j}
  \right),
```

\min_{\mathbf j\in\cG_{\mathbf b}}Use in source · 2 passages

Search aliases: min_ j∈cG_ b min_ j∈ G_ b min mathbf in cG mathbf min mathbf in mathcal mathbf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2422
r total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ G_ bw_ j . For scalar worst case loss, a minimax representative is the midrange a^ c,∞_a; b = frac12 ( max_ j∈ G_ bg_a; j + min_ j∈ G_ bg_a; j ), with optimal scalar fiber error ε^∞_a; b = frac12 ( max_ j∈ G_ bg_a; j - min_ j∈ G_ bg_a; j ). These formulas are local coefficient summaries. A global operator norm can couple several fibers and may require a joint optimization.
Exact mathematical passage
```
\label{eq:minimax-coarse-coeff}
  a^{\coarse,\infty}_{a;\mathbf b}
  =
  \frac12
  \left(
  \max_{\mathbf j\in\cG_{\mathbf b}}g_{a;\mathbf j}
  +
  \min_{\mathbf j\in\cG_{\mathbf b}}g_{a;\mathbf j}
  \right),
```

\min_{C,\{A_m^{\coarse}\},\{\cE_m^{\mathrm{keep}}\}}Use in source · 1 passage

Search aliases: min_C,{A_m^coarse},{cE_m^keep} min_C,{A_m^ c},{ E_m^keep} min coarse cE mathrm min mathrm mathcal mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2358
A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability conditions on C. The three terms measure failure of dynamical closure, computational cost, and loss of observational information. Their weights and th…
Exact mathematical passage
```
\label{eq:budgeted-design}
  \min_{C,\{A_m^{\coarse}\},\{\cE_m^{\mathrm{keep}}\}}
  \left[
  \sum_mw_m\norm{\Delta_m^{\mathrm{total}}}
  +\lambda\sum_m\operatorname{Cost}(\cE_m^{\mathrm{keep}})
  +\gamma\sum_y d_{\Bup,y}
  \right].
```

\min_{C,\{A_m^{\coarse}\},\{\lambda_y^{\coarse}\}}Use in source · 1 passage

Search aliases: min_C,{A_m^coarse},{λ_y^coarse} min_C,{A_m^ c},{λ_y^ c} min coarse lambda coarse min mathrm lambda mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2821
Approximate joint design. A finite approximate problem is min_C,{A_m^ c},{λ_y^ c} [ ∑_mw_mnormCA_m^ f-A_m^ cC^× m +∑_yv_yd_ B,y +γComplexity(C) ]. The main questions concern existence, tractable relaxations, and stability of the minimizers. Another question is whether the task weights can be learned without invalidating the certificate after a distribution shift.
Exact mathematical passage
```
  \min_{C,\{A_m^{\coarse}\},\{\lambda_y^{\coarse}\}}
  \left[
  \sum_mw_m\norm{CA_m^{\fine}-A_m^{\coarse}C^{\times m}}
  +\sum_yv_yd_{\Bup,y}
  +\gamma\operatorname{Complexity}(C)
  \right].
```

\min_{|\cE|=k}Use in source · 1 passage

Search aliases: min_|cE|=k min_| E|=k min cE min mathcal

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2286
[Optimal retention of k orthogonal hyperedges] Among all retained sets of cardinality k, the set {h_1,…,h_k} minimizes the Hilbert Schmidt residual. The minimum is min_| E|=k normR_m( E)_HS^2 = ∑_r>ke_h_r.
Exact mathematical passage
```
\label{eq:best-k-residual}
  \boxed{
  \min_{|\cE|=k}
  \norm{R_m(\cE)}_{\mathrm{HS}}^2
  =
  \sum_{r>k}e_{h_r}.}
```

\min_{\pi}Use in source · 1 passage

Search aliases: min_π min pi

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2804
Coarsest exact joint partition. For a finite interaction family (A_m^ f)_m≤ M and likelihood family (λ_y^ f)_y∈ Y, seek min_π|I_ c| subject to CA_m^ f=A_m^ cC^× m (m≤ M), and λ_y^ f=s_yC^Tλ_y^ c (y∈ Y). The partition determined by the observations alone groups states with identical likelihood profiles. The dynamical condition also depends on the partition because changing the partition changes the output aggregation. A terminating refinement algorithm, together with a useful complexity bound, re…
Exact mathematical passage
```
\label{eq:coarsest-joint-partition}
  \min_{\pi}|I_{\coarse}|
```

n^{m+1}Use in source · 1 passage

Search aliases: n^m+1

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2399
A dense implementation requires O(n^m+1) arithmetic to aggregate a tensor of order m on n states. If the fine tensor is stored as E_m active scalar hyperedges, the aggregation pass is O(E_m), followed by grouping and consistency checks over the represented coarse incidences.
Exact mathematical passage
```
O(n^{m+1})
```

N_{\mathrm{eff},m}Use in source · 1 passage

Search aliases: N_eff,m mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2311
Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The quantity deliberately depends on the resolution. It evaluates a representation rather than an invariant of the unresolved tensor alone.
Exact mathematical passage
```
  N_{\mathrm{eff},m}=e^{H_m}.
```

\Nrm_{\ell_{\coarse}}Use in source · 2 passages

Search aliases: Nrm_ℓ_coarse N_ℓ_ c Nrm ell coarse mathsf ell mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1125
malized transfer] Let L_ f:X_ f→ X_ f and L_ c:X_ c→ X_ c be positive linear operators. Suppose CL_ f=sL_ cC for some s>0, and suppose [source reference] holds. Then, on the common domain of normalization, C N_ℓ_ f(L_ f) = N_ℓ_ c(L_ c)C.
Exact mathematical passage
```
\label{eq:normalized-transfer-naturality}
  \boxed{
  C\Nrm_{\ell_{\fine}}(L_{\fine})
  =
  \Nrm_{\ell_{\coarse}}(L_{\coarse})C.}
```

\Nrm_{\ell_{\fine}}Use in source · 2 passages

Search aliases: Nrm_ℓ_fine N_ℓ_ f Nrm ell fine mathsf ell mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1125
malized transfer] Let L_ f:X_ f→ X_ f and L_ c:X_ c→ X_ c be positive linear operators. Suppose CL_ f=sL_ cC for some s>0, and suppose [source reference] holds. Then, on the common domain of normalization, C N_ℓ_ f(L_ f) = N_ℓ_ c(L_ c)C.
Exact mathematical passage
```
\label{eq:normalized-transfer-naturality}
  \boxed{
  C\Nrm_{\ell_{\fine}}(L_{\fine})
  =
  \Nrm_{\ell_{\coarse}}(L_{\coarse})C.}
```

OUse in source · 2 passages

Search aliases: O

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2399
A dense implementation requires O(n^m+1) arithmetic to aggregate a tensor of order m on n states. If the fine tensor is stored as E_m active scalar hyperedges, the aggregation pass is O(E_m), followed by grouping and consistency checks over the represented coarse incidences.
Exact mathematical passage
```
O(n^{m+1})
```

\mathrm{obs}Use in source · 2 passages

Search aliases: obs mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2943
A finite observational partition lemma. Let Y be a finite observation alphabet and let each state i have a strictly positive likelihood profile Λ(i) = (λ_y(i))_y∈ Y. Define i∼_obsj Longleftrightarrow Λ(i)=Λ(j).
Exact mathematical passage
```
  i\sim_{\mathrm{obs}}j
  \quad\Longleftrightarrow\quad
  \Lambda(i)=\Lambda(j).
```

\odotUse in source · 6 passages

Search aliases: odot

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 191
Why Bayesian observations add another nonlinear operation. Prediction is only one part of a filter. Given a probability vector μ and a positive likelihood λ, Bayesian conditioning is B_λ(μ) = fracλodotμλ^Tμ. Likelihood multiplication is linear, but the evidence in the denominator couples all packets. The full update is therefore nonlinear even when the likelihood acts separately on each state.
Exact mathematical passage
```
\label{eq:bayes-intro}
  \Bup_{\lambda}(\mu)
  =
  \frac{\lambda\odot\mu}{\lambda^T\mu}.
```

\one_{\coarse}^TUse in source · 1 passage

Search aliases: one_coarse^T 1_ c^T one coarse mathbf mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1058
Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.
Exact mathematical passage
```
  \one_{\coarse}^TC=\one_{\fine}^T.
```

\one_{\fine}^TUse in source · 1 passage

Search aliases: one_fine^T 1_ f^T one fine mathbf mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1058
Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.
Exact mathematical passage
```
  \one_{\coarse}^TC=\one_{\fine}^T.
```

\overline AUse in source · 2 passages

Search aliases: overline A overline

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2920
/Ker C be the canonical quotient map. Since C is surjective, there is a unique vector space isomorphism widetilde C:X_ f/Ker C→ X_ c satisfying C=widetilde Cq. The multilinear map CA_m^ f:X_ f^m→ X_ c descends if and only if its value is unchanged when any one input is altered by an element of Ker C. The induced quotient tensor is overline A_m ([x_1],…,[x_m]) =CA_m^ f(x_1,…,x_m), and A_m^ c = widetilde C overline A_m (widetilde C^-1,…,widetilde C^-1). This formulation shows that [source referenc…
Exact mathematical passage
```
  \overline A_m
  ([x_1],\ldots,[x_m])
  =CA_m^{\fine}(x_1,\ldots,x_m),
```

\overline A_mUse in source · 2 passages

Search aliases: overline A_m overline

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2920
/Ker C be the canonical quotient map. Since C is surjective, there is a unique vector space isomorphism widetilde C:X_ f/Ker C→ X_ c satisfying C=widetilde Cq. The multilinear map CA_m^ f:X_ f^m→ X_ c descends if and only if its value is unchanged when any one input is altered by an element of Ker C. The induced quotient tensor is overline A_m ([x_1],…,[x_m]) =CA_m^ f(x_1,…,x_m), and A_m^ c = widetilde C overline A_m (widetilde C^-1,…,widetilde C^-1). This formulation shows that [source referenc…
Exact mathematical passage
```
  \overline A_m
  ([x_1],\ldots,[x_m])
  =CA_m^{\fine}(x_1,\ldots,x_m),
```

\overline\lambdaUse in source · 3 passages

Search aliases: overlineλ overline lambda

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1259
fracλ_i^ fλ_k^ f = fracλ_a^ cλ_b^ c. Take another state j in the same fiber as i. Applying the same argument to j and k gives λ_j^ f=λ_i^ f. The fine likelihood is therefore constant on each fiber. Write its value on fiber a as overlineλ_a. Equation [source reference] now gives fracoverlineλ_aoverlineλ_b = fracλ_a^ cλ_b^ c for every pair a,b. Hence overlineλ=sλ^ c for one s>0, proving the result.
Exact mathematical passage
```
\overline\lambda_a
```

\overline\lambda_aUse in source · 2 passages

Search aliases: overlineλ_a overline lambda

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1259
fracλ_i^ fλ_k^ f = fracλ_a^ cλ_b^ c. Take another state j in the same fiber as i. Applying the same argument to j and k gives λ_j^ f=λ_i^ f. The fine likelihood is therefore constant on each fiber. Write its value on fiber a as overlineλ_a. Equation [source reference] now gives fracoverlineλ_aoverlineλ_b = fracλ_a^ cλ_b^ c for every pair a,b. Hence overlineλ=sλ^ c for one s>0, proving the result.
Exact mathematical passage
```
\overline\lambda_a
```

\overline\lambda_bUse in source · 1 passage

Search aliases: overlineλ_b overline lambda

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1260
λ_k^ f = fracλ_a^ cλ_b^ c. Take another state j in the same fiber as i. Applying the same argument to j and k gives λ_j^ f=λ_i^ f. The fine likelihood is therefore constant on each fiber. Write its value on fiber a as overlineλ_a. Equation [source reference] now gives fracoverlineλ_aoverlineλ_b = fracλ_a^ cλ_b^ c for every pair a,b. Hence overlineλ=sλ^ c for one s>0, proving the result.
Exact mathematical passage
```
  \frac{\overline\lambda_a}{\overline\lambda_b}
  =
  \frac{\lambda_a^{\coarse}}{\lambda_b^{\coarse}}
```

pUse in source · 3 passages

Search aliases: p

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2305
Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The quantity deliberately depends on the resolution. It evaluates a representation rather than an invariant of the unresolved tensor alone.
Exact mathematical passage
```
  p_h=\frac{e_h}{\sum_{\eta}e_{\eta}},
  \qquad
  H_m=-\sum_hp_h\log p_h,
```

P_C^2Use in source · 1 passage

Search aliases: P_C^2

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 339
Coarse maps, sections, and hidden directions. Let C:X_ f↠ X_ c be a surjective linear coarse map. Finite dimensionality guarantees a linear right inverse E:X_ c→ X_ f, CE=I_X_ c. Define P_C=EC, Q_C=I_X_ f-EC. Then P_C^2=P_C, Q_C^2=Q_C, Ran Q_C=Ker C. Every fine state has the exact decomposition x=ECx+Q_Cx. The first term is the chosen fine representative of the coarse state. The second lies in Ker C, the hidden subspace. Exact quotienting means that motion inside this subspace cannot change the …
Exact mathematical passage
```
  P_C^2=P_C,
  \qquad
  Q_C^2=Q_C,
  \qquad
  \Ran Q_C=\Ker C.
```

p_hUse in source · 1 passage

Search aliases: p_h

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2305
Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The quantity deliberately depends on the resolution. It evaluates a representation rather than an invariant of the unresolved tensor alone.
Exact mathematical passage
```
  p_h=\frac{e_h}{\sum_{\eta}e_{\eta}},
  \qquad
  H_m=-\sum_hp_h\log p_h,
```

\PerturbationSlopeUse in source · 1 passage

Search aliases: PerturbationSlope

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2696
[Perturbation scaling] The experiment used logarithmically spaced values with ε=η. The fitted slope of the final posterior discrepancy was PerturbationSlope. Every measured error remained below its recursive certificate.
Exact mathematical passage
```
\PerturbationSlope
```

\proptoUse in source · 1 passage

Search aliases: propto

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2833
Posterior distributions over resolutions. Instead of selecting one C, place a posterior on a finite family of candidate resolutions: p(Cmid D) propto p(Dmid C)p(C). Predictions then average both parameter and topology uncertainty: p(z_tmid D) = ∑_Cp(z_tmid C,D)p(Cmid D). A fuller model should distinguish uncertainty in the coefficients of a fixed hypergraph from uncertainty about the hypergraph and the quotient map themselves.
Exact mathematical passage
```
  p(C\mid D)
  \propto
  p(D\mid C)p(C).
```

\mathfrak QUse in source · 1 passage

Search aliases: Q mathfrak

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1510
Best coarse interaction and quotient defect. Let Q_m(C) = {B_mC^× m:B_m∈ L^m(X_ c;X_ c)} be the subspace of outputs from multilinear maps of order m that factor through the coarse space.
Exact mathematical passage
```
  \mathfrak Q_m(C)
  =
  \{B_mC^{\times m}:B_m\in\cL^m(X_{\coarse};X_{\coarse})\}
```

Q_C^2Use in source · 1 passage

Search aliases: Q_C^2

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 339
Coarse maps, sections, and hidden directions. Let C:X_ f↠ X_ c be a surjective linear coarse map. Finite dimensionality guarantees a linear right inverse E:X_ c→ X_ f, CE=I_X_ c. Define P_C=EC, Q_C=I_X_ f-EC. Then P_C^2=P_C, Q_C^2=Q_C, Ran Q_C=Ker C. Every fine state has the exact decomposition x=ECx+Q_Cx. The first term is the chosen fine representative of the coarse state. The second lies in Ker C, the hidden subspace. Exact quotienting means that motion inside this subspace cannot change the …
Exact mathematical passage
```
  P_C^2=P_C,
  \qquad
  Q_C^2=Q_C,
  \qquad
  \Ran Q_C=\Ker C.
```

Q_{\coarse}Use in source · 2 passages

Search aliases: Q_coarse Q_ c coarse mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2749
Markov lumpability and Markov functions. For unary stochastic dynamics, the identity CQ_ f=Q_ cC is a standard exact lumping condition. It implies Ce^tQ_ f=e^tQ_ cC and means that the coarse process is Markov for every fine initial distribution compatible with the quotient. Classical treatments distinguish several forms of strong, ordinary, exact, and weak lumpability, depending on the admissible initial laws and the direction of the stochastic convention [source reference]. Here we fix one colu…
Exact mathematical passage
```
  CQ_{\fine}=Q_{\coarse}C
```

Q_{\fine}Use in source · 3 passages

Search aliases: Q_fine Q_ f fine mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 636
Let Q_ f be the generator of a continuous time Markov chain, using the column vector convention. The left side of [source reference] is the total transition rate from fine state j into coarse packet a. Corollary~[source reference] is therefore the standard criterion for strong lumpability.
Exact mathematical passage
```
Q_{\fine}
```

\mathfrak Q_mUse in source · 1 passage

Search aliases: Q_m mathfrak

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1510
Best coarse interaction and quotient defect. Let Q_m(C) = {B_mC^× m:B_m∈ L^m(X_ c;X_ c)} be the subspace of outputs from multilinear maps of order m that factor through the coarse space.
Exact mathematical passage
```
  \mathfrak Q_m(C)
  =
  \{B_mC^{\times m}:B_m\in\cL^m(X_{\coarse};X_{\coarse})\}
```

rUse in source · 43 passages

Search aliases: r

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 403
Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.
Exact mathematical passage
```
  \norm{A}
  =
  \sup_{\norm{x_r}\leq1}
  \norm{A(x_1,\ldots,x_m)}.
```

\R^{2q}Use in source · 1 passage

Search aliases: R^2q R mathbb

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 662
[A quotient with invisible internal interactions] Let X_ c= R^q, X_ f= R^2q, and C=I_q · I_q, E=frac12I_q
Exact mathematical passage
```
X_{\fine}=\R^{2q}
```

R^mUse in source · 2 passages

Search aliases: R^m

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1625
[Analytic defect majorant] If normΔ_m≤ε_m and the scalar series d_F(R)=∑_m≥0ε_mR^m converges, then sup_normx≤ Rnormδ_F(x) ≤ d_F(R).
Exact mathematical passage
```
  \mathfrak d_F(R)=\sum_{m\geq0}\varepsilon_mR^m
```

r^mUse in source · 1 passage

Search aliases: r^m

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 803
[Analytic interaction descent] Let C:X_ f↠ X_ c have a bounded right inverse E. Suppose the coefficient series converges normally on every smaller ball, ∑_m=0^∞normA_m^ fr^m<∞ (0≤ r<R), and, for every m≥0, there is a coarse coefficient A_m^ c satisfying CA_m^ f=A_m^ cC^× m. Then F_ c(z) = ∑_m=0^∞A_m^ c[z^⊗ m] converges absolutely whenever normEnormz<R, and CF_ f(x)=F_ c(Cx) for every x in the fine convergence domain whose coarse image lies in the coarse domain.
Exact mathematical passage
```
\label{eq:fine-normal-summability}
  \sum_{m=0}^{\infty}\norm{A_m^{\fine}}r^m<\infty
  \qquad(0\leq r<R),
```

\R_+^nUse in source · 1 passage

Search aliases: R_+^n R mathbb

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1050
Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.
Exact mathematical passage
```
  \Delta_{\fine}
  =\{\mu\in\R_+^n:\one^T\mu=1\},
  \qquad
  \Delta_{\coarse}
  =\{\nu\in\R_+^q:\one^T\nu=1\}.
```

\R_+^qUse in source · 2 passages

Search aliases: R_+^q R mathbb

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1050
Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.
Exact mathematical passage
```
  \Delta_{\fine}
  =\{\mu\in\R_+^n:\one^T\mu=1\},
  \qquad
  \Delta_{\coarse}
  =\{\nu\in\R_+^q:\one^T\nu=1\}.
```

r_0Use in source · 1 passage

Search aliases: r_0

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2547
0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x),
Exact mathematical passage
```
  h_{12}(x)
  &=
  (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
  \label{eq:h12}\\
  h_{34}(x)
  &=
  (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3,
  \label{eq:h34}
```

r_1Use in source · 1 passage

Search aliases: r_1

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2547
0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x),
Exact mathematical passage
```
  h_{12}(x)
  &=
  (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
  \label{eq:h12}\\
  h_{34}(x)
  &=
  (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3,
  \label{eq:h34}
```

\R_{>0}^nUse in source · 1 passage

Search aliases: R_>0^n R mathbb

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1197
[Sufficient condition for Bayesian refinement] Let λ^ f∈ R_>0^n and λ^ c∈ R_>0^q. If λ^ f=sC^Tλ^ c for some s>0, then C B_λ^ f = B_λ^ cC. Moreover, Z_λ^ f(μ) = sZ_λ^ c(Cμ).
Exact mathematical passage
```
\lambda^{\fine}\in\R_{>0}^n
```

\R_{>0}^qUse in source · 2 passages

Search aliases: R_>0^q R mathbb

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1197
[Sufficient condition for Bayesian refinement] Let λ^ f∈ R_>0^n and λ^ c∈ R_>0^q. If λ^ f=sC^Tλ^ c for some s>0, then C B_λ^ f = B_λ^ cC. Moreover, Z_λ^ f(μ) = sZ_λ^ c(Cμ).
Exact mathematical passage
```
\lambda^{\coarse}\in\R_{>0}^q
```

R_kUse in source · 3 passages

Search aliases: R_k

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2101
From interaction coefficients to posterior error. For a flow interval of length T_k, suppose the fine trajectory remains in a ball of radius R_k and the coarse field has Lipschitz constant L_F,k. By Corollary~[source reference], one may take d_Φ,k = d_F,k(R_k) frace^L_F,kT_k-1L_F,k, with the continuous value T_k d_F,k(R_k) at L_F,k=0. For the observation, Corollary~[source reference] gives d_ B,k ≤ frac2normr_λ,k_∞z_*,k. Substituting these bounds into [source reference] gives a certificate expre…
Exact mathematical passage
```
R_k
```

r_kUse in source · 4 passages

Search aliases: r_k

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2139
Sequential evidence error. For brevity, write λ_k^ f=λ_y_k^ f and λ_k^ c=λ_y_k^ c, and decompose λ_k^ f =s_kC^Tλ_k^ c+r_k. This gives z_k^ f-s_kz_k^ c · = s_k(λ_k^ c)^T (Cwidehatμ_k^ f-widehatμ_k^ c) +r_k^Twidehatμ_k^ f.
Exact mathematical passage
```
\label{eq:step-likelihood-residual}
  \lambda_k^{\fine}
  =s_kC^T\lambda_k^{\coarse}+r_k.
```

r_k^TUse in source · 1 passage

Search aliases: r_k^T

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2144
Sequential evidence error. For brevity, write λ_k^ f=λ_y_k^ f and λ_k^ c=λ_y_k^ c, and decompose λ_k^ f =s_kC^Tλ_k^ c+r_k. This gives z_k^ f-s_kz_k^ c · = s_k(λ_k^ c)^T (Cwidehatμ_k^ f-widehatμ_k^ c) +r_k^Twidehatμ_k^ f.
Exact mathematical passage
```
  z_k^{\fine}-s_kz_k^{\coarse}
  &=
  s_k(\lambda_k^{\coarse})^T
  (C\widehat\mu_k^{\fine}-\widehat\mu_k^{\coarse})
  +r_k^T\widehat\mu_k^{\fine}.
  \label{eq:evidence-error-identity}
```

R_m^{\fine}Use in source · 4 passages

Search aliases: R_m^fine R_m^ f fine mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2318
Combined quotient and pruning defect. Let the full fine coefficient split as A_m^ f =A_m^keep+R_m^ f. For a proposed coarse coefficient A_m^ c, define Δ_m^total =CA_m^ f-A_m^ cC^× m. Then Δ_m^total · = underbraceCA_m^keep-A_m^ cC^× m _Δ_m^keep +CR_m^ f. Therefore normΔ_m^total ≤ normΔ_m^keep +normCnormR_m^ f.
Exact mathematical passage
```
  A_m^{\fine}
  =A_m^{\mathrm{keep}}+R_m^{\fine}.
```

r_yUse in source · 1 passage

Search aliases: r_y

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2966
Certificate schemas and reproducibility. A minimal interaction certificate stores L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite ledger as JSON files and CSV tables. A certificate records a computation under specified inputs. The ledger provides cryptographic authenticity only …
Exact mathematical passage
```
\label{eq:bayes-certificate-schema}
  \mathsf B_y
  =
  (\lambda_y^{\fine},\lambda_y^{\coarse},s_y,r_y,
  z_{*,y},d_{\Bup,y}).
```

r_{\lambda}Use in source · 6 passages

Search aliases: r_λ lambda

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1836
Likelihood variation inside a fiber. For diagonal likelihood operators, write λ^ f = sC^Tλ^ c+r_λ. Then R_Lμ=C(r_λodotμ). Since block sum aggregation is an ℓ^1 contraction on signed vectors, normR_Lμ_1 ≤ ∑_iμ_iabsr_λ,i ≤ normr_λ_∞.
Exact mathematical passage
```
\label{eq:likelihood-residual}
  \lambda^{\fine}
  =
  sC^T\lambda^{\coarse}+r_{\lambda}.
```

r_{\lambda,i}Use in source · 2 passages

Search aliases: r_λ,i lambda

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1846
Likelihood variation inside a fiber. For diagonal likelihood operators, write λ^ f = sC^Tλ^ c+r_λ. Then R_Lμ=C(r_λodotμ). Since block sum aggregation is an ℓ^1 contraction on signed vectors, normR_Lμ_1 ≤ ∑_iμ_iabsr_λ,i ≤ normr_λ_∞.
Exact mathematical passage
```
\label{eq:likelihood-residual-l1}
  \norm{R_L\mu}_1
  \leq
  \sum_i\mu_i\abs{r_{\lambda,i}}
  \leq
  \norm{r_{\lambda}}_{\infty}.
```

r_{\lambda,k}Use in source · 2 passages

Search aliases: r_λ,k lambda

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2110
ield has Lipschitz constant L_F,k. By Corollary~[source reference], one may take d_Φ,k = d_F,k(R_k) frace^L_F,kT_k-1L_F,k, with the continuous value T_k d_F,k(R_k) at L_F,k=0. For the observation, Corollary~[source reference] gives d_ B,k ≤ frac2normr_λ,k_∞z_*,k. Substituting these bounds into [source reference] gives a certificate expressed directly in terms of interaction tensor defects and likelihood residuals within each fiber.
Exact mathematical passage
```
\label{eq:dbayes-coefficient}
  d_{\Bup,k}
  \leq
  \frac{2\norm{r_{\lambda,k}}_{\infty}}{z_{*,k}}.
```

\RanUse in source · 2 passages

Search aliases: Ran operatorname

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 339
Coarse maps, sections, and hidden directions. Let C:X_ f↠ X_ c be a surjective linear coarse map. Finite dimensionality guarantees a linear right inverse E:X_ c→ X_ f, CE=I_X_ c. Define P_C=EC, Q_C=I_X_ f-EC. Then P_C^2=P_C, Q_C^2=Q_C, Ran Q_C=Ker C. Every fine state has the exact decomposition x=ECx+Q_Cx. The first term is the chosen fine representative of the coarse state. The second lies in Ker C, the hidden subspace. Exact quotienting means that motion inside this subspace cannot change the …
Exact mathematical passage
```
  P_C^2=P_C,
  \qquad
  Q_C^2=Q_C,
  \qquad
  \Ran Q_C=\Ker C.
```

SUse in source · 3 passages

Search aliases: S

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1565
[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,
Exact mathematical passage
```
\label{eq:hidden-slot-sum}
  \boxed{
  \Delta_m^E(x_1,\ldots,x_m)
  =
  \sum_{\varnothing\neq S\subseteq\{1,\ldots,m\}}
  CA_m^{\fine}(y_1^S,\ldots,y_m^S),}
```

s_0Use in source · 1 passage

Search aliases: s_0

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2547
0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x),
Exact mathematical passage
```
  h_{12}(x)
  &=
  (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
  \label{eq:h12}\\
  h_{34}(x)
  &=
  (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3,
  \label{eq:h34}
```

s_1Use in source · 1 passage

Search aliases: s_1

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2547
0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x),
Exact mathematical passage
```
  h_{12}(x)
  &=
  (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
  \label{eq:h12}\\
  h_{34}(x)
  &=
  (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3,
  \label{eq:h34}
```

s_kUse in source · 11 passages

Search aliases: s_k

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1416
[Evidence factorization under exact refinement] Assume the hypotheses of [source reference], let μ_0^ c=Cμ_0^ f, and suppose λ_y_k^ f =s_kC^Tλ_y_k^ c (s_k>0). Then z_k^ f=s_kz_k^ c for every step, and the likelihood of the observation sequence satisfies Z_1:N^ f = (∏_k=1^Ns_k) Z_1:N^ c. Under the gauge s_k=1, the fine and coarse sequential evidences agree exactly.
Exact mathematical passage
```
  \lambda_{y_k}^{\fine}
  =s_kC^T\lambda_{y_k}^{\coarse}
  \qquad(s_k>0).
```

s_yUse in source · 5 passages

Search aliases: s_y

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1458
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f )
Exact mathematical passage
```
  \lambda_y^{\fine}=s_yC^T\lambda_y^{\coarse}.
```

\supUse in source · 4 passages

Search aliases: sup

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 403
Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.
Exact mathematical passage
```
  \norm{A}
  =
  \sup_{\norm{x_r}\leq1}
  \norm{A(x_1,\ldots,x_m)}.
```

\sup_{\norm{x_r}\leq1}Use in source · 1 passage

Search aliases: sup_normx_r≤1 sup norm leq

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 403
Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.
Exact mathematical passage
```
  \norm{A}
  =
  \sup_{\norm{x_r}\leq1}
  \norm{A(x_1,\ldots,x_m)}.
```

\sup_{\norm{x}\leq R}Use in source · 1 passage

Search aliases: sup_normx≤ R sup norm leq

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1629
[Analytic defect majorant] If normΔ_m≤ε_m and the scalar series d_F(R)=∑_m≥0ε_mR^m converges, then sup_normx≤ Rnormδ_F(x) ≤ d_F(R).
Exact mathematical passage
```
\label{eq:field-defect-majorant}
  \boxed{
  \sup_{\norm{x}\leq R}\norm{\delta_F(x)}
  \leq
  \mathfrak d_F(R).}
```

\sup_{\mu}Use in source · 2 passages

Search aliases: sup_μ sup mu

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1980
Assume the coarse prediction is Lipschitz with constant L_Φ,k and that its quotient defect is bounded by sup_μ normCΦ_k^ f(μ)-Φ_k^ c(Cμ)_1 ≤ d_Φ,k. Assume the coarse update is Lipschitz with constant L_ B,k and that its direct refinement defect is bounded by sup_μ normC B_k^ f(μ)- B_k^ c(Cμ)_1 ≤ d_ B,k. The suprema in [source reference] and [source reference] are taken over a declared admissible set that contains both recursions.
Exact mathematical passage
```
\label{eq:prediction-map-defect}
  \sup_{\mu}
  \norm{C\Phi_k^{\fine}(\mu)-\Phi_k^{\coarse}(C\mu)}_1
  \leq d_{\Phi,k}.
```

t^mUse in source · 1 passage

Search aliases: t^m

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 874
The reverse implication follows from [source reference], applied to a finite sum. Conversely, fix x∈ X_ f and evaluate the field identity at tx: ∑_m=0^Mt^mCA_m^ f[x^⊗ m] = ∑_m=0^Mt^mA_m^ c[(Cx)^⊗ m]. Equality of the coefficients of this polynomial with values in X_ c gives equality on the diagonal for the two symmetric multilinear maps of order m. The polarization identity recovers each multilinear map from its diagonal polynomial, so CA_m^ f=A_m^ cC^× m for every degree.
Exact mathematical passage
```
  \sum_{m=0}^{M}t^mCA_m^{\fine}[x^{\otimes m}]
  =
  \sum_{m=0}^{M}t^mA_m^{\coarse}[(Cx)^{\otimes m}].
```

t_0Use in source · 1 passage

Search aliases: t_0

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1310
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ cc
Exact mathematical passage
```
0=t_0<t_1<\cdots<t_N
```

t_1Use in source · 1 passage

Search aliases: t_1

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1310
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ cc
Exact mathematical passage
```
0=t_0<t_1<\cdots<t_N
```

T_kUse in source · 3 passages

Search aliases: T_k

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2101
From interaction coefficients to posterior error. For a flow interval of length T_k, suppose the fine trajectory remains in a ball of radius R_k and the coarse field has Lipschitz constant L_F,k. By Corollary~[source reference], one may take d_Φ,k = d_F,k(R_k) frace^L_F,kT_k-1L_F,k, with the continuous value T_k d_F,k(R_k) at L_F,k=0. For the observation, Corollary~[source reference] gives d_ B,k ≤ frac2normr_λ,k_∞z_*,k. Substituting these bounds into [source reference] gives a certificate expre…
Exact mathematical passage
```
T_k
```

t_kUse in source · 1 passage

Search aliases: t_k

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1311
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Phi_k^{\fine}=\Phi_{t_k-t_{k-1}}^{\fine},
  \qquad
  \Phi_k^{\coarse}=\Phi_{t_k-t_{k-1}}^{\coarse}.
```

t_{k-1}Use in source · 1 passage

Search aliases: t_k-1

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1311
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Phi_k^{\fine}=\Phi_{t_k-t_{k-1}}^{\fine},
  \qquad
  \Phi_k^{\coarse}=\Phi_{t_k-t_{k-1}}^{\coarse}.
```

t_NUse in source · 1 passage

Search aliases: t_N

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1310
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ cc
Exact mathematical passage
```
0=t_0<t_1<\cdots<t_N
```

\mathrm{total}Use in source · 4 passages

Search aliases: total mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2323
Combined quotient and pruning defect. Let the full fine coefficient split as A_m^ f =A_m^keep+R_m^ f. For a proposed coarse coefficient A_m^ c, define Δ_m^total =CA_m^ f-A_m^ cC^× m. Then Δ_m^total · = underbraceCA_m^keep-A_m^ cC^× m _Δ_m^keep +CR_m^ f. Therefore normΔ_m^total ≤ normΔ_m^keep +normCnormR_m^ f.
Exact mathematical passage
```
  \Delta_m^{\mathrm{total}}
  =CA_m^{\fine}-A_m^{\coarse}C^{\times m}.
```

\TVUse in source · 1 passage

Search aliases: TV operatorname

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2127
[Coarse observable error] For any coarse observable g∈ R^q, absg^TCμ_N^ f-g^Tμ_N^ c ≤ normg_∞e_N^+. In particular, TV(Cμ_N^ f,μ_N^ c) =frac12e_N^+.
Exact mathematical passage
```
  \TV(C\mu_N^{\fine},\mu_N^{\coarse})
  =\frac12e_N^+.
```

uUse in source · 18 passages

Search aliases: u

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1245
Both sides of [source reference] extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets a and b. Choose i∈π^-1(a) and k∈π^-1(b), then consider a prior supported on {i,k} with masses u,v>0. On the fine side, the ratio of the posterior masses of packets a and b is fracλ_i^ fuλ_k^ fv, whereas the coarse posterior ratio is fracλ_a^ cuλ_b^ cv. Equality for all u,v>0 gives fracλ_i^ fλ_k^ f = f…
Exact mathematical passage
```
u,v>0
```

u_0Use in source · 1 passage

Search aliases: u_0

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2547
0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x),
Exact mathematical passage
```
  h_{12}(x)
  &=
  (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
  \label{eq:h12}\\
  h_{34}(x)
  &=
  (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3,
  \label{eq:h34}
```

u_1Use in source · 1 passage

Search aliases: u_1

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2547
0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x),
Exact mathematical passage
```
  h_{12}(x)
  &=
  (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
  \label{eq:h12}\\
  h_{34}(x)
  &=
  (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3,
  \label{eq:h34}
```

vUse in source · 20 passages

Search aliases: v

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1245
Both sides of [source reference] extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets a and b. Choose i∈π^-1(a) and k∈π^-1(b), then consider a prior supported on {i,k} with masses u,v>0. On the fine side, the ratio of the posterior masses of packets a and b is fracλ_i^ fuλ_k^ fv, whereas the coarse posterior ratio is fracλ_a^ cuλ_b^ cv. Equality for all u,v>0 gives fracλ_i^ fλ_k^ f = f…
Exact mathematical passage
```
u,v>0
```

v_0Use in source · 1 passage

Search aliases: v_0

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2547
0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x),
Exact mathematical passage
```
  h_{12}(x)
  &=
  (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
  \label{eq:h12}\\
  h_{34}(x)
  &=
  (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3,
  \label{eq:h34}
```

v_1Use in source · 1 passage

Search aliases: v_1

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2547
0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x),
Exact mathematical passage
```
  h_{12}(x)
  &=
  (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
  \label{eq:h12}\\
  h_{34}(x)
  &=
  (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3,
  \label{eq:h34}
```

v_rUse in source · 1 passage

Search aliases: v_r

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2732
Interpretation for reaction networks. Consider a polynomial reaction system dot x=S v(x), v_r(x)=k_r∏_i x_i^α_ir. Each reaction monomial contributes one interaction hyperedge to every species changed by the corresponding stoichiometric column. The inputs are the reactants. A block sum aggregation is exact precisely when the monomial coefficients, after summation over each output fiber, are constant on the product fibers specified by [source reference]. This gives a coefficient test at every inte…
Exact mathematical passage
```
  \dot x=S v(x),
  \qquad
  v_r(x)=k_r\prod_i x_i^{\alpha_{ir}}.
```

v_yUse in source · 1 passage

Search aliases: v_y

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2821
Approximate joint design. A finite approximate problem is min_C,{A_m^ c},{λ_y^ c} [ ∑_mw_mnormCA_m^ f-A_m^ cC^× m +∑_yv_yd_ B,y +γComplexity(C) ]. The main questions concern existence, tractable relaxations, and stability of the minimizers. Another question is whether the task weights can be learned without invalidating the certificate after a distribution shift.
Exact mathematical passage
```
  \min_{C,\{A_m^{\coarse}\},\{\lambda_y^{\coarse}\}}
  \left[
  \sum_mw_m\norm{CA_m^{\fine}-A_m^{\coarse}C^{\times m}}
  +\sum_yv_yd_{\Bup,y}
  +\gamma\operatorname{Complexity}(C)
  \right].
```

wUse in source · 4 passages

Search aliases: w

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2358
A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability conditions on C. The three terms measure failure of dynamical closure, computational cost, and loss of observational information. Their weights and th…
Exact mathematical passage
```
\label{eq:budgeted-design}
  \min_{C,\{A_m^{\coarse}\},\{\cE_m^{\mathrm{keep}}\}}
  \left[
  \sum_mw_m\norm{\Delta_m^{\mathrm{total}}}
  +\lambda\sum_m\operatorname{Cost}(\cE_m^{\mathrm{keep}})
  +\gamma\sum_y d_{\Bup,y}
  \right].
```

w_{\mathbf j}Use in source · 2 passages

Search aliases: w_ j mathbf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2410
Approximate coarse coefficients. When exact fiber constancy fails, the approximation depends on a chosen loss and, when relevant, a task distribution. Under scalar squared loss with nonnegative weights w_ j on a fiber product, assume their total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ G_ bw_ j . For scalar worst case loss, a minimax representative is the midrange a^ c,∞_a; b = frac12 ( max_ j∈ G_ bg_a; j + min_ j∈ G_ bg_a; j ), with…
Exact mathematical passage
```
w_{\mathbf j}
```

w_mUse in source · 2 passages

Search aliases: w_m

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2358
A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability conditions on C. The three terms measure failure of dynamical closure, computational cost, and loss of observational information. Their weights and th…
Exact mathematical passage
```
\label{eq:budgeted-design}
  \min_{C,\{A_m^{\coarse}\},\{\cE_m^{\mathrm{keep}}\}}
  \left[
  \sum_mw_m\norm{\Delta_m^{\mathrm{total}}}
  +\lambda\sum_m\operatorname{Cost}(\cE_m^{\mathrm{keep}})
  +\gamma\sum_y d_{\Bup,y}
  \right].
```

\widehat AUse in source · 9 passages

Search aliases: widehat A widehat

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 454
item Let widehat A_m^ f:X_ f^⊗ m→ X_ c be the linearization of CA_m^ f. Then Ker(C^⊗ m) ⊆ Ker(widehat A_m^ f).
Exact mathematical passage
```
\widehat A_m^{\fine}:X_{\fine}^{\otimes m}\to X_{\coarse}
```

\widehat A_{i\mid j_1,\ldots,j_m}Use in source · 2 passages

Search aliases: widehat A_imid j_1,…,j_m widehat mid ldots

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2215
Hilbert Schmidt tensor blocks. Let X and Y be finite dimensional Hilbert spaces with orthogonal packet projectors (P_j) and (Q_i). Identify a multilinear Hilbert Schmidt tensor of order m with the Hilbert Schmidt linear map widehat A:X^⊗ m→ Y. Its packet block is widehat A_imid j_1,…,j_m = Q_iwidehat A(P_j_1⊗⋯⊗ P_j_m).
Exact mathematical passage
```
  \widehat A_{i\mid j_1,\ldots,j_m}
  =
  Q_i\widehat A(P_{j_1}\otimes\cdots\otimes P_{j_m}).
```

\widehat A_mUse in source · 1 passage

Search aliases: widehat A_m widehat

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2242
Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.
Exact mathematical passage
```
\label{eq:pruned-tensor}
  \widehat A_m^{\mathrm{keep}}
  =
  \sum_{h\in\cE_m^{\mathrm{keep}}}
  \widehat A_{m,h},
  \qquad
  R_m
  =
  \widehat A_m-\widehat A_m^{\mathrm{keep}}.
```

\widehat A_{m,h}Use in source · 3 passages

Search aliases: widehat A_m,h widehat

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2242
Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.
Exact mathematical passage
```
\label{eq:pruned-tensor}
  \widehat A_m^{\mathrm{keep}}
  =
  \sum_{h\in\cE_m^{\mathrm{keep}}}
  \widehat A_{m,h},
  \qquad
  R_m
  =
  \widehat A_m-\widehat A_m^{\mathrm{keep}}.
```

\widehat A_m^{\fine}Use in source · 3 passages

Search aliases: widehat A_m^fine widehat A_m^ f widehat fine widehat mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 454
item Let widehat A_m^ f:X_ f^⊗ m→ X_ c be the linearization of CA_m^ f. Then Ker(C^⊗ m) ⊆ Ker(widehat A_m^ f).
Exact mathematical passage
```
\widehat A_m^{\fine}:X_{\fine}^{\otimes m}\to X_{\coarse}
```

\widehat A_m^{\mathrm{keep}}Use in source · 1 passage

Search aliases: widehat A_m^keep widehat mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2242
Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.
Exact mathematical passage
```
\label{eq:pruned-tensor}
  \widehat A_m^{\mathrm{keep}}
  =
  \sum_{h\in\cE_m^{\mathrm{keep}}}
  \widehat A_{m,h},
  \qquad
  R_m
  =
  \widehat A_m-\widehat A_m^{\mathrm{keep}}.
```

\widehat\muUse in source · 12 passages

Search aliases: widehatμ widehat mu

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1388
Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.
Exact mathematical passage
```
  \widehat\mu_k^{\fine}
  =\Phi_k^{\fine}(\mu_{k-1}^{\fine}),
  \qquad
  \widehat\mu_k^{\coarse}
  =\Phi_k^{\coarse}(\mu_{k-1}^{\coarse}),
```

\widehat\mu_k^{\coarse}Use in source · 6 passages

Search aliases: widehatμ_k^coarse widehatμ_k^ c widehat mu coarse widehat mu mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1388
Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.
Exact mathematical passage
```
  \widehat\mu_k^{\fine}
  =\Phi_k^{\fine}(\mu_{k-1}^{\fine}),
  \qquad
  \widehat\mu_k^{\coarse}
  =\Phi_k^{\coarse}(\mu_{k-1}^{\coarse}),
```

\widehat\mu_k^{\fine}Use in source · 11 passages

Search aliases: widehatμ_k^fine widehatμ_k^ f widehat mu fine widehat mu mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1388
Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.
Exact mathematical passage
```
  \widehat\mu_k^{\fine}
  =\Phi_k^{\fine}(\mu_{k-1}^{\fine}),
  \qquad
  \widehat\mu_k^{\coarse}
  =\Phi_k^{\coarse}(\mu_{k-1}^{\coarse}),
```

\widetilde CUse in source · 3 passages

Search aliases: widetilde C widetilde

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2912
Formulation on the quotient space. Let q:X_ f→ X_ f/Ker C be the canonical quotient map. Since C is surjective, there is a unique vector space isomorphism widetilde C:X_ f/Ker C→ X_ c satisfying C=widetilde Cq. The multilinear map CA_m^ f:X_ f^m→ X_ c descends if and only if its value is unchanged when any one input is altered by an element of Ker C. The induced quotient tensor is overline A_m ([x_1],…,[x_m]) =CA_m^ f(x_1,…,x_m), and A_m^ c = widetilde C overline A_m (widetilde C^-1,…,widetilde …
Exact mathematical passage
```
  \widetilde C:X_{\fine}/\Ker C\to X_{\coarse}
```

\widetilde C^{-1}Use in source · 1 passage

Search aliases: widetilde C^-1 widetilde

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2926
que vector space isomorphism widetilde C:X_ f/Ker C→ X_ c satisfying C=widetilde Cq. The multilinear map CA_m^ f:X_ f^m→ X_ c descends if and only if its value is unchanged when any one input is altered by an element of Ker C. The induced quotient tensor is overline A_m ([x_1],…,[x_m]) =CA_m^ f(x_1,…,x_m), and A_m^ c = widetilde C overline A_m (widetilde C^-1,…,widetilde C^-1). This formulation shows that [source reference] is independent of coordinates and of the chosen right inverse.
Exact mathematical passage
```
  A_m^{\coarse}
  =
  \widetilde C\,
  \overline A_m
  (\widetilde C^{-1},\ldots,\widetilde C^{-1}).
```

X^{\otimes m}Use in source · 1 passage

Search aliases: X^⊗ m otimes

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2211
Hilbert Schmidt tensor blocks. Let X and Y be finite dimensional Hilbert spaces with orthogonal packet projectors (P_j) and (Q_i). Identify a multilinear Hilbert Schmidt tensor of order m with the Hilbert Schmidt linear map widehat A:X^⊗ m→ Y. Its packet block is widehat A_imid j_1,…,j_m = Q_iwidehat A(P_j_1⊗⋯⊗ P_j_m).
Exact mathematical passage
```
  \widehat A:X^{\otimes m}\to Y.
```

x^{\otimes m}Use in source · 7 passages

Search aliases: x^⊗ m otimes

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 790
Descent of polynomial and analytic vector fields. Let F_ f(x) = ∑_m=0^∞A_m^ f[x^⊗ m] be locally analytic, where A_m^ f is a symmetric multilinear map of order m for m≥1 and A_0^ f∈ X_ f. Throughout, A_m[x^⊗ m] =A_m(x,…,x).
Exact mathematical passage
```
\label{eq:fine-analytic-field}
  F_{\fine}(x)
  =
  \sum_{m=0}^{\infty}A_m^{\fine}[x^{\otimes m}]
```

x_1'Use in source · 1 passage

Search aliases: x_1'

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 448
item The map CA_m^ f is constant on products of fibers: if Cx_r=Cx_r' for all r, then CA_m^ f(x_1,…,x_m) = CA_m^ f(x_1',…,x_m').
Exact mathematical passage
```
\label{eq:fiber-constancy-abstract}
  CA_m^{\fine}(x_1,\ldots,x_m)
  =
  CA_m^{\fine}(x_1',\ldots,x_m').
```

X_2Use in source · 1 passage

Search aliases: X_2

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 709
Composition across refinement levels. Let X_2xrightarrowC_21X_1xrightarrowC_10X_0, C_20=C_10C_21.
Exact mathematical passage
```
  X_2\xrightarrow{C_{21}}X_1\xrightarrow{C_{10}}X_0,
  \qquad
  C_{20}=C_{10}C_{21}.
```

x_2Use in source · 8 passages

Search aliases: x_2

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2522
A nonlinear fine system with four states and a quotient with two states. Let x=(x_1,x_2,x_3,x_4)^T∈Δ_4, C= 1 · 1 · 0 · 0 0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
Exact mathematical passage
```
  x=(x_1,x_2,x_3,x_4)^T\in\Delta_4,
  \qquad
  C=
  \begin{bmatrix}
  1&1&0&0\\
  0&0&1&1
  \end{bmatrix},
```

x_3Use in source · 7 passages

Search aliases: x_3

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2522
A nonlinear fine system with four states and a quotient with two states. Let x=(x_1,x_2,x_3,x_4)^T∈Δ_4, C= 1 · 1 · 0 · 0 0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
Exact mathematical passage
```
  x=(x_1,x_2,x_3,x_4)^T\in\Delta_4,
  \qquad
  C=
  \begin{bmatrix}
  1&1&0&0\\
  0&0&1&1
  \end{bmatrix},
```

x_4Use in source · 7 passages

Search aliases: x_4

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2522
A nonlinear fine system with four states and a quotient with two states. Let x=(x_1,x_2,x_3,x_4)^T∈Δ_4, C= 1 · 1 · 0 · 0 0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
Exact mathematical passage
```
  x=(x_1,x_2,x_3,x_4)^T\in\Delta_4,
  \qquad
  C=
  \begin{bmatrix}
  1&1&0&0\\
  0&0&1&1
  \end{bmatrix},
```

X_cUse in source · 1 passage

Search aliases: X_c

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Use in source · README.md, line 19
For a surjective coarse map C:X_f→ X_c, the paper studies
Exact mathematical passage
```
C:X_f\to X_c
```

X_fUse in source · 1 passage

Search aliases: X_f

Link to this entry ·

Use in source · README.md, line 19
For a surjective coarse map C:X_f→ X_c, the paper studies
Exact mathematical passage
```
C:X_f\to X_c
```

X_{\fine}^{\otimes m}Use in source · 2 passages

Search aliases: X_fine^⊗ m X_ f^⊗ m fine otimes mathrm otimes

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 211
[label=Carabic*.,leftmargin=3.5em] item A quotient criterion for every interaction order. For a surjective map C, a multilinear fine interaction descends exactly when CA_m^ f vanishes whenever one input lies in Ker C. Equivalently, its linearization on X_ f^⊗ m factors through C^⊗ m.
Exact mathematical passage
```
X_{\fine}^{\otimes m}
```

X_{\fine}^{\otimes(m-r)}Use in source · 1 passage

Search aliases: X_fine^⊗(m-r) X_ f^⊗(m-r) fine otimes mathrm otimes

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 491
To compare (ii) and (iv), identify a multilinear map of order m with its linearization on the algebraic tensor product. Since C is surjective, Ker(C^⊗ m) = ∑_r=1^m X_ f^⊗(r-1)⊗Ker C⊗ X_ f^⊗(m-r). Choose a splitting X_ f=E(X_ c)⊕Ker C and expand the tensor power. Vanishing of widehat A_m^ f on this sum is exactly the slot condition. Thus (ii) and (iv) are equivalent. Formula [source reference] follows by choosing x_r=Ez_r.
Exact mathematical passage
```
  \Ker(C^{\otimes m})
  =
  \sum_{r=1}^m
  X_{\fine}^{\otimes(r-1)}\otimes\Ker C\otimes
  X_{\fine}^{\otimes(m-r)}.
```

X_{\fine}^{\otimes(r-1)}Use in source · 1 passage

Search aliases: X_fine^⊗(r-1) X_ f^⊗(r-1) fine otimes mathrm otimes

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 491
To compare (ii) and (iv), identify a multilinear map of order m with its linearization on the algebraic tensor product. Since C is surjective, Ker(C^⊗ m) = ∑_r=1^m X_ f^⊗(r-1)⊗Ker C⊗ X_ f^⊗(m-r). Choose a splitting X_ f=E(X_ c)⊕Ker C and expand the tensor power. Vanishing of widehat A_m^ f on this sum is exactly the slot condition. Thus (ii) and (iv) are equivalent. Formula [source reference] follows by choosing x_r=Ez_r.
Exact mathematical passage
```
  \Ker(C^{\otimes m})
  =
  \sum_{r=1}^m
  X_{\fine}^{\otimes(r-1)}\otimes\Ker C\otimes
  X_{\fine}^{\otimes(m-r)}.
```

x_i^{\alpha_{ir}}Use in source · 1 passage

Search aliases: x_i^α_ir alpha

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2732
Interpretation for reaction networks. Consider a polynomial reaction system dot x=S v(x), v_r(x)=k_r∏_i x_i^α_ir. Each reaction monomial contributes one interaction hyperedge to every species changed by the corresponding stoichiometric column. The inputs are the reactants. A block sum aggregation is exact precisely when the monomial coefficients, after summation over each output fiber, are constant on the product fibers specified by [source reference]. This gives a coefficient test at every inte…
Exact mathematical passage
```
  \dot x=S v(x),
  \qquad
  v_r(x)=k_r\prod_i x_i^{\alpha_{ir}}.
```

x_m'Use in source · 1 passage

Search aliases: x_m'

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 448
item The map CA_m^ f is constant on products of fibers: if Cx_r=Cx_r' for all r, then CA_m^ f(x_1,…,x_m) = CA_m^ f(x_1',…,x_m').
Exact mathematical passage
```
\label{eq:fiber-constancy-abstract}
  CA_m^{\fine}(x_1,\ldots,x_m)
  =
  CA_m^{\fine}(x_1',\ldots,x_m').
```

x_rUse in source · 10 passages

Search aliases: x_r

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 403
Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.
Exact mathematical passage
```
  \norm{A}
  =
  \sup_{\norm{x_r}\leq1}
  \norm{A(x_1,\ldots,x_m)}.
```

x_r'Use in source · 3 passages

Search aliases: x_r'

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 447
item The map CA_m^ f is constant on products of fibers: if Cx_r=Cx_r' for all r, then CA_m^ f(x_1,…,x_m) = CA_m^ f(x_1',…,x_m').
Exact mathematical passage
```
Cx_r=Cx_r'
```

x_{r+1}Use in source · 1 passage

Search aliases: x_r+1

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 442
item For every slot r∈{1,…,m}, CA_m^ f(x_1,…,x_r-1,k,x_r+1,…,x_m)=0 whenever k∈Ker C.
Exact mathematical passage
```
\label{eq:slot-kernel-condition}
  CA_m^{\fine}(x_1,\ldots,x_{r-1},k,x_{r+1},\ldots,x_m)=0
```

x_{r-1}Use in source · 1 passage

Search aliases: x_r-1

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 442
item For every slot r∈{1,…,m}, CA_m^ f(x_1,…,x_r-1,k,x_r+1,…,x_m)=0 whenever k∈Ker C.
Exact mathematical passage
```
\label{eq:slot-kernel-condition}
  CA_m^{\fine}(x_1,\ldots,x_{r-1},k,x_{r+1},\ldots,x_m)=0
```

yUse in source · 28 passages

Search aliases: y

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 503
[Nonlinear congruence interpretation] For a general map F:X_ f→ X_ f, descent through C is equivalent to Cx=Cy Longrightarrow CF(x)=CF(y). Thus the relation x∼_Cy, defined by Cx=Cy, must be a congruence for the coarse output of F. [source reference] turns that statement into a condition on each multilinear coefficient.
Exact mathematical passage
```
  Cx=Cy
  \quad\Longrightarrow\quad
  CF(x)=CF(y).
```

y_1Use in source · 2 passages

Search aliases: y_1

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1316
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
y_1,\ldots,y_N
```

y_{1:N}Use in source · 1 passage

Search aliases: y_1:N

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1461
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f )
Exact mathematical passage
```
y_{1:N}
```

y_1^SUse in source · 1 passage

Search aliases: y_1^S

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1565
[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,
Exact mathematical passage
```
\label{eq:hidden-slot-sum}
  \boxed{
  \Delta_m^E(x_1,\ldots,x_m)
  =
  \sum_{\varnothing\neq S\subseteq\{1,\ldots,m\}}
  CA_m^{\fine}(y_1^S,\ldots,y_m^S),}
```

y_kUse in source · 6 passages

Search aliases: y_k

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1317
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Bup_k^{\fine}=\Bup_{\lambda_{y_k}^{\fine}},
  \qquad
  \Bup_k^{\coarse}=\Bup_{\lambda_{y_k}^{\coarse}}.
```

y_m^SUse in source · 1 passage

Search aliases: y_m^S

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1565
[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,
Exact mathematical passage
```
\label{eq:hidden-slot-sum}
  \boxed{
  \Delta_m^E(x_1,\ldots,x_m)
  =
  \sum_{\varnothing\neq S\subseteq\{1,\ldots,m\}}
  CA_m^{\fine}(y_1^S,\ldots,y_m^S),}
```

y_NUse in source · 2 passages

Search aliases: y_N

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1316
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
y_1,\ldots,y_N
```

y_r^SUse in source · 1 passage

Search aliases: y_r^S

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1573
ensor is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S, P_Cx_r, · r∉ S.
Exact mathematical passage
```
  y_r^S=
  \begin{cases}
  Q_Cx_r,&r\in S,\\
  P_Cx_r,&r\notin S.
  \end{cases}
```

ZUse in source · 10 passages

Search aliases: Z

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1183
Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.
Exact mathematical passage
```
  Z_{\lambda}(\mu)=\lambda^T\mu.
```

z^{\otimes m}Use in source · 3 passages

Search aliases: z^⊗ m otimes

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 812
nt series converges normally on every smaller ball, ∑_m=0^∞normA_m^ fr^m<∞ (0≤ r<R), and, for every m≥0, there is a coarse coefficient A_m^ c satisfying CA_m^ f=A_m^ cC^× m. Then F_ c(z) = ∑_m=0^∞A_m^ c[z^⊗ m] converges absolutely whenever normEnormz<R, and CF_ f(x)=F_ c(Cx) for every x in the fine convergence domain whose coarse image lies in the coarse domain.
Exact mathematical passage
```
\label{eq:coarse-analytic-field}
  F_{\coarse}(z)
  =
  \sum_{m=0}^{\infty}A_m^{\coarse}[z^{\otimes m}]
```

z_*Use in source · 8 passages

Search aliases: z_*

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1873
refinement-defect Let z_ f=Z_λ^ f(μ), z_ c=sZ_λ^ c(Cμ). Then norm C B_λ^ f(μ) - B_λ^ c(Cμ) _1 ≤ frac2∑_iμ_iabsr_λ,i max{z_ f,z_ c}. If both evidences are at least z_*>0, then d_ B ≤ frac2normr_λ_∞z_*.
Exact mathematical passage
```
z_*>0
```

z_{*,y}Use in source · 1 passage

Search aliases: z_*,y

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2966
Certificate schemas and reproducibility. A minimal interaction certificate stores L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite ledger as JSON files and CSV tables. A certificate records a computation under specified inputs. The ledger provides cryptographic authenticity only …
Exact mathematical passage
```
\label{eq:bayes-certificate-schema}
  \mathsf B_y
  =
  (\lambda_y^{\fine},\lambda_y^{\coarse},s_y,r_y,
  z_{*,y},d_{\Bup,y}).
```

z_1Use in source · 6 passages

Search aliases: z_1

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 410
Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.
Exact mathematical passage
```
\label{eq:coarse-tensor-bound-preview}
  A_m^{\coarse}(z_1,\ldots,z_m)
  =
  CA_m^{\fine}(Ez_1,\ldots,Ez_m),
```

Z_{1:N}^{\coarse}Use in source · 4 passages

Search aliases: Z_1:N^coarse Z_1:N^ c coarse mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1404
Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.
Exact mathematical passage
```
  Z_{1:N}^{\fine}=\prod_{k=1}^N z_k^{\fine},
  \qquad
  Z_{1:N}^{\coarse}=\prod_{k=1}^N z_k^{\coarse}.
```

Z_{1:N}^{\fine}Use in source · 4 passages

Search aliases: Z_1:N^fine Z_1:N^ f fine mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1404
Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.
Exact mathematical passage
```
  Z_{1:N}^{\fine}=\prod_{k=1}^N z_k^{\fine},
  \qquad
  Z_{1:N}^{\coarse}=\prod_{k=1}^N z_k^{\coarse}.
```

z_2Use in source · 2 passages

Search aliases: z_2

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2532
0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
Exact mathematical passage
```
  z=Cx=(z_1,z_2)^T,
  \qquad
  z_1=x_1+x_2,
  \quad
  z_2=x_3+x_4.
```

z_{\coarse}Use in source · 2 passages

Search aliases: z_coarse z_ c coarse mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1856
[Bayesian refinement defect] Let z_ f=Z_λ^ f(μ), z_ c=sZ_λ^ c(Cμ). Then norm C B_λ^ f(μ) - B_λ^ c(Cμ) _1 ≤ frac2∑_iμ_iabsr_λ,i max{z_ f,z_ c}. If both evidences are at least z_*>0, then d_ B ≤ frac2normr_λ_∞z_*.
Exact mathematical passage
```
  z_{\fine}=Z_{\lambda^{\fine}}(\mu),
  \qquad
  z_{\coarse}=sZ_{\lambda^{\coarse}}(C\mu).
```

z_{\fine}Use in source · 2 passages

Search aliases: z_fine z_ f fine mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1856
[Bayesian refinement defect] Let z_ f=Z_λ^ f(μ), z_ c=sZ_λ^ c(Cμ). Then norm C B_λ^ f(μ) - B_λ^ c(Cμ) _1 ≤ frac2∑_iμ_iabsr_λ,i max{z_ f,z_ c}. If both evidences are at least z_*>0, then d_ B ≤ frac2normr_λ_∞z_*.
Exact mathematical passage
```
  z_{\fine}=Z_{\lambda^{\fine}}(\mu),
  \qquad
  z_{\coarse}=sZ_{\lambda^{\coarse}}(C\mu).
```

z_k^{\coarse}Use in source · 9 passages

Search aliases: z_k^coarse z_k^ c coarse mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1396
Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.
Exact mathematical passage
```
  z_k^{\fine}
  =(\lambda_{y_k}^{\fine})^T\widehat\mu_k^{\fine},
  \qquad
  z_k^{\coarse}
  =(\lambda_{y_k}^{\coarse})^T\widehat\mu_k^{\coarse}.
```

z_k^{\fine}Use in source · 9 passages

Search aliases: z_k^fine z_k^ f fine mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1396
Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.
Exact mathematical passage
```
  z_k^{\fine}
  =(\lambda_{y_k}^{\fine})^T\widehat\mu_k^{\fine},
  \qquad
  z_k^{\coarse}
  =(\lambda_{y_k}^{\coarse})^T\widehat\mu_k^{\coarse}.
```

z_mUse in source · 3 passages

Search aliases: z_m

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 410
Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.
Exact mathematical passage
```
\label{eq:coarse-tensor-bound-preview}
  A_m^{\coarse}(z_1,\ldots,z_m)
  =
  CA_m^{\fine}(Ez_1,\ldots,Ez_m),
```

z_rUse in source · 3 passages

Search aliases: z_r

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 482
Assume (iii). For z_r∈ X_ c choose arbitrary lifts x_r with Cx_r=z_r and set A_m^ c(z_1,…,z_m) = CA_m^ f(x_1,…,x_m). Fiber constancy makes the definition independent of the lifts. Multilinearity is inherited from A_m^ f, and the defining identity gives (i). Surjectivity of C also gives uniqueness: every coarse input tuple has a fine lift.
Exact mathematical passage
```
z_r\in X_{\coarse}
```

z_tUse in source · 1 passage

Search aliases: z_t

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2839
Posterior distributions over resolutions. Instead of selecting one C, place a posterior on a finite family of candidate resolutions: p(Cmid D) propto p(Dmid C)p(C). Predictions then average both parameter and topology uncertainty: p(z_tmid D) = ∑_Cp(z_tmid C,D)p(Cmid D). A fuller model should distinguish uncertainty in the coefficients of a fixed hypergraph from uncertainty about the hypergraph and the quotient map themselves.
Exact mathematical passage
```
  p(z_t\mid D)
  =
  \sum_Cp(z_t\mid C,D)p(C\mid D).
```

Z_{\lambda}Use in source · 3 passages

Search aliases: Z_λ lambda

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1183
Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.
Exact mathematical passage
```
  Z_{\lambda}(\mu)=\lambda^T\mu.
```

Z_{\lambda^{\coarse}}Use in source · 2 passages

Search aliases: Z_λ^coarse Z_λ^ c lambda coarse lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1209
ent] Let λ^ f∈ R_>0^n and λ^ c∈ R_>0^q. If λ^ f=sC^Tλ^ c for some s>0, then C B_λ^ f = B_λ^ cC. Moreover, Z_λ^ f(μ) = sZ_λ^ c(Cμ).
Exact mathematical passage
```
\label{eq:evidence-scaling}
  Z_{\lambda^{\fine}}(\mu)
  =
  sZ_{\lambda^{\coarse}}(C\mu).
```

Z_{\lambda^{\fine}}Use in source · 2 passages

Search aliases: Z_λ^fine Z_λ^ f lambda fine lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1209
ent] Let λ^ f∈ R_>0^n and λ^ c∈ R_>0^q. If λ^ f=sC^Tλ^ c for some s>0, then C B_λ^ f = B_λ^ cC. Moreover, Z_λ^ f(μ) = sZ_λ^ c(Cμ).
Exact mathematical passage
```
\label{eq:evidence-scaling}
  Z_{\lambda^{\fine}}(\mu)
  =
  sZ_{\lambda^{\coarse}}(C\mu).
```

\alphaUse in source · 13 passages

Search aliases: α alpha

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2022
Combining the two inequalities gives e_k^+ ≤ α_ke_k-1^++β_k, where α_k=L_ B,kL_Φ,k, β_k=L_ B,kd_Φ,k+d_ B,k.
Exact mathematical passage
```
\label{eq:combined-step}
  e_k^+
  \leq
  \alpha_ke_{k-1}^++\beta_k,
```

\alpha^NUse in source · 2 passages

Search aliases: α^N alpha

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2067
[label=(roman*),leftmargin=2.8em] item If 0≤α<1, then e_N^+ ≤ α^Ne_0^+ + βfrac1-α^N1-α, limsup_N→∞e_N^+ ≤fracβ1-α.
Exact mathematical passage
```
\label{eq:stable-filter-bound}
  e_N^+
  \leq
  \alpha^Ne_0^+
  +
  \beta\frac{1-\alpha^N}{1-\alpha},
  \qquad
  \limsup_{N\to\infty}e_N^+
  \leq\frac{\beta}{1-\alpha}.
```

\alpha_{ir}Use in source · 1 passage

Search aliases: α_ir alpha

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2732
Interpretation for reaction networks. Consider a polynomial reaction system dot x=S v(x), v_r(x)=k_r∏_i x_i^α_ir. Each reaction monomial contributes one interaction hyperedge to every species changed by the corresponding stoichiometric column. The inputs are the reactants. A block sum aggregation is exact precisely when the monomial coefficients, after summation over each output fiber, are constant on the product fibers specified by [source reference]. This gives a coefficient test at every inte…
Exact mathematical passage
```
  \dot x=S v(x),
  \qquad
  v_r(x)=k_r\prod_i x_i^{\alpha_{ir}}.
```

\alpha_jUse in source · 2 passages

Search aliases: α_j alpha

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2038
[Certified approximate filtering] Under the assumptions above, e_N^+ ≤ (∏_j=1^Nα_j)e_0^+ + ∑_k=1^N (∏_j=k+1^Nα_j) β_k. Here and below, an empty product equals one. With exact initial aggregation e_0^+=0, e_N^+ ≤ ∑_k=1^N (∏_j=k+1^Nα_j) (L_ B,kd_Φ,k+d_ B,k).
Exact mathematical passage
```
\label{eq:posterior-certificate}
  \boxed{
  e_N^+
  \leq
  \left(\prod_{j=1}^N\alpha_j\right)e_0^+
  +
  \sum_{k=1}^N
  \left(\prod_{j=k+1}^N\alpha_j\right)
  \beta_k.}
```

\alpha_kUse in source · 3 passages

Search aliases: α_k alpha

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2022
Combining the two inequalities gives e_k^+ ≤ α_ke_k-1^++β_k, where α_k=L_ B,kL_Φ,k, β_k=L_ B,kd_Φ,k+d_ B,k.
Exact mathematical passage
```
\label{eq:combined-step}
  e_k^+
  \leq
  \alpha_ke_{k-1}^++\beta_k,
```

\betaUse in source · 7 passages

Search aliases: β beta

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2022
Combining the two inequalities gives e_k^+ ≤ α_ke_k-1^++β_k, where α_k=L_ B,kL_Φ,k, β_k=L_ B,kd_Φ,k+d_ B,k.
Exact mathematical passage
```
\label{eq:combined-step}
  e_k^+
  \leq
  \alpha_ke_{k-1}^++\beta_k,
```

\beta_kUse in source · 4 passages

Search aliases: β_k beta

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2022
Combining the two inequalities gives e_k^+ ≤ α_ke_k-1^++β_k, where α_k=L_ B,kL_Φ,k, β_k=L_ B,kd_Φ,k+d_ B,k.
Exact mathematical passage
```
\label{eq:combined-step}
  e_k^+
  \leq
  \alpha_ke_{k-1}^++\beta_k,
```

\gammaUse in source · 2 passages

Search aliases: γ gamma

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2358
A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability conditions on C. The three terms measure failure of dynamical closure, computational cost, and loss of observational information. Their weights and th…
Exact mathematical passage
```
\label{eq:budgeted-design}
  \min_{C,\{A_m^{\coarse}\},\{\cE_m^{\mathrm{keep}}\}}
  \left[
  \sum_mw_m\norm{\Delta_m^{\mathrm{total}}}
  +\lambda\sum_m\operatorname{Cost}(\cE_m^{\mathrm{keep}})
  +\gamma\sum_y d_{\Bup,y}
  \right].
```

\Gamma_1^{\coarse}Use in source · 1 passage

Search aliases: Γ_1^coarse Γ_1^ c Gamma coarse Gamma mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1331
servations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
\label{eq:filter-cocycles}
  \Psi_{1:N}^{\fine}
  =\Gamma_N^{\fine}\circ\cdots\circ\Gamma_1^{\fine},
  \qquad
  \Psi_{1:N}^{\coarse}
  =\Gamma_N^{\coarse}\circ\cdots\circ\Gamma_1^{\coarse}.
```

\Gamma_1^{\fine}Use in source · 1 passage

Search aliases: Γ_1^fine Γ_1^ f Gamma fine Gamma mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1331
servations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
\label{eq:filter-cocycles}
  \Psi_{1:N}^{\fine}
  =\Gamma_N^{\fine}\circ\cdots\circ\Gamma_1^{\fine},
  \qquad
  \Psi_{1:N}^{\coarse}
  =\Gamma_N^{\coarse}\circ\cdots\circ\Gamma_1^{\coarse}.
```

\Gamma_k^{\coarse}Use in source · 2 passages

Search aliases: Γ_k^coarse Γ_k^ c Gamma coarse Gamma mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1323
observation sequences Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Gamma_k^{\fine}
  =\Bup_k^{\fine}\circ\Phi_k^{\fine},
  \qquad
  \Gamma_k^{\coarse}
  =\Bup_k^{\coarse}\circ\Phi_k^{\coarse},
```

\Gamma_k^{\fine}Use in source · 2 passages

Search aliases: Γ_k^fine Γ_k^ f Gamma fine Gamma mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1323
observation sequences Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Gamma_k^{\fine}
  =\Bup_k^{\fine}\circ\Phi_k^{\fine},
  \qquad
  \Gamma_k^{\coarse}
  =\Bup_k^{\coarse}\circ\Phi_k^{\coarse},
```

\Gamma_N^{\coarse}Use in source · 1 passage

Search aliases: Γ_N^coarse Γ_N^ c Gamma coarse Gamma mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1331
servations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
\label{eq:filter-cocycles}
  \Psi_{1:N}^{\fine}
  =\Gamma_N^{\fine}\circ\cdots\circ\Gamma_1^{\fine},
  \qquad
  \Psi_{1:N}^{\coarse}
  =\Gamma_N^{\coarse}\circ\cdots\circ\Gamma_1^{\coarse}.
```

\Gamma_N^{\fine}Use in source · 1 passage

Search aliases: Γ_N^fine Γ_N^ f Gamma fine Gamma mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1331
servations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
\label{eq:filter-cocycles}
  \Psi_{1:N}^{\fine}
  =\Gamma_N^{\fine}\circ\cdots\circ\Gamma_1^{\fine},
  \qquad
  \Psi_{1:N}^{\coarse}
  =\Gamma_N^{\coarse}\circ\cdots\circ\Gamma_1^{\coarse}.
```

\Delta_4Use in source · 1 passage

Search aliases: Δ_4 Delta

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2522
A nonlinear fine system with four states and a quotient with two states. Let x=(x_1,x_2,x_3,x_4)^T∈Δ_4, C= 1 · 1 · 0 · 0 0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
Exact mathematical passage
```
  x=(x_1,x_2,x_3,x_4)^T\in\Delta_4,
  \qquad
  C=
  \begin{bmatrix}
  1&1&0&0\\
  0&0&1&1
  \end{bmatrix},
```

\Delta_{\coarse}Use in source · 5 passages

Search aliases: Δ_coarse Δ_ c Delta coarse Delta mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1050
Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.
Exact mathematical passage
```
  \Delta_{\fine}
  =\{\mu\in\R_+^n:\one^T\mu=1\},
  \qquad
  \Delta_{\coarse}
  =\{\nu\in\R_+^q:\one^T\nu=1\}.
```

\delta_FUse in source · 10 passages

Search aliases: δ_F delta

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1611
Defect majorants for the full field. Suppose fine and proposed coarse fields have expansions F_ f(x)=∑_m≥0A_m^ f[x^⊗ m], F_ c(z)=∑_m≥0A_m^ c[z^⊗ m]. Define the field defect δ_F(x) = CF_ f(x)-F_ c(Cx). Whenever both series converge absolutely, δ_F(x) = ∑_m≥0Δ_m[x^⊗ m].
Exact mathematical passage
```
\label{eq:field-defect}
  \delta_F(x)
  =
  CF_{\fine}(x)-F_{\coarse}(Cx).
```

\Delta_{\fine}Use in source · 5 passages

Search aliases: Δ_fine Δ_ f Delta fine Delta mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1050
Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.
Exact mathematical passage
```
  \Delta_{\fine}
  =\{\mu\in\R_+^n:\one^T\mu=1\},
  \qquad
  \Delta_{\coarse}
  =\{\nu\in\R_+^q:\one^T\nu=1\}.
```

\Delta_m^{10}Use in source · 3 passages

Search aliases: Δ_m^10 Delta

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 746
[Interaction defect across two levels] Define Δ_m^21 =C_21A_m^(2)-A_m^(1)C_21^× m, Δ_m^10 =C_10A_m^(1)-A_m^(0)C_10^× m, and Δ_m^20 =C_20A_m^(2)-A_m^(0)C_20^× m. Then Δ_m^20 = C_10Δ_m^21 + Δ_m^10C_21^× m. Consequently, normΔ_m^20 ≤ normC_10normΔ_m^21 + normΔ_m^10normC_21^m.
Exact mathematical passage
```
  \Delta_m^{10}
  =C_{10}A_m^{(1)}-A_m^{(0)}C_{10}^{\times m},
```

\Delta_m^{20}Use in source · 4 passages

Search aliases: Δ_m^20 Delta

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 751
[Interaction defect across two levels] Define Δ_m^21 =C_21A_m^(2)-A_m^(1)C_21^× m, Δ_m^10 =C_10A_m^(1)-A_m^(0)C_10^× m, and Δ_m^20 =C_20A_m^(2)-A_m^(0)C_20^× m. Then Δ_m^20 = C_10Δ_m^21 + Δ_m^10C_21^× m. Consequently, normΔ_m^20 ≤ normC_10normΔ_m^21 + normΔ_m^10normC_21^m.
Exact mathematical passage
```
  \Delta_m^{20}
  =C_{20}A_m^{(2)}-A_m^{(0)}C_{20}^{\times m}.
```

\Delta_m^{21}Use in source · 3 passages

Search aliases: Δ_m^21 Delta

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 742
[Interaction defect across two levels] Define Δ_m^21 =C_21A_m^(2)-A_m^(1)C_21^× m, Δ_m^10 =C_10A_m^(1)-A_m^(0)C_10^× m, and Δ_m^20 =C_20A_m^(2)-A_m^(0)C_20^× m. Then Δ_m^20 = C_10Δ_m^21 + Δ_m^10C_21^× m. Consequently, normΔ_m^20 ≤ normC_10normΔ_m^21 + normΔ_m^10normC_21^m.
Exact mathematical passage
```
  \Delta_m^{21}
  =C_{21}A_m^{(2)}-A_m^{(1)}C_{21}^{\times m},
```

\Delta_m^EUse in source · 3 passages

Search aliases: Δ_m^E Delta

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1557
[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,
Exact mathematical passage
```
\label{eq:section-defect}
  \Delta_m^E(x_1,\ldots,x_m)
  =
  CA_m^{\fine}(x_1,\ldots,x_m)
  -
  CA_m^{\fine}(P_Cx_1,\ldots,P_Cx_m),
```

\Delta_m^{\mathrm{keep}}Use in source · 3 passages

Search aliases: Δ_m^keep Delta mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2328
Combined quotient and pruning defect. Let the full fine coefficient split as A_m^ f =A_m^keep+R_m^ f. For a proposed coarse coefficient A_m^ c, define Δ_m^total =CA_m^ f-A_m^ cC^× m. Then Δ_m^total · = underbraceCA_m^keep-A_m^ cC^× m _Δ_m^keep +CR_m^ f. Therefore normΔ_m^total ≤ normΔ_m^keep +normCnormR_m^ f.
Exact mathematical passage
```
  \Delta_m^{\mathrm{total}}
  &=
  \underbrace{CA_m^{\mathrm{keep}}-A_m^{\coarse}C^{\times m}}
  _{\Delta_m^{\mathrm{keep}}}
  +CR_m^{\fine}.
  \label{eq:combined-pruning-quotient}
```

\Delta_m^{L0}Use in source · 2 passages

Search aliases: Δ_m^L0 Delta

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1727
[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.
Exact mathematical passage
```
\label{eq:tower-defect}
  \Delta_m^{L0}
  =
  \sum_{\ell=0}^{L-1}
  C_{\ell0}\,
  \Delta_m^{\ell+1,\ell}
  C_{L,\ell+1}^{\times m},
```

\Delta_m^{\mathrm{total}}Use in source · 4 passages

Search aliases: Δ_m^total Delta mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2323
Combined quotient and pruning defect. Let the full fine coefficient split as A_m^ f =A_m^keep+R_m^ f. For a proposed coarse coefficient A_m^ c, define Δ_m^total =CA_m^ f-A_m^ cC^× m. Then Δ_m^total · = underbraceCA_m^keep-A_m^ cC^× m _Δ_m^keep +CR_m^ f. Therefore normΔ_m^total ≤ normΔ_m^keep +normCnormR_m^ f.
Exact mathematical passage
```
  \Delta_m^{\mathrm{total}}
  =CA_m^{\fine}-A_m^{\coarse}C^{\times m}.
```

\Delta_m^{\ell+1,\ell}Use in source · 2 passages

Search aliases: Δ_m^ℓ+1,ℓ Delta ell ell

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1727
[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.
Exact mathematical passage
```
\label{eq:tower-defect}
  \Delta_m^{L0}
  =
  \sum_{\ell=0}^{L-1}
  C_{\ell0}\,
  \Delta_m^{\ell+1,\ell}
  C_{L,\ell+1}^{\times m},
```

\Delta_nUse in source · 1 passage

Search aliases: Δ_n Delta

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1238
[Necessary condition for Bayesian refinement] Assume q≥2, every fiber of π is nonempty, and λ^ f,λ^ c are strictly positive. If C B_λ^ f(μ) = B_λ^ c(Cμ) for every μ in the interior of Δ_n, then there exists s>0 such that λ^ f=sC^Tλ^ c.
Exact mathematical passage
```
\Delta_n
```

\varepsilon^{\infty}_{a;\mathbf b}Use in source · 1 passage

Search aliases: ε^∞_a; b varepsilon infty mathbf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2433
athbf bw_ j . For scalar worst case loss, a minimax representative is the midrange a^ c,∞_a; b = frac12 ( max_ j∈ G_ bg_a; j + min_ j∈ G_ bg_a; j ), with optimal scalar fiber error ε^∞_a; b = frac12 ( max_ j∈ G_ bg_a; j - min_ j∈ G_ bg_a; j ). These formulas are local coefficient summaries. A global operator norm can couple several fibers and may require a joint optimization.
Exact mathematical passage
```
\label{eq:minimax-fiber-error}
  \varepsilon^{\infty}_{a;\mathbf b}
  =
  \frac12
  \left(
  \max_{\mathbf j\in\cG_{\mathbf b}}g_{a;\mathbf j}
  -
  \min_{\mathbf j\in\cG_{\mathbf b}}g_{a;\mathbf j}
  \right).
```

\varepsilon_FUse in source · 2 passages

Search aliases: ε_F varepsilon

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1670
d Let dot x=F_ f(x), dot z=F_ c(z), and suppose both solutions remain in domains on which normF_ c(u)-F_ c(v) ≤ Lnormu-v. Set e(t)=normCx(t)-z(t). Then e(t) ≤ e^Lte(0) + ∫_0^t e^L(t-s)normδ_F(x(s)) d s. If e(0)=0 and normδ_F(x(s))≤ε_F on [0,t], then e(t) ≤ displaystyle ε_Ffrace^Lt-1L, · L>0,
Exact mathematical passage
```
\norm{\delta_F(x(s))}\leq\varepsilon_F
```

\varepsilon_mUse in source · 3 passages

Search aliases: ε_m varepsilon

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1624
[Analytic defect majorant] If normΔ_m≤ε_m and the scalar series d_F(R)=∑_m≥0ε_mR^m converges, then sup_normx≤ Rnormδ_F(x) ≤ d_F(R).
Exact mathematical passage
```
\norm{\Delta_m}\leq\varepsilon_m
```

\etaUse in source · 13 passages

Search aliases: η eta

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2305
Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The quantity deliberately depends on the resolution. It evaluates a representation rather than an invariant of the unresolved tensor alone.
Exact mathematical passage
```
  p_h=\frac{e_h}{\sum_{\eta}e_{\eta}},
  \qquad
  H_m=-\sum_hp_h\log p_h,
```

\LambdaUse in source · 2 passages

Search aliases: Λ Lambda

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2937
A finite observational partition lemma. Let Y be a finite observation alphabet and let each state i have a strictly positive likelihood profile Λ(i) = (λ_y(i))_y∈ Y. Define i∼_obsj Longleftrightarrow Λ(i)=Λ(j).
Exact mathematical passage
```
  \Lambda(i)
  =
  (\lambda_y(i))_{y\in\cY}.
```

\lambda^{\coarse}Use in source · 18 passages

Search aliases: λ^coarse λ^ c lambda coarse lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 199
Classical sufficiency theory asks whether a statistic preserves all information relevant to a chosen observation family [source reference]. Here the operator question is C B_λ^ f = B_λ^ cC. When both the flow identity [source reference] and the Bayesian identity [source reference] hold at each observation time, the whole filtering sequence commutes with aggregation. Hidden Markov filtering is the linear unary case [source reference]. The same argument applies when the prediction step comes from …
Exact mathematical passage
```
\label{eq:bayes-question}
  C\Bup_{\lambda^{\fine}}
  =
  \Bup_{\lambda^{\coarse}}C.
```

\lambda^{\fine,\eta}Use in source · 2 passages

Search aliases: λ^fine,η λ^ f,η lambda fine eta lambda mathrm eta

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2660
A controlled observation violation. Perturb the fine likelihood inside one fiber: λ^ f,η = C^Tλ^ c + η(1,-1,0,0)^T. Assume absη<λ_1^ c, so every component of λ^ f,η remains strictly positive. The fiber oscillation is 2absη and normr_λ_∞=absη. If the evidence is bounded below by z_*>0, then d_ B ≤ frac2absηz_*. The sweep in [source reference] varies only the observation model and isolates this effect at one interior prior. The measured discrepancy is first order in η and remains below the likelih…
Exact mathematical passage
```
\label{eq:likelihood-perturbation-example}
  \lambda^{\fine,\eta}
  =
  C^T\lambda^{\coarse}
  +
  \eta(1,-1,0,0)^T.
```

\lambda^TUse in source · 4 passages

Search aliases: λ^T lambda

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 191
Why Bayesian observations add another nonlinear operation. Prediction is only one part of a filter. Given a probability vector μ and a positive likelihood λ, Bayesian conditioning is B_λ(μ) = fracλodotμλ^Tμ. Likelihood multiplication is linear, but the evidence in the denominator couples all packets. The full update is therefore nonlinear even when the likelihood acts separately on each state.
Exact mathematical passage
```
\label{eq:bayes-intro}
  \Bup_{\lambda}(\mu)
  =
  \frac{\lambda\odot\mu}{\lambda^T\mu}.
```

\lambda_1Use in source · 4 passages

Search aliases: λ_1 lambda

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1170
Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.
Exact mathematical passage
```
  L_{\lambda}=\diag(\lambda_1,\ldots,\lambda_n),
  \qquad
  \lambda_i>0.
```

\lambda_1^{\coarse}Use in source · 1 passage

Search aliases: λ_1^coarse λ_1^ c lambda coarse lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2667
A controlled observation violation. Perturb the fine likelihood inside one fiber: λ^ f,η = C^Tλ^ c + η(1,-1,0,0)^T. Assume absη<λ_1^ c, so every component of λ^ f,η remains strictly positive. The fiber oscillation is 2absη and normr_λ_∞=absη. If the evidence is bounded below by z_*>0, then d_ B ≤ frac2absηz_*. The sweep in [source reference] varies only the observation model and isolates this effect at one interior prior. The measured discrepancy is first order in η and remains below the likelih…
Exact mathematical passage
```
\abs{\eta}<\lambda_1^{\coarse}
```

\lambda_2Use in source · 3 passages

Search aliases: λ_2 lambda

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1187
Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.
Exact mathematical passage
```
\label{eq:bayes-composition}
  \Bup_{\lambda_2}\circ\Bup_{\lambda_1}
  =
  \Bup_{\lambda_1\odot\lambda_2}.
```

\lambda_a^{\coarse}Use in source · 4 passages

Search aliases: λ_a^coarse λ_a^ c lambda coarse lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1250
ivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets a and b. Choose i∈π^-1(a) and k∈π^-1(b), then consider a prior supported on {i,k} with masses u,v>0. On the fine side, the ratio of the posterior masses of packets a and b is fracλ_i^ fuλ_k^ fv, whereas the coarse posterior ratio is fracλ_a^ cuλ_b^ cv. Equality for all u,v>0 gives fracλ_i^ fλ_k^ f = fracλ_a^ cλ_b^ c. Take another state j in the same fiber as i. Applying the same argument to j and k g…
Exact mathematical passage
```
  \frac{\lambda_a^{\coarse}u}{\lambda_b^{\coarse}v}.
```

\lambda_b^{\coarse}Use in source · 3 passages

Search aliases: λ_b^coarse λ_b^ c lambda coarse lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1250
ivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets a and b. Choose i∈π^-1(a) and k∈π^-1(b), then consider a prior supported on {i,k} with masses u,v>0. On the fine side, the ratio of the posterior masses of packets a and b is fracλ_i^ fuλ_k^ fv, whereas the coarse posterior ratio is fracλ_a^ cuλ_b^ cv. Equality for all u,v>0 gives fracλ_i^ fλ_k^ f = fracλ_a^ cλ_b^ c. Take another state j in the same fiber as i. Applying the same argument to j and k g…
Exact mathematical passage
```
  \frac{\lambda_a^{\coarse}u}{\lambda_b^{\coarse}v}.
```

\lambda_iUse in source · 1 passage

Search aliases: λ_i lambda

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1170
Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.
Exact mathematical passage
```
  L_{\lambda}=\diag(\lambda_1,\ldots,\lambda_n),
  \qquad
  \lambda_i>0.
```

\lambda_j^{\fine}Use in source · 1 passage

Search aliases: λ_j^fine λ_j^ f lambda fine lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1259
whereas the coarse posterior ratio is fracλ_a^ cuλ_b^ cv. Equality for all u,v>0 gives fracλ_i^ fλ_k^ f = fracλ_a^ cλ_b^ c. Take another state j in the same fiber as i. Applying the same argument to j and k gives λ_j^ f=λ_i^ f. The fine likelihood is therefore constant on each fiber. Write its value on fiber a as overlineλ_a. Equation [source reference] now gives fracoverlineλ_aoverlineλ_b = fracλ_a^ cλ_b^ c for every pair a,b. Hence overlineλ=sλ^ c for one s>0, proving the result.
Exact mathematical passage
```
\lambda_j^{\fine}=\lambda_i^{\fine}
```

\lambda_k^{\coarse}Use in source · 6 passages

Search aliases: λ_k^coarse λ_k^ c lambda coarse lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2138
Sequential evidence error. For brevity, write λ_k^ f=λ_y_k^ f and λ_k^ c=λ_y_k^ c, and decompose λ_k^ f =s_kC^Tλ_k^ c+r_k. This gives z_k^ f-s_kz_k^ c · = s_k(λ_k^ c)^T (Cwidehatμ_k^ f-widehatμ_k^ c) +r_k^Twidehatμ_k^ f.
Exact mathematical passage
```
\lambda_k^{\coarse}=\lambda_{y_k}^{\coarse}
```

\lambda_k^{\fine}Use in source · 4 passages

Search aliases: λ_k^fine λ_k^ f lambda fine lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1246
es-equality-all-priors extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets a and b. Choose i∈π^-1(a) and k∈π^-1(b), then consider a prior supported on {i,k} with masses u,v>0. On the fine side, the ratio of the posterior masses of packets a and b is fracλ_i^ fuλ_k^ fv, whereas the coarse posterior ratio is fracλ_a^ cuλ_b^ cv. Equality for all u,v>0 gives fracλ_i^ fλ_k^ f = fracλ_a^ cλ…
Exact mathematical passage
```
  \frac{\lambda_i^{\fine}u}{\lambda_k^{\fine}v},
```

\lambda_nUse in source · 1 passage

Search aliases: λ_n lambda

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1170
Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.
Exact mathematical passage
```
  L_{\lambda}=\diag(\lambda_1,\ldots,\lambda_n),
  \qquad
  \lambda_i>0.
```

\lambda_yUse in source · 1 passage

Search aliases: λ_y lambda

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2937
A finite observational partition lemma. Let Y be a finite observation alphabet and let each state i have a strictly positive likelihood profile Λ(i) = (λ_y(i))_y∈ Y. Define i∼_obsj Longleftrightarrow Λ(i)=Λ(j).
Exact mathematical passage
```
  \Lambda(i)
  =
  (\lambda_y(i))_{y\in\cY}.
```

\lambda_y^{\coarse}Use in source · 5 passages

Search aliases: λ_y^coarse λ_y^ c lambda coarse lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1458
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f )
Exact mathematical passage
```
  \lambda_y^{\fine}=s_yC^T\lambda_y^{\coarse}.
```

\lambda_y^{\fine}Use in source · 5 passages

Search aliases: λ_y^fine λ_y^ f lambda fine lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1458
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f )
Exact mathematical passage
```
  \lambda_y^{\fine}=s_yC^T\lambda_y^{\coarse}.
```

\lambda_{y_1}^{\coarse}Use in source · 1 passage

Search aliases: λ_y_1^coarse λ_y_1^ c lambda coarse lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1462
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.
Exact mathematical passage
```
&C\bigl(
\Bup_{\lambda_{y_N}^{\fine}}\circ M_{\fine}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\fine}}\circ M_{\fine}
\bigr)\\
&\qquad=
\bigl(
\Bup_{\lambda_{y_N}^{\coarse}}\circ M_{\coarse}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\coarse}}\circ M_{\coarse}
\bigr)C.
```

\lambda_{y_1}^{\fine}Use in source · 1 passage

Search aliases: λ_y_1^fine λ_y_1^ f lambda fine lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1462
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.
Exact mathematical passage
```
&C\bigl(
\Bup_{\lambda_{y_N}^{\fine}}\circ M_{\fine}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\fine}}\circ M_{\fine}
\bigr)\\
&\qquad=
\bigl(
\Bup_{\lambda_{y_N}^{\coarse}}\circ M_{\coarse}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\coarse}}\circ M_{\coarse}
\bigr)C.
```

\lambda_{y_k}^{\coarse}Use in source · 5 passages

Search aliases: λ_y_k^coarse λ_y_k^ c lambda coarse lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1317
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Bup_k^{\fine}=\Bup_{\lambda_{y_k}^{\fine}},
  \qquad
  \Bup_k^{\coarse}=\Bup_{\lambda_{y_k}^{\coarse}}.
```

\lambda_{y_k}^{\fine}Use in source · 4 passages

Search aliases: λ_y_k^fine λ_y_k^ f lambda fine lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1317
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Bup_k^{\fine}=\Bup_{\lambda_{y_k}^{\fine}},
  \qquad
  \Bup_k^{\coarse}=\Bup_{\lambda_{y_k}^{\coarse}}.
```

\lambda_{y_N}^{\coarse}Use in source · 1 passage

Search aliases: λ_y_N^coarse λ_y_N^ c lambda coarse lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1462
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.
Exact mathematical passage
```
&C\bigl(
\Bup_{\lambda_{y_N}^{\fine}}\circ M_{\fine}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\fine}}\circ M_{\fine}
\bigr)\\
&\qquad=
\bigl(
\Bup_{\lambda_{y_N}^{\coarse}}\circ M_{\coarse}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\coarse}}\circ M_{\coarse}
\bigr)C.
```

\lambda_{y_N}^{\fine}Use in source · 1 passage

Search aliases: λ_y_N^fine λ_y_N^ f lambda fine lambda mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1462
[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.
Exact mathematical passage
```
&C\bigl(
\Bup_{\lambda_{y_N}^{\fine}}\circ M_{\fine}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\fine}}\circ M_{\fine}
\bigr)\\
&\qquad=
\bigl(
\Bup_{\lambda_{y_N}^{\coarse}}\circ M_{\coarse}
\circ\cdots\circ
\Bup_{\lambda_{y_1}^{\coarse}}\circ M_{\coarse}
\bigr)C.
```

\muUse in source · 47 passages

Search aliases: μ mu

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 190
Why Bayesian observations add another nonlinear operation. Prediction is only one part of a filter. Given a probability vector μ and a positive likelihood λ, Bayesian conditioning is B_λ(μ) = fracλodotμλ^Tμ. Likelihood multiplication is linear, but the evidence in the denominator couples all packets. The full update is therefore nonlinear even when the likelihood acts separately on each state.
Exact mathematical passage
```
\mu
```

\mu_0^{\coarse}Use in source · 6 passages

Search aliases: μ_0^coarse μ_0^ c mu coarse mu mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1370
[Exact posterior aggregation] If μ_0^ c=Cμ_0^ f, then Cμ_N^ f=μ_N^ c for every finite observation sequence, where μ_N^ f=Ψ_1:N^ f(μ_0^ f), μ_N^ c=Ψ_1:N^ c(μ_0^ c).
Exact mathematical passage
```
\mu_0^{\coarse}=C\mu_0^{\fine}
```

\mu_0^{\fine}Use in source · 7 passages

Search aliases: μ_0^fine μ_0^ f mu fine mu mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1370
[Exact posterior aggregation] If μ_0^ c=Cμ_0^ f, then Cμ_N^ f=μ_N^ c for every finite observation sequence, where μ_N^ f=Ψ_1:N^ f(μ_0^ f), μ_N^ c=Ψ_1:N^ c(μ_0^ c).
Exact mathematical passage
```
\mu_0^{\coarse}=C\mu_0^{\fine}
```

\mu_iUse in source · 2 passages

Search aliases: μ_i mu

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1846
Likelihood variation inside a fiber. For diagonal likelihood operators, write λ^ f = sC^Tλ^ c+r_λ. Then R_Lμ=C(r_λodotμ). Since block sum aggregation is an ℓ^1 contraction on signed vectors, normR_Lμ_1 ≤ ∑_iμ_iabsr_λ,i ≤ normr_λ_∞.
Exact mathematical passage
```
\label{eq:likelihood-residual-l1}
  \norm{R_L\mu}_1
  \leq
  \sum_i\mu_i\abs{r_{\lambda,i}}
  \leq
  \norm{r_{\lambda}}_{\infty}.
```

\mu_{k-1}^{\coarse}Use in source · 2 passages

Search aliases: μ_k-1^coarse μ_k-1^ c mu coarse mu mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1388
Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.
Exact mathematical passage
```
  \widehat\mu_k^{\fine}
  =\Phi_k^{\fine}(\mu_{k-1}^{\fine}),
  \qquad
  \widehat\mu_k^{\coarse}
  =\Phi_k^{\coarse}(\mu_{k-1}^{\coarse}),
```

\mu_{k-1}^{\fine}Use in source · 3 passages

Search aliases: μ_k-1^fine μ_k-1^ f mu fine mu mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1388
Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.
Exact mathematical passage
```
  \widehat\mu_k^{\fine}
  =\Phi_k^{\fine}(\mu_{k-1}^{\fine}),
  \qquad
  \widehat\mu_k^{\coarse}
  =\Phi_k^{\coarse}(\mu_{k-1}^{\coarse}),
```

\mu_k^{\coarse}Use in source · 3 passages

Search aliases: μ_k^coarse μ_k^ c mu coarse mu mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1959
Prediction and update errors. Let the fine recursion be widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), μ_k^ f = B_k^ f(widehatμ_k^ f), and the coarse recursion be widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), μ_k^ c = B_k^ c(widehatμ_k^ c). Define the discrepancies before and after each update by e_k^- =normCwidehatμ_k^ f-widehatμ_k^ c_1, e_k^+ =normCμ_k^ f-μ_k^ c_1. At the initial time, set e_0^+=normCμ_0^ f-μ_0^ c_1.
Exact mathematical passage
```
\label{eq:coarse-filter-recursion}
  \widehat\mu_k^{\coarse}
  =\Phi_k^{\coarse}(\mu_{k-1}^{\coarse}),
  \qquad
  \mu_k^{\coarse}
  =\Bup_k^{\coarse}(\widehat\mu_k^{\coarse}).
```

\mu_k^{\fine}Use in source · 3 passages

Search aliases: μ_k^fine μ_k^ f mu fine mu mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1951
Prediction and update errors. Let the fine recursion be widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), μ_k^ f = B_k^ f(widehatμ_k^ f), and the coarse recursion be widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), μ_k^ c = B_k^ c(widehatμ_k^ c). Define the discrepancies before and after each update by e_k^- =normCwidehatμ_k^ f-widehatμ_k^ c_1, e_k^+ =normCμ_k^ f-μ_k^ c_1. At the initial time, set e_0^+=normCμ_0^ f-μ_0^ c_1.
Exact mathematical passage
```
\label{eq:fine-filter-recursion}
  \widehat\mu_k^{\fine}
  =\Phi_k^{\fine}(\mu_{k-1}^{\fine}),
  \qquad
  \mu_k^{\fine}
  =\Bup_k^{\fine}(\widehat\mu_k^{\fine}),
```

\mu_N^{\coarse}Use in source · 4 passages

Search aliases: μ_N^coarse μ_N^ c mu coarse mu mathrm

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1371
[Exact posterior aggregation] If μ_0^ c=Cμ_0^ f, then Cμ_N^ f=μ_N^ c for every finite observation sequence, where μ_N^ f=Ψ_1:N^ f(μ_0^ f), μ_N^ c=Ψ_1:N^ c(μ_0^ c).
Exact mathematical passage
```
  \boxed{C\mu_N^{\fine}=\mu_N^{\coarse}}
```

\mu_N^{\fine}Use in source · 4 passages

Search aliases: μ_N^fine μ_N^ f mu fine mu mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1371
[Exact posterior aggregation] If μ_0^ c=Cμ_0^ f, then Cμ_N^ f=μ_N^ c for every finite observation sequence, where μ_N^ f=Ψ_1:N^ f(μ_0^ f), μ_N^ c=Ψ_1:N^ c(μ_0^ c).
Exact mathematical passage
```
  \boxed{C\mu_N^{\fine}=\mu_N^{\coarse}}
```

\nuUse in source · 8 passages

Search aliases: ν nu

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1050
Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.
Exact mathematical passage
```
  \Delta_{\fine}
  =\{\mu\in\R_+^n:\one^T\mu=1\},
  \qquad
  \Delta_{\coarse}
  =\{\nu\in\R_+^q:\one^T\nu=1\}.
```

\pi^{-1}Use in source · 2 passages

Search aliases: π^-1 pi

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1245
Both sides of [source reference] extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets a and b. Choose i∈π^-1(a) and k∈π^-1(b), then consider a prior supported on {i,k} with masses u,v>0. On the fine side, the ratio of the posterior masses of packets a and b is fracλ_i^ fuλ_k^ fv, whereas the coarse posterior ratio is fracλ_a^ cuλ_b^ cv. Equality for all u,v>0 gives fracλ_i^ fλ_k^ f = f…
Exact mathematical passage
```
i\in\pi^{-1}(a)
```

\rhoUse in source · 2 passages

Search aliases: ρ rho

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2691
[H] centering includegraphics[width=0.78linewidth]posterior_error_certificate.pdf captionJoint dynamical and likelihood perturbation sweep with ε=η=ρ. For small ρ, the final posterior discrepancy grows linearly, while the recursive certificate remains above every measured value.
Exact mathematical passage
```
\varepsilon=\eta=\rho
```

\Phi_k^{\coarse}Use in source · 9 passages

Search aliases: Φ_k^coarse Φ_k^ c Phi coarse Phi mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1311
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Phi_k^{\fine}=\Phi_{t_k-t_{k-1}}^{\fine},
  \qquad
  \Phi_k^{\coarse}=\Phi_{t_k-t_{k-1}}^{\coarse}.
```

\Phi_k^{\fine}Use in source · 8 passages

Search aliases: Φ_k^fine Φ_k^ f Phi fine Phi mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1311
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Phi_k^{\fine}=\Phi_{t_k-t_{k-1}}^{\fine},
  \qquad
  \Phi_k^{\coarse}=\Phi_{t_k-t_{k-1}}^{\coarse}.
```

\Phi_t^cUse in source · 2 passages

Search aliases: Φ_t^c Phi

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Use in source · MANUSCRIPT_STATUS.md, line 56
CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.
Exact mathematical passage
```
CA_m^f=A_m^cC^{\times m}
\Rightarrow
CF_f=F_cC
\Rightarrow
C\Phi_t^f=\Phi_t^cC.
```

\Phi_T^{\coarse}Use in source · 2 passages

Search aliases: Φ_T^coarse Φ_T^ c Phi coarse Phi mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1700
[Coefficient certificate for a flow segment] If normx(s)≤ R and Proposition~[source reference] applies, then a flow interval of length T satisfies normCΦ_T^ f(x_0)-Φ_T^ c(Cx_0) ≤ d_F(R)frace^LT-1L for L>0, with T d_F(R) at L=0.
Exact mathematical passage
```
  \norm{C\Phi_T^{\fine}(x_0)-\Phi_T^{\coarse}(Cx_0)}
  \leq
  \mathfrak d_F(R)\frac{e^{LT}-1}{L}
```

\Phi_t^fUse in source · 2 passages

Search aliases: Φ_t^f Phi

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Use in source · MANUSCRIPT_STATUS.md, line 56
CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.
Exact mathematical passage
```
CA_m^f=A_m^cC^{\times m}
\Rightarrow
CF_f=F_cC
\Rightarrow
C\Phi_t^f=\Phi_t^cC.
```

\Phi_T^{\fine}Use in source · 1 passage

Search aliases: Φ_T^fine Φ_T^ f Phi fine Phi mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1700
[Coefficient certificate for a flow segment] If normx(s)≤ R and Proposition~[source reference] applies, then a flow interval of length T satisfies normCΦ_T^ f(x_0)-Φ_T^ c(Cx_0) ≤ d_F(R)frace^LT-1L for L>0, with T d_F(R) at L=0.
Exact mathematical passage
```
  \norm{C\Phi_T^{\fine}(x_0)-\Phi_T^{\coarse}(Cx_0)}
  \leq
  \mathfrak d_F(R)\frac{e^{LT}-1}{L}
```

\Phi_T^{\fine,\varepsilon}Use in source · 1 passage

Search aliases: Φ_T^fine,ε Φ_T^ f,ε Phi fine varepsilon Phi mathrm varepsilon

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2646
1, so, on the simplex, normδ_F(x)_1 =2ε x_1 ≤2ε. The coarse Markov flow on two states is an ℓ^1 contraction. Over one interval, this gives normCΦ_T^ f,ε(x)-Φ_T^ c(Cx)_1 ≤2ε T.
Exact mathematical passage
```
  \norm{C\Phi_T^{\fine,\varepsilon}(x)-\Phi_T^{\coarse}(Cx)}_1
  \leq2\varepsilon T.
```

\Phi_{t_k-t_{k-1}}^{\coarse}Use in source · 1 passage

Search aliases: Φ_t_k-t_k-1^coarse Φ_t_k-t_k-1^ c Phi coarse Phi mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1311
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Phi_k^{\fine}=\Phi_{t_k-t_{k-1}}^{\fine},
  \qquad
  \Phi_k^{\coarse}=\Phi_{t_k-t_{k-1}}^{\coarse}.
```

\Phi_{t_k-t_{k-1}}^{\fine}Use in source · 1 passage

Search aliases: Φ_t_k-t_k-1^fine Φ_t_k-t_k-1^ f Phi fine Phi mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1311
Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
  \Phi_k^{\fine}=\Phi_{t_k-t_{k-1}}^{\fine},
  \qquad
  \Phi_k^{\coarse}=\Phi_{t_k-t_{k-1}}^{\coarse}.
```

\Psi_{1:N}^cUse in source · 2 passages

Search aliases: Ψ_1:N^c Psi

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Use in source · MANUSCRIPT_STATUS.md, line 66
CL_y^f=s_yL_y^cC ⇒ C B_y^f= B_y^cC ⇒ CΨ_1:N^f=Ψ_1:N^cC.
Exact mathematical passage
```
CL_y^f=s_yL_y^cC
\Rightarrow
C\mathsf B_y^f=\mathsf B_y^cC
\Rightarrow
C\Psi_{1:N}^f=\Psi_{1:N}^cC.
```

\Psi_{1:N}^{\coarse}Use in source · 3 passages

Search aliases: Ψ_1:N^coarse Ψ_1:N^ c Psi coarse Psi mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1331
servations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
\label{eq:filter-cocycles}
  \Psi_{1:N}^{\fine}
  =\Gamma_N^{\fine}\circ\cdots\circ\Gamma_1^{\fine},
  \qquad
  \Psi_{1:N}^{\coarse}
  =\Gamma_N^{\coarse}\circ\cdots\circ\Gamma_1^{\coarse}.
```

\Psi_{1:N}^fUse in source · 2 passages

Search aliases: Ψ_1:N^f Psi

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Use in source · MANUSCRIPT_STATUS.md, line 66
CL_y^f=s_yL_y^cC ⇒ C B_y^f= B_y^cC ⇒ CΨ_1:N^f=Ψ_1:N^cC.
Exact mathematical passage
```
CL_y^f=s_yL_y^cC
\Rightarrow
C\mathsf B_y^f=\mathsf B_y^cC
\Rightarrow
C\Psi_{1:N}^f=\Psi_{1:N}^cC.
```

\Psi_{1:N}^{\fine}Use in source · 3 passages

Search aliases: Ψ_1:N^fine Ψ_1:N^ f Psi fine Psi mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1331
servations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.
Exact mathematical passage
```
\label{eq:filter-cocycles}
  \Psi_{1:N}^{\fine}
  =\Gamma_N^{\fine}\circ\cdots\circ\Gamma_1^{\fine},
  \qquad
  \Psi_{1:N}^{\coarse}
  =\Gamma_N^{\coarse}\circ\cdots\circ\Gamma_1^{\coarse}.
```

\ell^1Use in source · 4 passages

Search aliases: ℓ^1 ell

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1845
Likelihood variation inside a fiber. For diagonal likelihood operators, write λ^ f = sC^Tλ^ c+r_λ. Then R_Lμ=C(r_λodotμ). Since block sum aggregation is an ℓ^1 contraction on signed vectors, normR_Lμ_1 ≤ ∑_iμ_iabsr_λ,i ≤ normr_λ_∞.
Exact mathematical passage
```
\ell^1
```

\ell^\inftyUse in source · 1 passage

Search aliases: ℓ^∞ ell infty

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2134
Apply the duality between ℓ^1 and ℓ^∞. The total variation identity is standard for finite probability vectors.
Exact mathematical passage
```
\ell^\infty
```

\ell_{\coarse}Use in source · 5 passages

Search aliases: ℓ_coarse ℓ_ c ell coarse ell mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1089
Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.
Exact mathematical passage
```
  \ell_{\fine}:X_{\fine}\to\R,
  \qquad
  \ell_{\coarse}:X_{\coarse}\to\R
```

\ell_{\fine}Use in source · 5 passages

Search aliases: ℓ_fine ℓ_ f ell fine ell mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1089
Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.
Exact mathematical passage
```
  \ell_{\fine}:X_{\fine}\to\R,
  \qquad
  \ell_{\coarse}:X_{\coarse}\to\R
```

\RightarrowUse in source · 2 passages

Search aliases: ⇒ Rightarrow

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Use in source · MANUSCRIPT_STATUS.md, line 56
CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.
Exact mathematical passage
```
CA_m^f=A_m^cC^{\times m}
\Rightarrow
CF_f=F_cC
\Rightarrow
C\Phi_t^f=\Phi_t^cC.
```

\existsUse in source · 1 passage

Search aliases: ∃ exists

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 182
At order m, the central question is ∃ A_m^ c such that C A_m^ f=A_m^ cC^× m? If this question has an answer at every order, the quotient relation can be carried from individual interactions to the full field, its derivatives, its interaction trees, and its flow.
Exact mathematical passage
```
\label{eq:arity-question}
  \exists A_m^{\coarse}
  \quad\text{such that}\quad
  C A_m^{\fine}=A_m^{\coarse}C^{\times m}?
```

\varnothingUse in source · 1 passage

Search aliases: ∅ varnothing

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1565
[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,
Exact mathematical passage
```
\label{eq:hidden-slot-sum}
  \boxed{
  \Delta_m^E(x_1,\ldots,x_m)
  =
  \sum_{\varnothing\neq S\subseteq\{1,\ldots,m\}}
  CA_m^{\fine}(y_1^S,\ldots,y_m^S),}
```

\notinUse in source · 2 passages

Search aliases: ∉ notin

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1573
ensor is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S, P_Cx_r, · r∉ S.
Exact mathematical passage
```
  y_r^S=
  \begin{cases}
  Q_Cx_r,&r\in S,\\
  P_Cx_r,&r\notin S.
  \end{cases}
```

\prodUse in source · 7 passages

Search aliases: ∏ prod

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1404
Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.
Exact mathematical passage
```
  Z_{1:N}^{\fine}=\prod_{k=1}^N z_k^{\fine},
  \qquad
  Z_{1:N}^{\coarse}=\prod_{k=1}^N z_k^{\coarse}.
```

\prod_iUse in source · 1 passage

Search aliases: ∏_i prod

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2732
Interpretation for reaction networks. Consider a polynomial reaction system dot x=S v(x), v_r(x)=k_r∏_i x_i^α_ir. Each reaction monomial contributes one interaction hyperedge to every species changed by the corresponding stoichiometric column. The inputs are the reactants. A block sum aggregation is exact precisely when the monomial coefficients, after summation over each output fiber, are constant on the product fibers specified by [source reference]. This gives a coefficient test at every inte…
Exact mathematical passage
```
  \dot x=S v(x),
  \qquad
  v_r(x)=k_r\prod_i x_i^{\alpha_{ir}}.
```

\prod_{j=1}^NUse in source · 1 passage

Search aliases: ∏_j=1^N prod

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2038
[Certified approximate filtering] Under the assumptions above, e_N^+ ≤ (∏_j=1^Nα_j)e_0^+ + ∑_k=1^N (∏_j=k+1^Nα_j) β_k. Here and below, an empty product equals one. With exact initial aggregation e_0^+=0, e_N^+ ≤ ∑_k=1^N (∏_j=k+1^Nα_j) (L_ B,kd_Φ,k+d_ B,k).
Exact mathematical passage
```
\label{eq:posterior-certificate}
  \boxed{
  e_N^+
  \leq
  \left(\prod_{j=1}^N\alpha_j\right)e_0^+
  +
  \sum_{k=1}^N
  \left(\prod_{j=k+1}^N\alpha_j\right)
  \beta_k.}
```

\prod_{j=k+1}^NUse in source · 2 passages

Search aliases: ∏_j=k+1^N prod

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2038
[Certified approximate filtering] Under the assumptions above, e_N^+ ≤ (∏_j=1^Nα_j)e_0^+ + ∑_k=1^N (∏_j=k+1^Nα_j) β_k. Here and below, an empty product equals one. With exact initial aggregation e_0^+=0, e_N^+ ≤ ∑_k=1^N (∏_j=k+1^Nα_j) (L_ B,kd_Φ,k+d_ B,k).
Exact mathematical passage
```
\label{eq:posterior-certificate}
  \boxed{
  e_N^+
  \leq
  \left(\prod_{j=1}^N\alpha_j\right)e_0^+
  +
  \sum_{k=1}^N
  \left(\prod_{j=k+1}^N\alpha_j\right)
  \beta_k.}
```

\prod_kUse in source · 2 passages

Search aliases: ∏_k prod

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2185
[Certificate for cumulative log evidence] Let Z_1:N^ f=∏_kz_k^ f and Z_1:N^ c=∏_kz_k^ c. Then abs log Z_1:N^ f -∑_k=1^Nlog s_k -log Z_1:N^ c ≤ ∑_k=1^N frac s_knormλ_k^ c_∞e_k^- +normr_k_∞ z_*,k.
Exact mathematical passage
```
Z_{1:N}^{\fine}=\prod_kz_k^{\fine}
```

\prod_{k=1}^NUse in source · 2 passages

Search aliases: ∏_k=1^N prod

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1404
Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.
Exact mathematical passage
```
  Z_{1:N}^{\fine}=\prod_{k=1}^N z_k^{\fine},
  \qquad
  Z_{1:N}^{\coarse}=\prod_{k=1}^N z_k^{\coarse}.
```

\sum_{\mathbf j\in\cG_{\mathbf b}}Use in source · 1 passage

Search aliases: ∑_ j∈cG_ b ∑_ j∈ G_ b sum mathbf in cG mathbf sum mathbf in mathcal mathbf

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2411
Approximate coarse coefficients. When exact fiber constancy fails, the approximation depends on a chosen loss and, when relevant, a task distribution. Under scalar squared loss with nonnegative weights w_ j on a fiber product, assume their total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ G_ bw_ j . For scalar worst case loss, a minimax representative is the midrange a^ c,∞_a; b = frac12 ( max_ j∈ G_ bg_a; j + min_ j∈ G_ bg_a; j ), with…
Exact mathematical passage
```
\label{eq:least-squares-coarse-coeff}
  a^{\coarse,\mathrm{LS}}_{a;\mathbf b}
  =
  \frac{
  \sum_{\mathbf j\in\cG_{\mathbf b}}
  w_{\mathbf j}g_{a;\mathbf j}
  }{
  \sum_{\mathbf j\in\cG_{\mathbf b}}w_{\mathbf j}
  }.
```

\sum_{a=1}^qUse in source · 1 passage

Search aliases: ∑_a=1^q sum

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 582
Applying C to [source reference] gives CA_m^ f(e_j_1,…,e_j_m) = ∑_a=1^q g^(m)_a;j_1,…,j_mf_a. If a coarse tensor exists, this vector equals A_m^ c(f_π(j_1),…,f_π(j_m)), which depends only on the coarse input labels. This proves necessity. Conversely, constancy on each fiber makes [source reference] well defined. Equality on every tuple of fine basis vectors then gives CA_m^ f=A_m^ cC^× m by multilinearity.
Exact mathematical passage
```
  CA_m^{\fine}(e_{j_1},\ldots,e_{j_m})
  =
  \sum_{a=1}^q
  g^{(m)}_{a;j_1,\ldots,j_m}f_a.
```

\sum_CUse in source · 1 passage

Search aliases: ∑_C sum

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2839
Posterior distributions over resolutions. Instead of selecting one C, place a posterior on a finite family of candidate resolutions: p(Cmid D) propto p(Dmid C)p(C). Predictions then average both parameter and topology uncertainty: p(z_tmid D) = ∑_Cp(z_tmid C,D)p(Cmid D). A fuller model should distinguish uncertainty in the coefficients of a fixed hypergraph from uncertainty about the hypergraph and the quotient map themselves.
Exact mathematical passage
```
  p(z_t\mid D)
  =
  \sum_Cp(z_t\mid C,D)p(C\mid D).
```

\sum_hUse in source · 1 passage

Search aliases: ∑_h sum

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2305
Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The quantity deliberately depends on the resolution. It evaluates a representation rather than an invariant of the unresolved tensor alone.
Exact mathematical passage
```
  p_h=\frac{e_h}{\sum_{\eta}e_{\eta}},
  \qquad
  H_m=-\sum_hp_h\log p_h,
```

\sum_{h\in\cE_m^{\mathrm{keep}}}Use in source · 1 passage

Search aliases: ∑_h∈cE_m^keep ∑_h∈ E_m^keep sum in cE mathrm sum in mathcal mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2242
Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.
Exact mathematical passage
```
\label{eq:pruned-tensor}
  \widehat A_m^{\mathrm{keep}}
  =
  \sum_{h\in\cE_m^{\mathrm{keep}}}
  \widehat A_{m,h},
  \qquad
  R_m
  =
  \widehat A_m-\widehat A_m^{\mathrm{keep}}.
```

\sum_{h\notin\cE_m^{\mathrm{keep}}}Use in source · 1 passage

Search aliases: ∑_h∉cE_m^keep ∑_h∉ E_m^keep sum notin cE mathrm sum notin mathcal mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2255
[Exact pruning residual] Under the hypotheses of [source reference], normR_m_HS^2 = ∑_h∉ E_m^keep normwidehat A_m,h_HS^2.
Exact mathematical passage
```
\label{eq:pruning-residual}
  \boxed{
  \norm{R_m}_{\mathrm{HS}}^2
  =
  \sum_{h\notin\cE_m^{\mathrm{keep}}}
  \norm{\widehat A_{m,h}}_{\mathrm{HS}}^2.}
```

\sum_iUse in source · 3 passages

Search aliases: ∑_i sum

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 319
Insert I_X=∑_jP_j in every input slot and I_Y=∑_iQ_i after applying A. Multilinearity distributes all finite sums and gives [source reference].
Exact mathematical passage
```
I_Y=\sum_iQ_i
```

\sum_{i,j_1,\ldots,j_m}Use in source · 2 passages

Search aliases: ∑_i,j_1,…,j_m sum ldots

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 309
[Exact multilinear packet reconstruction] Every multilinear map A:X^m→ Y of order m satisfies A(x_1,…,x_m) = ∑_i,j_1,…,j_m A_imid j_1,…,j_m (P_j_1x_1,…,P_j_mx_m).
Exact mathematical passage
```
\label{eq:packet-reconstruction}
  A(x_1,\ldots,x_m)
  =
  \sum_{i,j_1,\ldots,j_m}
  A_{i\mid j_1,\ldots,j_m}
  (P_{j_1}x_1,\ldots,P_{j_m}x_m).
```

\sum_{i:\pi(i)=a}Use in source · 4 passages

Search aliases: ∑_i:π(i)=a sum pi

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 551
R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.
Exact mathematical passage
```
\label{eq:aggregated-coefficient}
  g^{(m)}_{a;j_1,\ldots,j_m}
  =
  \sum_{i:\pi(i)=a}
  a^{\fine,(m)}_{i;j_1,\ldots,j_m}.
```

\sum_jUse in source · 1 passage

Search aliases: ∑_j sum

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 319
Insert I_X=∑_jP_j in every input slot and I_Y=∑_iQ_i after applying A. Multilinearity distributes all finite sums and gives [source reference].
Exact mathematical passage
```
I_X=\sum_jP_j
```

\sum_{k=1}^NUse in source · 3 passages

Search aliases: ∑_k=1^N sum

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2038
[Certified approximate filtering] Under the assumptions above, e_N^+ ≤ (∏_j=1^Nα_j)e_0^+ + ∑_k=1^N (∏_j=k+1^Nα_j) β_k. Here and below, an empty product equals one. With exact initial aggregation e_0^+=0, e_N^+ ≤ ∑_k=1^N (∏_j=k+1^Nα_j) (L_ B,kd_Φ,k+d_ B,k).
Exact mathematical passage
```
\label{eq:posterior-certificate}
  \boxed{
  e_N^+
  \leq
  \left(\prod_{j=1}^N\alpha_j\right)e_0^+
  +
  \sum_{k=1}^N
  \left(\prod_{j=k+1}^N\alpha_j\right)
  \beta_k.}
```

\sum_mUse in source · 3 passages

Search aliases: ∑_m sum

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2346
For analytic fields on a ball of radius R, d_F(R) ≤ ∑_m normΔ_m^keepR^m + normC∑_mnormR_m^ fR^m. The flow and posterior bounds use this combined quantity. A pruning decision is justified only when the omitted part remains explicit and is carried through the later estimates like any other model error.
Exact mathematical passage
```
\label{eq:combined-field-majorant}
  \mathfrak d_F(R)
  \leq
  \sum_m
  \norm{\Delta_m^{\mathrm{keep}}}R^m
  +
  \norm{C}\sum_m\norm{R_m^{\fine}}R^m.
```

\sum_{m=0}^{M}Use in source · 2 passages

Search aliases: ∑_m=0^M sum

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 856
[Coefficientwise characterization of polynomial descent] Let F_ f(x)=∑_m=0^MA_m^ f[x^⊗ m], F_ c(z)=∑_m=0^MA_m^ c[z^⊗ m], where every A_m^ f and A_m^ c is symmetric. Then CF_ f=F_ cC if and only if CA_m^ f=A_m^ cC^× m (0≤ m≤ M).
Exact mathematical passage
```
  F_{\fine}(x)=\sum_{m=0}^{M}A_m^{\fine}[x^{\otimes m}],
  \qquad
  F_{\coarse}(z)=\sum_{m=0}^{M}A_m^{\coarse}[z^{\otimes m}],
```

\sum_{m=0}^{\infty}Use in source · 3 passages

Search aliases: ∑_m=0^∞ sum infty

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 790
Descent of polynomial and analytic vector fields. Let F_ f(x) = ∑_m=0^∞A_m^ f[x^⊗ m] be locally analytic, where A_m^ f is a symmetric multilinear map of order m for m≥1 and A_0^ f∈ X_ f. Throughout, A_m[x^⊗ m] =A_m(x,…,x).
Exact mathematical passage
```
\label{eq:fine-analytic-field}
  F_{\fine}(x)
  =
  \sum_{m=0}^{\infty}A_m^{\fine}[x^{\otimes m}]
```

\sum_{m\geq0}Use in source · 6 passages

Search aliases: ∑_m≥0 sum geq

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 171
Why hypergraphs appear. Suppose a local analytic vector field has the expansion F_ f(x)=∑_m≥0A_m^ f[x,…,x]. The linear coefficient A_1^ f moves one input packet into one output packet. The quadratic coefficient A_2^ f combines two input packets before producing an output. In general, A_m^ f combines m inputs. We record this incidence by the directed hyperedge (j_1,…,j_m)⟶ i.
Exact mathematical passage
```
\label{eq:analytic-expansion-intro}
  F_{\fine}(x)=\sum_{m\geq0}A_m^{\fine}[x,\ldots,x].
```

\sum_{r=1}^mUse in source · 1 passage

Search aliases: ∑_r=1^m sum

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 491
To compare (ii) and (iv), identify a multilinear map of order m with its linearization on the algebraic tensor product. Since C is surjective, Ker(C^⊗ m) = ∑_r=1^m X_ f^⊗(r-1)⊗Ker C⊗ X_ f^⊗(m-r). Choose a splitting X_ f=E(X_ c)⊕Ker C and expand the tensor power. Vanishing of widehat A_m^ f on this sum is exactly the slot condition. Thus (ii) and (iv) are equivalent. Formula [source reference] follows by choosing x_r=Ez_r.
Exact mathematical passage
```
  \Ker(C^{\otimes m})
  =
  \sum_{r=1}^m
  X_{\fine}^{\otimes(r-1)}\otimes\Ker C\otimes
  X_{\fine}^{\otimes(m-r)}.
```

\sum_{r>k}Use in source · 1 passage

Search aliases: ∑_r>k sum

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2286
[Optimal retention of k orthogonal hyperedges] Among all retained sets of cardinality k, the set {h_1,…,h_k} minimizes the Hilbert Schmidt residual. The minimum is min_| E|=k normR_m( E)_HS^2 = ∑_r>ke_h_r.
Exact mathematical passage
```
\label{eq:best-k-residual}
  \boxed{
  \min_{|\cE|=k}
  \norm{R_m(\cE)}_{\mathrm{HS}}^2
  =
  \sum_{r>k}e_{h_r}.}
```

\sum_yUse in source · 2 passages

Search aliases: ∑_y sum

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2358
A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability conditions on C. The three terms measure failure of dynamical closure, computational cost, and loss of observational information. Their weights and th…
Exact mathematical passage
```
\label{eq:budgeted-design}
  \min_{C,\{A_m^{\coarse}\},\{\cE_m^{\mathrm{keep}}\}}
  \left[
  \sum_mw_m\norm{\Delta_m^{\mathrm{total}}}
  +\lambda\sum_m\operatorname{Cost}(\cE_m^{\mathrm{keep}})
  +\gamma\sum_y d_{\Bup,y}
  \right].
```

\sum_{\eta}Use in source · 1 passage

Search aliases: ∑_η sum eta

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2305
Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The quantity deliberately depends on the resolution. It evaluates a representation rather than an invariant of the unresolved tensor alone.
Exact mathematical passage
```
  p_h=\frac{e_h}{\sum_{\eta}e_{\eta}},
  \qquad
  H_m=-\sum_hp_h\log p_h,
```

\sum_{\ell=0}^{L-1}Use in source · 2 passages

Search aliases: ∑_ℓ=0^L-1 sum ell

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1727
[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.
Exact mathematical passage
```
\label{eq:tower-defect}
  \Delta_m^{L0}
  =
  \sum_{\ell=0}^{L-1}
  C_{\ell0}\,
  \Delta_m^{\ell+1,\ell}
  C_{L,\ell+1}^{\times m},
```

\sum_{\varnothing\neq S\subseteq\{1,\ldots,m\}}Use in source · 1 passage

Search aliases: ∑_∅≠ S⊆{1,…,m} sum varnothing neq subseteq ldots

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1565
[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,
Exact mathematical passage
```
\label{eq:hidden-slot-sum}
  \boxed{
  \Delta_m^E(x_1,\ldots,x_m)
  =
  \sum_{\varnothing\neq S\subseteq\{1,\ldots,m\}}
  CA_m^{\fine}(y_1^S,\ldots,y_m^S),}
```

\circUse in source · 7 passages

Search aliases: ∘ circ

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 982
[Compatibility with Fa a di Bruno composition] Let F_ f and F_ c be smooth maps with CF_ f=F_ cC, and let G_ f and G_ c satisfy the same relation. Then each rooted tree in the Fa a di Bruno expansion, written in packet coordinates, descends to the corresponding coarse tree. Moreover, C(G_ f∘ F_ f) =(G_ c∘ F_ c)C.
Exact mathematical passage
```
  C(G_{\fine}\circ F_{\fine})
  =(G_{\coarse}\circ F_{\coarse})C.
```

\inftyUse in source · 20 passages

Search aliases: ∞ infty

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 790
Descent of polynomial and analytic vector fields. Let F_ f(x) = ∑_m=0^∞A_m^ f[x^⊗ m] be locally analytic, where A_m^ f is a symmetric multilinear map of order m for m≥1 and A_0^ f∈ X_ f. Throughout, A_m[x^⊗ m] =A_m(x,…,x).
Exact mathematical passage
```
\label{eq:fine-analytic-field}
  F_{\fine}(x)
  =
  \sum_{m=0}^{\infty}A_m^{\fine}[x^{\otimes m}]
```

\intUse in source · 3 passages

Search aliases: ∫ int

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1662
[Approximate quotient flow bound] Let dot x=F_ f(x), dot z=F_ c(z), and suppose both solutions remain in domains on which normF_ c(u)-F_ c(v) ≤ Lnormu-v. Set e(t)=normCx(t)-z(t). Then e(t) ≤ e^Lte(0) + ∫_0^t e^L(t-s)normδ_F(x(s)) d s. If e(0)=0 and normδ_F(x(s))≤ε_F on [0,t], then e(t) ≤ displaystyle ε_Ffrace^Lt-1L, · L>0,
Exact mathematical passage
```
\label{eq:approx-flow-general}
  \boxed{
  e(t)
  \leq
  e^{Lt}e(0)
  +
  \int_0^t e^{L(t-s)}\norm{\delta_F(x(s))}\dd s.}
```

\int_0^tUse in source · 3 passages

Search aliases: ∫_0^t int

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1662
[Approximate quotient flow bound] Let dot x=F_ f(x), dot z=F_ c(z), and suppose both solutions remain in domains on which normF_ c(u)-F_ c(v) ≤ Lnormu-v. Set e(t)=normCx(t)-z(t). Then e(t) ≤ e^Lte(0) + ∫_0^t e^L(t-s)normδ_F(x(s)) d s. If e(0)=0 and normδ_F(x(s))≤ε_F on [0,t], then e(t) ≤ displaystyle ε_Ffrace^Lt-1L, · L>0,
Exact mathematical passage
```
\label{eq:approx-flow-general}
  \boxed{
  e(t)
  \leq
  e^{Lt}e(0)
  +
  \int_0^t e^{L(t-s)}\norm{\delta_F(x(s))}\dd s.}
```

\simUse in source · 3 passages

Search aliases: ∼ sim

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 508
[Nonlinear congruence interpretation] For a general map F:X_ f→ X_ f, descent through C is equivalent to Cx=Cy Longrightarrow CF(x)=CF(y). Thus the relation x∼_Cy, defined by Cx=Cy, must be a congruence for the coarse output of F. [source reference] turns that statement into a condition on each multilinear coefficient.
Exact mathematical passage
```
x\sim_Cy
```

\sim_CUse in source · 1 passage

Search aliases: ∼_C sim

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 508
[Nonlinear congruence interpretation] For a general map F:X_ f→ X_ f, descent through C is equivalent to Cx=Cy Longrightarrow CF(x)=CF(y). Thus the relation x∼_Cy, defined by Cx=Cy, must be a congruence for the coarse output of F. [source reference] turns that statement into a condition on each multilinear coefficient.
Exact mathematical passage
```
x\sim_Cy
```

\sim_{\mathrm{obs}}Use in source · 2 passages

Search aliases: ∼_obs sim mathrm

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 2943
A finite observational partition lemma. Let Y be a finite observation alphabet and let each state i have a strictly positive likelihood profile Λ(i) = (λ_y(i))_y∈ Y. Define i∼_obsj Longleftrightarrow Λ(i)=Λ(j).
Exact mathematical passage
```
  i\sim_{\mathrm{obs}}j
  \quad\Longleftrightarrow\quad
  \Lambda(i)=\Lambda(j).
```

\neqUse in source · 1 passage

Search aliases: ≠ neq

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1565
[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,
Exact mathematical passage
```
\label{eq:hidden-slot-sum}
  \boxed{
  \Delta_m^E(x_1,\ldots,x_m)
  =
  \sum_{\varnothing\neq S\subseteq\{1,\ldots,m\}}
  CA_m^{\fine}(y_1^S,\ldots,y_m^S),}
```

\leqUse in source · 86 passages

Search aliases: ≤ leq

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Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 403
Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.
Exact mathematical passage
```
  \norm{A}
  =
  \sup_{\norm{x_r}\leq1}
  \norm{A(x_1,\ldots,x_m)}.
```

\oplusUse in source · 1 passage

Search aliases: ⊕ oplus

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 498
To compare (ii) and (iv), identify a multilinear map of order m with its linearization on the algebraic tensor product. Since C is surjective, Ker(C^⊗ m) = ∑_r=1^m X_ f^⊗(r-1)⊗Ker C⊗ X_ f^⊗(m-r). Choose a splitting X_ f=E(X_ c)⊕Ker C and expand the tensor power. Vanishing of widehat A_m^ f on this sum is exactly the slot condition. Thus (ii) and (iv) are equivalent. Formula [source reference] follows by choosing x_r=Ez_r.
Exact mathematical passage
```
X_{\fine}=E(X_{\coarse})\oplus\Ker C
```

\longmapstoUse in source · 1 passage

Search aliases: ⟼ longmapsto

Link to this entry ·

Use in source · Hypergraph_Lumpability_Bayesian_Refinement.tex, line 1533
The map B_m⟼ B_mC^× m is injective because C is surjective. Its range is a finite dimensional linear subspace of L^m(X_ f;X_ c), hence it is closed. In a finite dimensional normed space, the distance to a nonempty closed subspace attains its minimum. A nearest point in this range therefore exists. Any such point has a unique preimage under the displayed map, and that preimage is a minimizing coarse tensor.
Exact mathematical passage
```
  B_m\longmapsto B_mC^{\times m}
```

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