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Paper VI: Transfer Semigroups: Stochastic, Bayesian, Fractional, and PDE Realizations
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1094 symbol spellings in 2618 mathematical passages across 45 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.
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+Definition passage · 210 passages
Search aliases: +
Definition passage: Relative information · Read passage · Paper_VI_Traditional_Final.tex, line 3643
[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.
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+\infty
=Definition passage · 935 passages
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Definition passage: Transfer-semigroup seed system · Read passage · Paper_VI_Traditional_Final.tex, line 326
[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.
Exact mathematical passage
\mathfrak S=(X,s,A,\{P_\alpha\}_{\alpha\in I},\tau),
ADefinition passage · 295 passages
Search aliases: A
Definition passage: Transfer-semigroup seed system · Read passage · Paper_VI_Traditional_Final.tex, line 326
[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.
Exact mathematical passage
\mathfrak S=(X,s,A,\{P_\alpha\}_{\alpha\in I},\tau),
BDefinition passage · 133 passages
Search aliases: B
Definition passage: Packet blocks · Read passage · Paper_VI_Traditional_Final.tex, line 2062
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
B\in\mathcal B(X)
\mathcal BDefinition passage · 16 passages
Search aliases: B mathcal
Definition passage: Packet blocks · Read passage · Paper_VI_Traditional_Final.tex, line 2062
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
B\in\mathcal B(X)
B_{\beta\alpha}Definition passage · 3 passages
Search aliases: B_βα beta alpha
Definition passage: Packet blocks · Read passage · Paper_VI_Traditional_Final.tex, line 2063
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
B_{\beta\alpha}=P_\beta BP_\alpha.
CDefinition passage · 236 passages
Search aliases: C
Definition passage: Reconstructed fine evolution · Read passage · Paper_VI_Traditional_Final.tex, line 2536
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
U_{m,n}(t)=C_{m,n}T_n(t)R_{n,m}.
C_jDefinition passage · 12 passages
Search aliases: C_j
Definition passage: Projective limit space · Read passage · Paper_VI_Traditional_Final.tex, line 5291
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
(X_j,C_j)
C_{m,n}Definition passage · 8 passages
Search aliases: C_m,n
Definition passage: Reconstructed fine evolution · Read passage · Paper_VI_Traditional_Final.tex, line 2536
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
U_{m,n}(t)=C_{m,n}T_n(t)R_{n,m}.
DDefinition passage · 88 passages
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Definition passage: Relative information · Read passage · Paper_VI_Traditional_Final.tex, line 3638
[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.
Exact mathematical passage
D(q\Vert p) = \sum_iq_i\log\frac{q_i}{p_i}.
\eDefinition passage · 130 passages
Search aliases: e mathrm
Definition passage: Packet blocks · Read passage · Paper_VI_Traditional_Final.tex, line 2066
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
T(t)=\e^{tA}
eDefinition passage · 155 passages
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Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3559
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
\e^{tA}Definition passage · 22 passages
Search aliases: e^tA e mathrm
Definition passage: Packet blocks · Read passage · Paper_VI_Traditional_Final.tex, line 2066
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
T(t)=\e^{tA}
hDefinition passage · 98 passages
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Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3559
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
\hboxDefinition passage · 5 passages
Search aliases: hbox
Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3559
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
IDefinition passage · 174 passages
Search aliases: I
Definition passage: Transfer-semigroup seed system · Read passage · Paper_VI_Traditional_Final.tex, line 326
[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.
Exact mathematical passage
\mathfrak S=(X,s,A,\{P_\alpha\}_{\alpha\in I},\tau),
iDefinition passage · 177 passages
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Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3559
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
I_\alphaDefinition passage · 27 passages
Search aliases: I_α alpha
Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3559
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
jDefinition passage · 144 passages
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Definition passage: Projective limit space · Read passage · Paper_VI_Traditional_Final.tex, line 5291
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
(X_j,C_j)
KDefinition passage · 120 passages
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Definition passage: Canonical lumping obstruction · Read passage · Paper_VI_Traditional_Final.tex, line 2852
[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).
Exact mathematical passage
K
\KerDefinition passage · 4 passages
Search aliases: Ker operatorname
Definition passage: Canonical lumping obstruction · Read passage · Paper_VI_Traditional_Final.tex, line 2853
[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).
Exact mathematical passage
\Ob_C(K) = (CK)|_{\Ker C} \in \mathcal B(\Ker C,\RR^m).
\logDefinition passage · 30 passages
Search aliases: log
Definition passage: Relative information · Read passage · Paper_VI_Traditional_Final.tex, line 3638
[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.
Exact mathematical passage
D(q\Vert p) = \sum_iq_i\log\frac{q_i}{p_i}.
mDefinition passage · 118 passages
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Definition passage: Reconstructed fine evolution · Read passage · Paper_VI_Traditional_Final.tex, line 2535
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
m
nDefinition passage · 170 passages
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Definition passage: Reconstructed fine evolution · Read passage · Paper_VI_Traditional_Final.tex, line 2535
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
T_n(t)
\ObDefinition passage · 3 passages
Search aliases: Ob operatorname
Definition passage: Canonical lumping obstruction · Read passage · Paper_VI_Traditional_Final.tex, line 2853
[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).
Exact mathematical passage
\Ob_C(K) = (CK)|_{\Ker C} \in \mathcal B(\Ker C,\RR^m).
\Ob_CDefinition passage · 3 passages
Search aliases: Ob_C Ob operatorname
Definition passage: Canonical lumping obstruction · Read passage · Paper_VI_Traditional_Final.tex, line 2853
[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).
Exact mathematical passage
\Ob_C(K) = (CK)|_{\Ker C} \in \mathcal B(\Ker C,\RR^m).
\overline\lambdaDefinition passage · 13 passages
Search aliases: overlineλ overline lambda
Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3559
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
\overline\lambda_\alphaDefinition passage · 12 passages
Search aliases: overlineλ_α overline lambda alpha
Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3559
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
PDefinition passage · 179 passages
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Definition passage: Transfer-semigroup seed system · Read passage · Paper_VI_Traditional_Final.tex, line 326
[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.
Exact mathematical passage
\mathfrak S=(X,s,A,\{P_\alpha\}_{\alpha\in I},\tau),
pDefinition passage · 228 passages
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Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3558
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
p
p_iDefinition passage · 48 passages
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Definition passage: Relative information · Read passage · Paper_VI_Traditional_Final.tex, line 3638
[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.
Exact mathematical passage
D(q\Vert p) = \sum_iq_i\log\frac{q_i}{p_i}.
P_\alphaDefinition passage · 32 passages
Search aliases: P_α alpha
Definition passage: Transfer-semigroup seed system · Read passage · Paper_VI_Traditional_Final.tex, line 326
[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.
Exact mathematical passage
\mathfrak S=(X,s,A,\{P_\alpha\}_{\alpha\in I},\tau),
P_\betaDefinition passage · 20 passages
Search aliases: P_β beta
Definition passage: Packet blocks · Read passage · Paper_VI_Traditional_Final.tex, line 2063
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
B_{\beta\alpha}=P_\beta BP_\alpha.
\PcalDefinition passage · 2 passages
Search aliases: Pcal P mathcal
Definition passage: Packet resolved transfer system · Read passage · Paper_VI_Traditional_Final.tex, line 2055
[Packet resolved transfer system] A packet resolved transfer system is a triple (X,A, P) consisting of a Banach state space, a bounded generator, and a finite packet system. A marked transfer system also includes an initial state or source s∈ X. The topology and all limiting operations are supplied by the ambient Banach space structure.
Exact mathematical passage
(X,A,\Pcal)
qDefinition passage · 81 passages
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Definition passage: Relative information · Read passage · Paper_VI_Traditional_Final.tex, line 3637
[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.
Exact mathematical passage
q
q_iDefinition passage · 10 passages
Search aliases: q_i
Definition passage: Relative information · Read passage · Paper_VI_Traditional_Final.tex, line 3638
[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.
Exact mathematical passage
D(q\Vert p) = \sum_iq_i\log\frac{q_i}{p_i}.
RDefinition passage · 54 passages
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Definition passage: Reconstructed fine evolution · Read passage · Paper_VI_Traditional_Final.tex, line 2536
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
U_{m,n}(t)=C_{m,n}T_n(t)R_{n,m}.
rDefinition passage · 137 passages
Search aliases: r
Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3566
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\sum_{i\in I_\alpha}r_{\alpha,i}\eta_i(p)=0.
R_{n,m}Definition passage · 11 passages
Search aliases: R_n,m
Definition passage: Reconstructed fine evolution · Read passage · Paper_VI_Traditional_Final.tex, line 2536
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
U_{m,n}(t)=C_{m,n}T_n(t)R_{n,m}.
r_{\alpha,i}Definition passage · 10 passages
Search aliases: r_α,i alpha
Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3566
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\sum_{i\in I_\alpha}r_{\alpha,i}\eta_i(p)=0.
\RRDefinition passage · 16 passages
Search aliases: RR R mathbb
Definition passage: Canonical lumping obstruction · Read passage · Paper_VI_Traditional_Final.tex, line 2853
[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).
Exact mathematical passage
\Ob_C(K) = (CK)|_{\Ker C} \in \mathcal B(\Ker C,\RR^m).
\RR^mDefinition passage · 1 passage
Search aliases: RR^m R^m RR mathbb
Definition passage: Canonical lumping obstruction · Read passage · Paper_VI_Traditional_Final.tex, line 2853
[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).
Exact mathematical passage
\Ob_C(K) = (CK)|_{\Ker C} \in \mathcal B(\Ker C,\RR^m).
\mathfrak SDefinition passage · 1 passage
Search aliases: S mathfrak
Definition passage: Transfer-semigroup seed system · Read passage · Paper_VI_Traditional_Final.tex, line 326
[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.
Exact mathematical passage
\mathfrak S=(X,s,A,\{P_\alpha\}_{\alpha\in I},\tau),
sDefinition passage · 121 passages
Search aliases: s
Definition passage: Transfer-semigroup seed system · Read passage · Paper_VI_Traditional_Final.tex, line 326
[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.
Exact mathematical passage
\mathfrak S=(X,s,A,\{P_\alpha\}_{\alpha\in I},\tau),
TDefinition passage · 202 passages
Search aliases: T
Definition passage: Packet blocks · Read passage · Paper_VI_Traditional_Final.tex, line 2066
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
T(t)=\e^{tA}
tDefinition passage · 492 passages
Search aliases: t
Definition passage: Packet blocks · Read passage · Paper_VI_Traditional_Final.tex, line 2066
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
T(t)=\e^{tA}
T_nDefinition passage · 14 passages
Search aliases: T_n
Definition passage: Reconstructed fine evolution · Read passage · Paper_VI_Traditional_Final.tex, line 2535
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
T_n(t)
T_{\beta\alpha}Definition passage · 12 passages
Search aliases: T_βα beta alpha
Definition passage: Packet blocks · Read passage · Paper_VI_Traditional_Final.tex, line 2067
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
T_{\beta\alpha}(t)=P_\beta T(t)P_\alpha.
UDefinition passage · 24 passages
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Definition passage: Reconstructed fine evolution · Read passage · Paper_VI_Traditional_Final.tex, line 2536
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
U_{m,n}(t)=C_{m,n}T_n(t)R_{n,m}.
U_{m,n}Definition passage · 8 passages
Search aliases: U_m,n
Definition passage: Reconstructed fine evolution · Read passage · Paper_VI_Traditional_Final.tex, line 2536
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
U_{m,n}(t)=C_{m,n}T_n(t)R_{n,m}.
\VertDefinition passage · 10 passages
Search aliases: Vert
Definition passage: Relative information · Read passage · Paper_VI_Traditional_Final.tex, line 3638
[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.
Exact mathematical passage
D(q\Vert p) = \sum_iq_i\log\frac{q_i}{p_i}.
wDefinition passage · 32 passages
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Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3559
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
XDefinition passage · 86 passages
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Definition passage: Transfer-semigroup seed system · Read passage · Paper_VI_Traditional_Final.tex, line 326
[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.
Exact mathematical passage
\mathfrak S=(X,s,A,\{P_\alpha\}_{\alpha\in I},\tau),
xDefinition passage · 138 passages
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Definition passage: Projective limit space · Read passage · Paper_VI_Traditional_Final.tex, line 5292
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
X_\infty = \left\{ (x_j)_{j\geq0}\in\prod_{j\geq0}X_j: C_jx_{j+1}=x_j\text{ for every }j \right\}.
X_jDefinition passage · 6 passages
Search aliases: X_j
Definition passage: Projective limit space · Read passage · Paper_VI_Traditional_Final.tex, line 5291
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
(X_j,C_j)
x_jDefinition passage · 17 passages
Search aliases: x_j
Definition passage: Projective limit space · Read passage · Paper_VI_Traditional_Final.tex, line 5292
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
X_\infty = \left\{ (x_j)_{j\geq0}\in\prod_{j\geq0}X_j: C_jx_{j+1}=x_j\text{ for every }j \right\}.
X_{j+1}Definition passage · 2 passages
Search aliases: X_j+1
Definition passage: Projective limit space · Read passage · Paper_VI_Traditional_Final.tex, line 5291
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
C_j:X_{j+1}\to X_j
x_{j+1}Definition passage · 2 passages
Search aliases: x_j+1
Definition passage: Projective limit space · Read passage · Paper_VI_Traditional_Final.tex, line 5292
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
X_\infty = \left\{ (x_j)_{j\geq0}\in\prod_{j\geq0}X_j: C_jx_{j+1}=x_j\text{ for every }j \right\}.
X_nDefinition passage · 9 passages
Search aliases: X_n
Definition passage: Reconstructed fine evolution · Read passage · Paper_VI_Traditional_Final.tex, line 2535
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
X_n
X_\inftyDefinition passage · 9 passages
Search aliases: X_∞ infty
Definition passage: Projective limit space · Read passage · Paper_VI_Traditional_Final.tex, line 5292
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
X_\infty = \left\{ (x_j)_{j\geq0}\in\prod_{j\geq0}X_j: C_jx_{j+1}=x_j\text{ for every }j \right\}.
|Definition passage · 23 passages
Search aliases: |
Definition passage: Canonical lumping obstruction · Read passage · Paper_VI_Traditional_Final.tex, line 2853
[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).
Exact mathematical passage
\Ob_C(K) = (CK)|_{\Ker C} \in \mathcal B(\Ker C,\RR^m).
|_{\Ker C}Definition passage · 1 passage
Search aliases: |_Ker C Ker operatorname
Definition passage: Canonical lumping obstruction · Read passage · Paper_VI_Traditional_Final.tex, line 2853
[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).
Exact mathematical passage
\Ob_C(K) = (CK)|_{\Ker C} \in \mathcal B(\Ker C,\RR^m).
\alphaDefinition passage · 192 passages
Search aliases: α alpha
Definition passage: Transfer-semigroup seed system · Read passage · Paper_VI_Traditional_Final.tex, line 326
[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.
Exact mathematical passage
\mathfrak S=(X,s,A,\{P_\alpha\}_{\alpha\in I},\tau),
\betaDefinition passage · 62 passages
Search aliases: β beta
Definition passage: Packet blocks · Read passage · Paper_VI_Traditional_Final.tex, line 2063
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
B_{\beta\alpha}=P_\beta BP_\alpha.
\etaDefinition passage · 35 passages
Search aliases: η eta
Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3559
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
\eta_iDefinition passage · 6 passages
Search aliases: η_i eta
Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3559
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
\lambdaDefinition passage · 156 passages
Search aliases: λ lambda
Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3559
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
\lambda_iDefinition passage · 42 passages
Search aliases: λ_i lambda
Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3559
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
\tauDefinition passage · 6 passages
Search aliases: τ tau
Definition passage: Transfer-semigroup seed system · Read passage · Paper_VI_Traditional_Final.tex, line 326
[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.
Exact mathematical passage
\mathfrak S=(X,s,A,\{P_\alpha\}_{\alpha\in I},\tau),
\toDefinition passage · 22 passages
Search aliases: → to
Definition passage: Projective limit space · Read passage · Paper_VI_Traditional_Final.tex, line 5291
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
C_j:X_{j+1}\to X_j
\inDefinition passage · 102 passages
Search aliases: ∈ in
Definition passage: Transfer-semigroup seed system · Read passage · Paper_VI_Traditional_Final.tex, line 326
[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.
Exact mathematical passage
\mathfrak S=(X,s,A,\{P_\alpha\}_{\alpha\in I},\tau),
\prodDefinition passage · 5 passages
Search aliases: ∏ prod
Definition passage: Projective limit space · Read passage · Paper_VI_Traditional_Final.tex, line 5292
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
X_\infty = \left\{ (x_j)_{j\geq0}\in\prod_{j\geq0}X_j: C_jx_{j+1}=x_j\text{ for every }j \right\}.
\prod_{j\geq0}Definition passage · 1 passage
Search aliases: ∏_j≥0 prod geq
Definition passage: Projective limit space · Read passage · Paper_VI_Traditional_Final.tex, line 5292
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
X_\infty = \left\{ (x_j)_{j\geq0}\in\prod_{j\geq0}X_j: C_jx_{j+1}=x_j\text{ for every }j \right\}.
\sumDefinition passage · 143 passages
Search aliases: ∑ sum
Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3566
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\sum_{i\in I_\alpha}r_{\alpha,i}\eta_i(p)=0.
\sum_iDefinition passage · 16 passages
Search aliases: ∑_i sum
Definition passage: Relative information · Read passage · Paper_VI_Traditional_Final.tex, line 3638
[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.
Exact mathematical passage
D(q\Vert p) = \sum_iq_i\log\frac{q_i}{p_i}.
\sum_{i\in I_\alpha}Definition passage · 8 passages
Search aliases: ∑_i∈ I_α sum in alpha
Definition passage: Within packet information loss · Read passage · Paper_VI_Traditional_Final.tex, line 3566
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\sum_{i\in I_\alpha}r_{\alpha,i}\eta_i(p)=0.
\inftyDefinition passage · 93 passages
Search aliases: ∞ infty
Definition passage: Relative information · Read passage · Paper_VI_Traditional_Final.tex, line 3643
[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.
Exact mathematical passage
+\infty
\geqDefinition passage · 60 passages
Search aliases: ≥ geq
Definition passage: Projective limit space · Read passage · Paper_VI_Traditional_Final.tex, line 5292
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
X_\infty = \left\{ (x_j)_{j\geq0}\in\prod_{j\geq0}X_j: C_jx_{j+1}=x_j\text{ for every }j \right\}.
A_cIntroduction passage · 42 passages
Search aliases: A_c
Introduction passage · atlas-pages.tex, line 570
vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONTraditional Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct multiplication gives C_rA_f=A_cC_r. For F(z)=1/(1+z), which is defined at both spectra, compute the matrices by inversion: F(A_f)=mat1/3 · -1/9 · 0
Exact mathematical passage
A_f=\mat{2&1&0\\0&2&0\\0&0&9},\qquad A_c=\mat{2&1\\0&2},\qquad C_r=\mat{1&0&0\\0&1&0}.
A_fIntroduction passage · 41 passages
Search aliases: A_f
Introduction passage · atlas-pages.tex, line 570
vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONTraditional Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct multiplication gives C_rA_f=A_cC_r. For F(z)=1/(1+z), which is defined at both spectra, compute the matrices by inversion: F(A_f)=mat1/3 · -1/9 · 0
Exact mathematical passage
A_f=\mat{2&1&0\\0&2&0\\0&0&9},\qquad A_c=\mat{2&1\\0&2},\qquad C_r=\mat{1&0&0\\0&1&0}.
A_{QQ}Introduction passage · 32 passages
Search aliases: A_QQ
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
Exact mathematical passage
B=A_{QQ}
\bar MIntroduction passage · 1 passage
Search aliases: bar M bar
Introduction passage · atlas-pages.tex, line 746
More generally, for a surjective aggregation map C, a linear map M descends exactly if CM vanishes on ker C. The descended map is defined by bar M(Cx)=CMx; the kernel condition makes this independent of the representative x.
Exact mathematical passage
\bar M(Cx)=CMx
cIntroduction passage · 97 passages
Search aliases: c
Introduction passage · atlas-pages.tex, line 570
vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONTraditional Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct multiplication gives C_rA_f=A_cC_r. For F(z)=1/(1+z), which is defined at both spectra, compute the matrices by inversion: F(A_f)=mat1/3 · -1/9 · 0
Exact mathematical passage
A_f=\mat{2&1&0\\0&2&0\\0&0&9},\qquad A_c=\mat{2&1\\0&2},\qquad C_r=\mat{1&0&0\\0&1&0}.
C_rIntroduction passage · 3 passages
Search aliases: C_r
Introduction passage · atlas-pages.tex, line 570
vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONTraditional Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct multiplication gives C_rA_f=A_cC_r. For F(z)=1/(1+z), which is defined at both spectra, compute the matrices by inversion: F(A_f)=mat1/3 · -1/9 · 0
Exact mathematical passage
A_f=\mat{2&1&0\\0&2&0\\0&0&9},\qquad A_c=\mat{2&1\\0&2},\qquad C_r=\mat{1&0&0\\0&1&0}.
\CCIntroduction passage · 2 passages
Search aliases: CC C mathbb
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3018
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
\CC^n
\CC^nIntroduction passage · 1 passage
Search aliases: CC^n C^n CC mathbb
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3018
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
\CC^n
\e^BIntroduction passage · 1 passage
Search aliases: e^B e mathrm
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
Exact mathematical passage
r(\e^B)=\e^{s(B)}<\e^\beta
\e^{nB}Introduction passage · 1 passage
Search aliases: e^nB e mathrm
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^…
Exact mathematical passage
\norm{\e^{nB}}\leq M_\beta\e^{\beta n}
\e^{s(B)}Introduction passage · 1 passage
Search aliases: e^s(B) e mathrm
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
Exact mathematical passage
r(\e^B)=\e^{s(B)}<\e^\beta
\e^\betaIntroduction passage · 1 passage
Search aliases: e^β e beta mathrm beta
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
Exact mathematical passage
r(\e^B)=\e^{s(B)}<\e^\beta
\e^{\beta n}Introduction passage · 1 passage
Search aliases: e^β n e beta mathrm beta
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^…
Exact mathematical passage
\norm{\e^{nB}}\leq M_\beta\e^{\beta n}
FIntroduction passage · 40 passages
Search aliases: F
Introduction passage · atlas-pages.tex, line 575
0 · 1 · 0. Direct multiplication gives C_rA_f=A_cC_r. For F(z)=1/(1+z), which is defined at both spectra, compute the matrices by inversion: F(A_f)=mat1/3 · -1/9 · 0
Exact mathematical passage
F(z)=1/(1+z)
fIntroduction passage · 176 passages
Search aliases: f
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3012
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
(P_\mathcal Qf)_i = \frac{\sum_{j\in B(i)}\pi_jf_j} {\sum_{j\in B(i)}\pi_j}.
f_iIntroduction passage · 4 passages
Search aliases: f_i
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3019
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
\ip{f}{g}_\pi = \sum_i\pi_if_i\overline{g_i}.
f_jIntroduction passage · 6 passages
Search aliases: f_j
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3012
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
(P_\mathcal Qf)_i = \frac{\sum_{j\in B(i)}\pi_jf_j} {\sum_{j\in B(i)}\pi_j}.
gIntroduction passage · 42 passages
Search aliases: g
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3019
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
\ip{f}{g}_\pi = \sum_i\pi_if_i\overline{g_i}.
\ipIntroduction passage · 13 passages
Search aliases: ip
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3019
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
\ip{f}{g}_\pi = \sum_i\pi_if_i\overline{g_i}.
\kerIntroduction passage · 2 passages
Search aliases: ker
Introduction passage · atlas-pages.tex, line 746
More generally, for a surjective aggregation map C, a linear map M descends exactly if CM vanishes on ker C. The descended map is defined by bar M(Cx)=CMx; the kernel condition makes this independent of the representative x.
Exact mathematical passage
\ker C
MIntroduction passage · 51 passages
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Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^…
Exact mathematical passage
\norm{\e^{nB}}\leq M_\beta\e^{\beta n}
M_\betaIntroduction passage · 3 passages
Search aliases: M_β beta
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^…
Exact mathematical passage
\norm{\e^{nB}}\leq M_\beta\e^{\beta n}
\matIntroduction passage · 17 passages
Search aliases: mat
Introduction passage · atlas-pages.tex, line 570
vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONTraditional Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct multiplication gives C_rA_f=A_cC_r. For F(z)=1/(1+z), which is defined at both spectra, compute the matrices by inversion: F(A_f)=mat1/3 · -1/9 · 0
Exact mathematical passage
A_f=\mat{2&1&0\\0&2&0\\0&0&9},\qquad A_c=\mat{2&1\\0&2},\qquad C_r=\mat{1&0&0\\0&1&0}.
\maxIntroduction passage · 18 passages
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Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
Exact mathematical passage
s(B)=\max\operatorname{Re}\Spec(B)
\normIntroduction passage · 109 passages
Search aliases: norm
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^…
Exact mathematical passage
\norm{\e^{nB}}\leq M_\beta\e^{\beta n}
\overline{g_i}Introduction passage · 2 passages
Search aliases: overlineg_i overline
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3019
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
\ip{f}{g}_\pi = \sum_i\pi_if_i\overline{g_i}.
P_\mathcalIntroduction passage · 6 passages
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Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3012
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
(P_\mathcal Qf)_i = \frac{\sum_{j\in B(i)}\pi_jf_j} {\sum_{j\in B(i)}\pi_j}.
QIntroduction passage · 136 passages
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Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3012
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
(P_\mathcal Qf)_i = \frac{\sum_{j\in B(i)}\pi_jf_j} {\sum_{j\in B(i)}\pi_j}.
\mathcal QIntroduction passage · 19 passages
Search aliases: Q mathcal
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3011
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
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\mathcal Q
\operatorname{Re}Introduction passage · 16 passages
Search aliases: Re operatorname
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
Exact mathematical passage
s(B)=\max\operatorname{Re}\Spec(B)
\SpecIntroduction passage · 3 passages
Search aliases: Spec operatorname
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
Exact mathematical passage
s(B)=\max\operatorname{Re}\Spec(B)
X_QIntroduction passage · 7 passages
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Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
Exact mathematical passage
X_Q=\{0\}
zIntroduction passage · 115 passages
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Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
Exact mathematical passage
s(B)<\beta<\operatorname{Re}z
\piIntroduction passage · 77 passages
Search aliases: π pi
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3011
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
\pi_i
\pi_iIntroduction passage · 9 passages
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Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3011
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
\pi_i
\pi_jIntroduction passage · 4 passages
Search aliases: π_j pi
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3012
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
(P_\mathcal Qf)_i = \frac{\sum_{j\in B(i)}\pi_jf_j} {\sum_{j\in B(i)}\pi_j}.
\OmegaIntroduction passage · 2 passages
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Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3011
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
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\Omega=\{1,\ldots,n\}
\ldotsIntroduction passage · 10 passages
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Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3011
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
\Omega=\{1,\ldots,n\}
\sum_{j\in B(i)}Introduction passage · 2 passages
Search aliases: ∑_j∈ B(i) sum in
Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 3012
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
(P_\mathcal Qf)_i = \frac{\sum_{j\in B(i)}\pi_jf_j} {\sum_{j\in B(i)}\pi_j}.
\leqIntroduction passage · 104 passages
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Introduction passage · Read passage · Paper_VI_Traditional_Final.tex, line 4578
ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^…
Exact mathematical passage
\norm{\e^{nB}}\leq M_\beta\e^{\beta n}
*Use in source · 4 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1120
Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.
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L=L^*\ge0
\bm1Use in source · 8 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 554
Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.
Exact mathematical passage
\bm 1^TQ=0
\bm1^TUse in source · 8 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 554
Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.
Exact mathematical passage
\bm 1^TQ=0
\mathrm AUse in source · 4 passages
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Use in source · atlas-pages.tex, line 90
Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.
Exact mathematical passage
P_b e^{(s+t)Q}P_a=\sum_{c\in\{\mathrm A,\mathrm B\}}(P_b e^{sQ}P_c)(P_c e^{tQ}P_a).
aUse in source · 127 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 839
Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero.
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I_a
A^2Use in source · 2 passages
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Use in source · atlas-pages.tex, line 669
vpage29Acyclic paths stop after finitely many transfersDIRECTED TRANSFER / A FINITE DYSON EXAMPLETraditional Part II: Theorem E2.8 and Corollary E2.9. Acyclicity concerns the transfer graph. Let three scalar packets form the directed chain 0→1→2, with transfer coefficients 2 and 3. The generator and its powers are A=mat0 · 0 · 0 2 · 0 · 0 0 · 3 · 0, A^2=mat0 · 0 · 0 0 · 0 · 0 6 · 0 · 0, A^3=0. Therefore the exponential is the exact polynomial e^tA=I+tA+fract^22A^2. vfigext_paths platecaptionOnly…
Exact mathematical passage
A=\mat{0&0&0\\2&0&0\\0&3&0},\qquad A^2=\mat{0&0&0\\0&0&0\\6&0&0},\qquad A^3=0.
A^3Use in source · 1 passage
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Use in source · atlas-pages.tex, line 669
vpage29Acyclic paths stop after finitely many transfersDIRECTED TRANSFER / A FINITE DYSON EXAMPLETraditional Part II: Theorem E2.8 and Corollary E2.9. Acyclicity concerns the transfer graph. Let three scalar packets form the directed chain 0→1→2, with transfer coefficients 2 and 3. The generator and its powers are A=mat0 · 0 · 0 2 · 0 · 0 0 · 3 · 0, A^2=mat0 · 0 · 0 0 · 0 · 0 6 · 0 · 0, A^3=0. Therefore the exponential is the exact polynomial e^tA=I+tA+fract^22A^2. vfigext_paths platecaptionOnly…
Exact mathematical passage
A=\mat{0&0&0\\2&0&0\\0&3&0},\qquad A^2=\mat{0&0&0\\0&0&0\\6&0&0},\qquad A^3=0.
A^mUse in source · 2 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 419
The hypothesis makes A block diagonal. Every power A^m is block diagonal with respect to the same projectors, and therefore so is the exponential series.
Exact mathematical passage
A^m
A^{m+n}Use in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 342
Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.
Exact mathematical passage
e^{sA}e^{tA} =\sum_{m,n\ge0}\frac{s^mt^n}{m!n!}A^{m+n} =\sum_{r\ge0}\frac{(s+t)^r}{r!}A^r.
A^rUse in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 342
Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.
Exact mathematical passage
e^{sA}e^{tA} =\sum_{m,n\ge0}\frac{s^mt^n}{m!n!}A^{m+n} =\sum_{r\ge0}\frac{(s+t)^r}{r!}A^r.
a^TUse in source · 12 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1592
Let A be a finite subgenerator: its off-diagonal entries are nonnegative and its column sums are nonpositive. These sign conditions alone allow nonabsorbing states; eventual absorption is a separate hypothesis. Let p_0 be an initial transient-state distribution. Define p(t)=e^tAp_0, S(t)=1^Tp(t). The exit-rate row vector is a^T=-1^TA≥0.
Exact mathematical passage
a^T=-\bm1^TA\ge0.
A^\alphaUse in source · 8 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1193
Let A≥0 act on a fine Hilbert space, let B≥0 act on a coarse Hilbert space, and let C be a bounded coarsening map. Set D=CA-BC, D_α=CA^α-B^α C, 0<α<1.
Exact mathematical passage
D=CA-BC, \qquad D_\alpha=CA^\alpha-B^\alpha C, \qquad 0<\alpha<1.
a_1Use in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5428
[Normalization bound] Let a,b∈[0,∞)^m be nonzero, and set A=norm a_1 and B=norm b_1. Then norma/A-b/B_1 ≤ frac2norma-b_1max{A,B}.
Exact mathematical passage
A=\norm a_1
A_jUse in source · 7 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5273
[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.
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A_j
A_{j+1}Use in source · 2 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5274
[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.
Exact mathematical passage
\delta_j=\norm{C_jA_{j+1}-A_jC_j}.
A_nUse in source · 9 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2800
[A usable closure condition for refinement towers] Assume the contractive summability hypothesis of [source reference] holds on every bounded time interval. Suppose every P_m,n is an orthogonal projection on a Hilbert space and T_n(t)= e^tA_n is contractive. If, for every fixed m, norm(I-P_m,n)A_nP_m,n ⟶0, then the limiting family U_m is a semigroup.
Exact mathematical passage
T_n(t)=\e^{tA_n}
a_nUse in source · 1 passage
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Use in source · atlas-pages.tex, line 712
vpage31A tail bound states what refinement buysSUMMABLE DEFECTS / A SCALAR COMPLETIONTraditional Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps. Take X_n= R, C_n=R_n=1, and a_n=1-2^-n, A_n=-a_n, T_n(t)=e^-a_nt (n≥0). Each semigroup is contractive for t≥0. The exact adjacent generator-defect magnitude is |A_n+1-A_n|=2^-(n+1), ∑_j=n^∞|A_j+1-A_j|=2^-n. vfigext_tails platecaptionThe bars show the certified tail 2^-n at horizon T=1. They are bounds on a whole time…
Exact mathematical passage
a_n=1-2^{-n},\qquad A_n=-a_n,\qquad T_n(t)=e^{-a_nt}\quad(n\ge0).
A_{n+1}Use in source · 1 passage
Search aliases: A_n+1
Use in source · atlas-pages.tex, line 716
vpage31A tail bound states what refinement buysSUMMABLE DEFECTS / A SCALAR COMPLETIONTraditional Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps. Take X_n= R, C_n=R_n=1, and a_n=1-2^-n, A_n=-a_n, T_n(t)=e^-a_nt (n≥0). Each semigroup is contractive for t≥0. The exact adjacent generator-defect magnitude is |A_n+1-A_n|=2^-(n+1), ∑_j=n^∞|A_j+1-A_j|=2^-n. vfigext_tails platecaptionThe bars show the certified tail 2^-n at horizon T=1. They are bounds on a whole time…
Exact mathematical passage
|A_{n+1}-A_n|=2^{-(n+1)},\qquad \sum_{j=n}^{\infty}|A_{j+1}-A_j|=2^{-n}.
A_n^{1/2}Use in source · 2 passages
Search aliases: A_n^1/2
Use in source · atlas-pages.tex, line 621
at each level is not summable across levelsREFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVETraditional Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data. For each n≥1, compare the two scalar operators A_n=0 and B_n=1/n^2 through the identity. Their ordinary defects have sizes 1/n^2, while their square-root defects have sizes 1/n: ∑_n|A_n-B_n|<∞, ∑_n|A_n^1/2-B_n^1/2|=∞. These are independent scalar comparisons, not a constructed inverse system. They…
Exact mathematical passage
\sum_n|A_n-B_n|<\infty, \qquad \sum_n|A_n^{1/2}-B_n^{1/2}|=\infty.
A_{PP}Use in source · 11 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1486
Let X=P X⊕ Q X and write A= A_PP · A_PQ A_QP · A_QQ .
Exact mathematical passage
A= \begin{pmatrix} A_{PP}&A_{PQ}\\ A_{QP}&A_{QQ} \end{pmatrix}.
A_{PQ}Use in source · 18 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1486
Let X=P X⊕ Q X and write A= A_PP · A_PQ A_QP · A_QQ .
Exact mathematical passage
A= \begin{pmatrix} A_{PP}&A_{PQ}\\ A_{QP}&A_{QQ} \end{pmatrix}.
A_{QP}Use in source · 19 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1486
Let X=P X⊕ Q X and write A= A_PP · A_PQ A_QP · A_QQ .
Exact mathematical passage
A= \begin{pmatrix} A_{PP}&A_{PQ}\\ A_{QP}&A_{QQ} \end{pmatrix}.
A_{QQ}^kUse in source · 2 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4598
sure] Assume x_Q(0)=0 and d=dim X_Q<∞. The following conditions are equivalent. item For every retained initial state, P e^tAP = e^tA_PPP for all t≥0. item Σ_P(z)=0 for every sufficiently large z. item K(t)=0 for every t≥0. item If d=dim X_Q, then A_PQA_QQ^kA_QP=0 for 0≤ k<d.
Exact mathematical passage
A_{PQ}A_{QQ}^kA_{QP}=0
a_r^TUse in source · 3 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5038
[Competing risk packets] Suppose the exit row splits into nonnegative cause rows a^T = ∑_r=1^q a_r^T. Define f_r(t)=a_r^T e^tAp_0. Then f(t)=∑_r=1^qf_r(t). The cause specific hazard is h_r(t)=fracf_r(t)S(t), and h(t)=∑_r=1^qh_r(t). If the spectrum of A lies in the open left half-plane, the eventual probability of cause r is ∫_0^∞ f_r(t) d t = a_r^T(-A)^-1p_0.
Exact mathematical passage
a^T = \sum_{r=1}^q a_r^T.
A_{\alpha\alpha}Use in source · 1 passage
Search aliases: A_αα alpha alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2189
thm:reducing_packets The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.
Exact mathematical passage
T(t)|_{P_\alpha X} = \e^{tA_{\alpha\alpha}}.
A_{\alpha\gamma}Use in source · 1 passage
Search aliases: A_αγ alpha gamma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2198
The first two conditions are equivalent because the packet resolution gives AP_α=∑_β A_βα, P_α A=∑_γ A_αγ. A block diagonal generator has block diagonal powers, so the second condition implies the third. Differentiating the off diagonal blocks at zero proves the converse. The equivalence with invariance follows from the definition of the blocks. The final identity follows by restricting the exponential series.
Exact mathematical passage
AP_\alpha=\sum_\beta A_{\beta\alpha}, \qquad P_\alpha A=\sum_\gamma A_{\alpha\gamma}.
A_{\beta\alpha}Use in source · 2 passages
Search aliases: A_βα beta alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2184
[Reducing packet criterion] The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.
Exact mathematical passage
A_{\beta\alpha}=0
A_{\beta\gamma}Use in source · 1 passage
Search aliases: A_βγ beta gamma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2092
[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.
Exact mathematical passage
\frac{\dd}{\dd t}T_{\beta\alpha}(t) = \sum_{\gamma\in I} A_{\beta\gamma}T_{\gamma\alpha}(t) = \sum_{\gamma\in I} T_{\beta\gamma}(t)A_{\gamma\alpha}.
A_{\gamma\alpha}Use in source · 1 passage
Search aliases: A_γα gamma alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2092
[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.
Exact mathematical passage
\frac{\dd}{\dd t}T_{\beta\alpha}(t) = \sum_{\gamma\in I} A_{\beta\gamma}T_{\gamma\alpha}(t) = \sum_{\gamma\in I} T_{\beta\gamma}(t)A_{\gamma\alpha}.
A_\ellUse in source · 9 passages
Search aliases: A_ℓ ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1328
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces; no operator sum across unrelated spaces…
Exact mathematical passage
D_\ell=C_\ell A_{\ell+1}-A_\ell C_\ell
A_{\ell+1}Use in source · 4 passages
Search aliases: A_ℓ+1 ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1328
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces; no operator sum across unrelated spaces…
Exact mathematical passage
D_\ell=C_\ell A_{\ell+1}-A_\ell C_\ell
A_{\ell+1}^{\alpha}Use in source · 1 passage
Search aliases: A_ℓ+1^α ell alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1333
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces; no operator sum across unrelated spaces…
Exact mathematical passage
C_\ell A_{\ell+1}^{\alpha}-A_\ell^{\alpha}C_\ell
A_\ell^{\alpha}Use in source · 1 passage
Search aliases: A_ℓ^α ell alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1333
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces; no operator sum across unrelated spaces…
Exact mathematical passage
C_\ell A_{\ell+1}^{\alpha}-A_\ell^{\alpha}C_\ell
A_\inftyUse in source · 2 passages
Search aliases: A_∞ infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2490
If T_ℓ(t)= e^tA_ℓ, the generators are bounded, and sup_ℓnormA_ℓ<∞, C_ℓ A_ℓ+1=A_ℓ C_ℓ, then the continuity condition holds and A_∞(x_ℓ)_ℓ=(A_ℓ x_ℓ)_ℓ is the bounded generator of T_∞.
Exact mathematical passage
A_\infty(x_\ell)_\ell=(A_\ell x_\ell)_\ell
\absUse in source · 16 passages
Search aliases: abs
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2388
a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that Γ lies in the common resolvent set. Thus its winding number is one about every spectral point. If g is holomorphic on a neighborhood of the closure of the union, then Cg(A_f)-g(A_c)C = frac12π i ∫_Γ g(z)R_c(z)Δ R_f(z) d z. In particular, normCg(A_f)-g(A_c)C ≤ · fraclength(Γ)2π sup_z∈Γabsg(z) sup_z∈ΓnormR_c(z) sup_z∈ΓnormR_f(z) normΔ.
Exact mathematical passage
\begin{aligned} \norm{Cg(A_f)-g(A_c)C} \leq{}& \frac{\operatorname{length}(\Gamma)}{2\pi} \sup_{z\in\Gamma}\abs{g(z)} \sup_{z\in\Gamma}\norm{R_c(z)} \sup_{z\in\Gamma}\norm{R_f(z)} \norm\Delta. \end{aligned}
\mathrm BUse in source · 4 passages
Search aliases: B mathrm
Use in source · atlas-pages.tex, line 90
Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.
Exact mathematical passage
P_b e^{(s+t)Q}P_a=\sum_{c\in\{\mathrm A,\mathrm B\}}(P_b e^{sQ}P_c)(P_c e^{tQ}P_a).
bUse in source · 41 passages
Search aliases: b
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 865
For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.
Exact mathematical passage
\bigl(C_\pi\cB_\lambda(p)\bigr)_a = \frac{\sum_{i\in I_a}\lambda_ip_i}{Z} = \frac{P_ar_a(p)}{\sum_bP_br_b(p)} = \bigl(\cB_{r(p)}(P)\bigr)_a.
B^\alphaUse in source · 7 passages
Search aliases: B^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1193
Let A≥0 act on a fine Hilbert space, let B≥0 act on a coarse Hilbert space, and let C be a bounded coarsening map. Set D=CA-BC, D_α=CA^α-B^α C, 0<α<1.
Exact mathematical passage
D=CA-BC, \qquad D_\alpha=CA^\alpha-B^\alpha C, \qquad 0<\alpha<1.
B_1Use in source · 6 passages
Search aliases: B_1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 648
Let Ω={1,…,n} carry strictly positive probabilities π_i, and let Q={B_1,…,B_m} be a partition. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j∑_j∈ B(i)π_j, where B(i) is the block containing i.
Exact mathematical passage
\mathcal Q=\{B_1,\ldots,B_m\}
b_1Use in source · 2 passages
Search aliases: b_1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5428
[Normalization bound] Let a,b∈[0,∞)^m be nonzero, and set A=norm a_1 and B=norm b_1. Then norma/A-b/B_1 ≤ frac2norma-b_1max{A,B}.
Exact mathematical passage
B=\norm b_1
B_2Use in source · 4 passages
Search aliases: B_2
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5820
[A four state lumpable chain] Partition four states into B_1={1,2} and B_2={3,4}. For positive rates a,b,u,v, set Q= -u-a · u · b/2 · b/2
Exact mathematical passage
B_2=\{3,4\}
B_aUse in source · 14 passages
Search aliases: B_a
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5348
The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement. FloatBarrier Reversible coarse dynamics. Let Q be a finite Markov generator acting on column probability vectors, let (B_a) be a partition into nonempty blocks, and let C_ai= 1_i∈ B_a. Strong lumpability means CQ=overline Q C for a coarse Markov generator. For observables, the backward generator is Q^T; weighted L^2 in…
Exact mathematical passage
(B_a)
B_bUse in source · 1 passage
Search aliases: B_b
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5375
Stationarity follows from overline Qoverlineπ =overline QCπ =CQπ=0. For reversibility, use the aggregate rate formula: overline Q_aboverlineπ_b · =∑_j∈ B_bπ_j∑_i∈ B_aQ_ij · =∑_i∈ B_a∑_j∈ B_bπ_iQ_ji · =overline Q_baoverlineπ_a.
Exact mathematical passage
\begin{aligned} \overline Q_{ab}\overline\pi_b &=\sum_{j\in B_b}\pi_j\sum_{i\in B_a}Q_{ij}\\ &=\sum_{i\in B_a}\sum_{j\in B_b}\pi_iQ_{ji}\\ &=\overline Q_{ba}\overline\pi_a. \end{aligned}
\mathcal B_{\bar\lambda}Use in source · 2 passages
Search aliases: B_barλ mathcal bar lambda
Use in source · atlas-pages.tex, line 227
For strictly positive fine and coarse likelihoods, and at least two coarse blocks, the manuscript proves the all-prior criterion C B_λ(p)= B_barλ(Cp) hboxfor every p Longleftrightarrow λ_i=cbarλ_a (i∈ I_a) for one global c>0. The global scale is invisible after normalization. A different arbitrary scale for each packet is not invisible: it changes posterior odds.
Exact mathematical passage
C\mathcal B_\lambda(p)=\mathcal B_{\bar\lambda}(Cp)\quad\hbox{for every }p \quad\Longleftrightarrow\quad \lambda_i=c\bar\lambda_a\quad(i\in I_a)
B_jUse in source · 6 passages
Search aliases: B_j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4758
[Normal hidden dynamics] Suppose X_Q is a finite dimensional Hilbert space. If A_QQ is normal with spectral resolution A_QQ=∑_j=1^rμ_jQ_j, then K(t)=∑_j=1^r e^μ_jtB_j, Σ_P(z)=∑_j=1^rfracB_jz-μ_j, where B_j=A_PQQ_jA_QP. If Reμ_j<0, then norm K- K_J_L^1(0,∞) ≤ ∑_j∉ J fracnormB_j-Reμ_j.
Exact mathematical passage
\Kcal(t)=\sum_{j=1}^r\e^{\mu_jt}B_j, \qquad \Sigma_P(z)=\sum_{j=1}^r\frac{B_j}{z-\mu_j},
B_{j,k}Use in source · 6 passages
Search aliases: B_j,k
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4649
mory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the compl…
Exact mathematical passage
B_{j,k} = A_{PQ}N_j^kQ_jA_{QP} \qquad (0\leq k<\nu_j).
B_{j,\nu_j-1}Use in source · 1 passage
Search aliases: B_j,ν_j-1 nu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4781
[Pole order and hidden Jordan depth] Every finite hidden block produces a strictly proper rational self energy. At an eigenvalue μ_j, its pole order is at most the size of the largest Jordan block of A_QQ at μ_j. Equality holds when the corresponding highest coefficient B_j,ν_j-1 is nonzero.
Exact mathematical passage
B_{j,\nu_j-1}
B_mUse in source · 2 passages
Search aliases: B_m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 648
Let Ω={1,…,n} carry strictly positive probabilities π_i, and let Q={B_1,…,B_m} be a partition. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j∑_j∈ B(i)π_j, where B(i) is the block containing i.
Exact mathematical passage
\mathcal Q=\{B_1,\ldots,B_m\}
B_nUse in source · 3 passages
Search aliases: B_n
Use in source · atlas-pages.tex, line 620
vpage27Small at each level is not summable across levelsREFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVETraditional Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data. For each n≥1, compare the two scalar operators A_n=0 and B_n=1/n^2 through the identity. Their ordinary defects have sizes 1/n^2, while their square-root defects have sizes 1/n: ∑_n|A_n-B_n|<∞, ∑_n|A_n^1/2-B_n^1/2|=∞. These are independent scalar comparisons, not a constructed inverse…
Exact mathematical passage
B_n=1/n^2
B_n^{1/2}Use in source · 2 passages
Search aliases: B_n^1/2
Use in source · atlas-pages.tex, line 621
at each level is not summable across levelsREFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVETraditional Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data. For each n≥1, compare the two scalar operators A_n=0 and B_n=1/n^2 through the identity. Their ordinary defects have sizes 1/n^2, while their square-root defects have sizes 1/n: ∑_n|A_n-B_n|<∞, ∑_n|A_n^1/2-B_n^1/2|=∞. These are independent scalar comparisons, not a constructed inverse system. They…
Exact mathematical passage
\sum_n|A_n-B_n|<\infty, \qquad \sum_n|A_n^{1/2}-B_n^{1/2}|=\infty.
\mathcal B_rUse in source · 1 passage
Search aliases: B_r mathcal
Use in source · atlas-pages.tex, line 242
page11The exact information-loss vectorBAYESIAN DEFECT / MORE THAN AN ERROR BARTraditional Part I: Theorems 8.1 and 8.4, and Corollary 8.6. For a prior p, let P=Cp and let r_a(p) be the conditional average of the fine likelihood inside packet a. For the uniform prior and λ=(4,1,1,1), P=(1/2,1/2)^ T, r=(5/2,1)^ T. The exact coarse posterior is B_r(P)=(5/7,2/7)^ T. Suppose instead that the declared coarse likelihood is barλ=(1,1). vfigloss_vector platecaptionThe posterior error lies along the mass…
Exact mathematical passage
\mathcal B_r(P)=(5/7,2/7)^{\mathsf T}
B_\alphaUse in source · 6 passages
Search aliases: B_α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2847
Let {B_1,…,B_m} be a partition of the fine state set. Its aggregation matrix is C_α i= 1_{i∈ B_α}.
Exact mathematical passage
C_{\alpha i}=\one_{\{i\in B_\alpha\}}.
B_\betaUse in source · 3 passages
Search aliases: B_β beta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2915
[Characterization of exact discrete lumping] Let K be column stochastic. A coarse column stochastic matrix overline K satisfying CK=overline KC exists if and only if, for every source block B_β, every pair i,j∈ B_β, and every target block B_α, ∑_r∈ B_αK_ri = ∑_r∈ B_αK_rj. When it exists, overline K is unique and overline K_αβ = ∑_r∈ B_αK_ri for any i∈ B_β.
Exact mathematical passage
B_\beta
\mathcal B_\lambdaUse in source · 4 passages
Search aliases: B_λ mathcal lambda
Use in source · atlas-pages.tex, line 35
B_λ · diag(λ)p/(λ^ Tp). · Nonnegative likelihood and positive total evidence.
Exact mathematical passage
\mathcal B_\lambda
\bar bUse in source · 6 passages
Search aliases: bar b bar
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 914
When sufficiency fails, the defect still has an exact finite factorization. Fix a declared coarse likelihood barλ, put P=C_π p, δ(p)=r(p)-barλ, bar b= B_barλ(P), Z_r=⟨ r(p),P⟩.
Exact mathematical passage
P=C_\pi p, \qquad \delta(p)=r(p)-\bar\lambda, \qquad \bar b=\cB_{\bar\lambda}(P), \qquad Z_r=\langle r(p),P\rangle.
\bar KUse in source · 21 passages
Search aliases: bar K bar
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 590
Let K be a column-stochastic fine transition matrix, let C aggregate fine states into coarse packets, and let bar K be a coarse transition matrix. Define Δ=CK-bar KC.
Exact mathematical passage
\bar K
\bar K^nUse in source · 9 passages
Search aliases: bar K^n bar
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 597
[Exact lumpability] If Δ=0, then CK^n=bar K^nC for every n≥0. Thus fine evolution followed by coarse reconstruction equals coarse evolution from the beginning.
Exact mathematical passage
\boxed{CK^n=\bar K^nC}
\bar K^{n+1}Use in source · 1 passage
Search aliases: bar K^n+1 bar
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 605
The result is immediate for n=0. If CK^n=bar K^nC, then CK^n+1=bar K^nCK=bar K^n+1C.
Exact mathematical passage
CK^{n+1}=\bar K^nCK=\bar K^{n+1}C.
\bar K^{n-1-j}Use in source · 3 passages
Search aliases: bar K^n-1-j bar
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 612
[Finite-horizon defect formula] For arbitrary Δ=CK-bar KC, CK^n-bar K^nC = ∑_j=0^n-1bar K^n-1-jΔ K^j. If K and bar K are contractions in the chosen norm, then normCK^n-bar K^nC≤ nnormΔ.
Exact mathematical passage
\boxed{ CK^n-\bar K^nC = \sum_{j=0}^{n-1}\bar K^{n-1-j}\Delta K^j. }
\bar K^{n-j}Use in source · 1 passage
Search aliases: bar K^n-j bar
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 627
The identity follows by telescoping: bar K^n-1-jCK^j+1-bar K^n-jCK^j =bar K^n-1-j(CK-bar KC)K^j. Summing from j=0 to n-1 leaves CK^n-bar K^nC. The norm bound follows termwise.
Exact mathematical passage
\bar K^{n-1-j}CK^{j+1}-\bar K^{n-j}CK^j =\bar K^{n-1-j}(CK-\bar KC)K^j.
\bar QUse in source · 2 passages
Search aliases: bar Q bar
Use in source · atlas-pages.tex, line 115
The continuous generator also closes: CQ=bar Q C, bar Q=-2 · 2 2 · -2. The Duhamel identity therefore gives Ce^tQ=e^tbar QC. This is a second consequence of the corresponding generator identity, not an inference that K is a fixed-time sample of Q.
Exact mathematical passage
CQ=\bar Q C,\qquad \bar Q=\begin{pmatrix}-2&2\\2&-2\end{pmatrix}.
\bar\lambdaUse in source · 22 passages
Search aliases: barλ bar lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 880
[Bayesian refinement criterion] Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive. The identity C_π B_λ(p) = B_barλ(C_π p) holds for every fine prior p if and only if there exists one scalar c>0 such that λ_i=c barλ_a (i∈ I_a). Equivalently, the likelihood is constant on each coarse packet, with the coarse values determined up to the globally invisible Bayesian scaling c.
Exact mathematical passage
\boxed{ C_\pi\cB_\lambda(p) = \cB_{\bar\lambda}(C_\pi p)}
\bar\lambda_aUse in source · 7 passages
Search aliases: barλ_a bar lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 887
[Bayesian refinement criterion] Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive. The identity C_π B_λ(p) = B_barλ(C_π p) holds for every fine prior p if and only if there exists one scalar c>0 such that λ_i=c barλ_a (i∈ I_a). Equivalently, the likelihood is constant on each coarse packet, with the coarse values determined up to the globally invisible Bayesian scaling c.
Exact mathematical passage
\boxed{ \lambda_i=c\,\bar\lambda_a \qquad(i\in I_a).}
\bar\lambda_bUse in source · 1 passage
Search aliases: barλ_b bar lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 901
Now use priors supported on one state of packet a and one state of packet b. Equality of the two coarse posterior odds gives fracλ_aλ_b = fracbarλ_abarλ_b. This holds for every pair of coarse packets, so λ_a=cbarλ_a for one common c>0.
Exact mathematical passage
\frac{\lambda_a}{\lambda_b} = \frac{\bar\lambda_a}{\bar\lambda_b}.
\bar\lambda_{\pi(i)}Use in source · 1 passage
Search aliases: barλ_π(i) bar lambda pi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 945
tes barλ. This records the common-scale null direction at the declared coarse likelihood; the full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together. Moreover, C_π B_λ(p)- B_barλ(P) _1 ≤ frac2∑_aP_a|δ_a(p)| min{Z_r,Z_barλ} and ∑_aP_a|δ_a(p)| ≤ ∑_i p_i|λ_i-barλ_π(i)|.
Exact mathematical passage
\sum_aP_a|\delta_a(p)| \le \sum_i p_i\left|\lambda_i-\bar\lambda_{\pi(i)}\right|.
\bbPUse in source · 1 passage
Search aliases: bbP P mathbb
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 574
[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.
Exact mathematical passage
\bbP(X_n=k)= 2^{-n}\binom{n}{(n+k)/2}
\bbRUse in source · 1 passage
Search aliases: bbR R mathbb
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 333
[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.
Exact mathematical passage
s,t\in\bbR
\BcalUse in source · 22 passages
Search aliases: Bcal B mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2028
Bayesian update · B_λ · likelihood constant inside each coarse packet
Exact mathematical passage
\Bcal_\lambda
\Bcal_{c\lambda}Use in source · 1 passage
Search aliases: Bcal_cλ B_cλ Bcal lambda mathcal lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3352
[Scale invariance and composition] For every c>0, B_cλ= B_λ. If all relevant evidences are positive, then B_μ( B_λ(p)) = B_λodotμ(p).
Exact mathematical passage
\Bcal_{c\lambda}=\Bcal_\lambda.
\Bcal_LUse in source · 4 passages
Search aliases: Bcal_L B_L Bcal mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5401
FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
\Bcal_L(p)=L\odot p/\ip{L}{p}
\Bcal_{L_y}Use in source · 1 passage
Search aliases: Bcal_L_y B_L_y Bcal mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5534
For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
\Phi_y=\Bcal_{L_y}\circ K
\Bcal_{\overline L}Use in source · 2 passages
Search aliases: Bcal_overline L B_overline L Bcal overline mathcal overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5410
ctor p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
C\Bcal_L(p) = \Bcal_{\overline L}(q).
\Bcal_{\overline L_y}Use in source · 2 passages
Search aliases: Bcal_overline L_y B_overline L_y Bcal overline mathcal overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5534
For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
\overline\Phi_y=\Bcal_{\overline L_y}\circ\overline K
\Bcal_{\overline\lambda(p)}Use in source · 1 passage
Search aliases: Bcal_overlineλ(p) B_overlineλ(p) Bcal overline lambda mathcal overline lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3483
[Exact two stage Bayesian update] For every strictly positive prior, C B_λ(p) = B_overlineλ(p)(Cp). For every packet with overlineλ_α(p)>0, the posterior conditional distribution is q_α,i = fracλ_ir_α,i overlineλ_α(p). A packet with zero average likelihood has zero posterior mass. Thus a fine Bayesian update consists of a coarse update by packet average likelihoods followed by independent local updates inside the packets that survive.
Exact mathematical passage
C\Bcal_\lambda(p) = \Bcal_{\overline\lambda(p)}(Cp).
\Bcal_{\widetilde\lambda}Use in source · 1 passage
Search aliases: Bcal_widetildeλ B_widetildeλ Bcal widetilde lambda mathcal widetilde lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3628
Block constancy gives C B_widetildeλ(p) = B_ℓ(Cp). Aggregation is an ℓ^1 contraction. Apply [source reference] to the two fine posteriors and then aggregate.
Exact mathematical passage
C\Bcal_{\widetilde\lambda}(p) = \Bcal_\ell(Cp).
\Bcal_\lambdaUse in source · 13 passages
Search aliases: Bcal_λ B_λ Bcal lambda mathcal lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2028
Bayesian update · B_λ · likelihood constant inside each coarse packet
Exact mathematical passage
\Bcal_\lambda
\Bcal_{\lambda^{(r)}}Use in source · 1 passage
Search aliases: Bcal_λ^(r) B_λ^(r) Bcal lambda mathcal lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3860
Let K be column stochastic and let λ^(r) be the likelihood of the rth observation. The hidden Markov filter is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r). A standard account of finite hidden Markov models appears in [source reference].
Exact mathematical passage
\widehat p_r=Kp_{r-1}, \qquad p_r=\Bcal_{\lambda^{(r)}}(\widehat p_r).
\Bcal_{\lambda\odot\mu}Use in source · 1 passage
Search aliases: Bcal_λodotμ B_λodotμ Bcal lambda odot mu mathcal lambda odot mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3356
[Scale invariance and composition] For every c>0, B_cλ= B_λ. If all relevant evidences are positive, then B_μ( B_λ(p)) = B_λodotμ(p).
Exact mathematical passage
\Bcal_\mu(\Bcal_\lambda(p)) = \Bcal_{\lambda\odot\mu}(p).
\Bcal_\muUse in source · 3 passages
Search aliases: Bcal_μ B_μ Bcal mu mathcal mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3356
[Scale invariance and composition] For every c>0, B_cλ= B_λ. If all relevant evidences are positive, then B_μ( B_λ(p)) = B_λodotμ(p).
Exact mathematical passage
\Bcal_\mu(\Bcal_\lambda(p)) = \Bcal_{\lambda\odot\mu}(p).
\Bcal_\ellUse in source · 3 passages
Search aliases: Bcal_ℓ B_ℓ Bcal ell mathcal ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3514
[Characterization of fixed coarse sufficiency] Assume λ_i>0 for every state. The following conditions are equivalent. item There is a positive coarse likelihood ℓ∈(0,∞)^m such that C B_λ(p) = B_ℓ(Cp) for every strictly positive prior p, and the posterior conditional distribution inside every packet equals its prior conditional distribution. item The likelihood is constant on every packet. When these conditions hold, ℓ_α can be chosen as the common value of λ on I_α, and it is unique up to a comm…
Exact mathematical passage
C\Bcal_\lambda(p) = \Bcal_\ell(Cp)
\Bcal_{\ell(p)}Use in source · 1 passage
Search aliases: Bcal_ℓ(p) B_ℓ(p) Bcal ell mathcal ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5454
For an arbitrary prior p, put q=Cp and Z(p,L)=ipLp. Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0; choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_aabsℓ_a(p)-overline L_a. Then norm C B_L(p)- B_overline L(q) _1 ≤ frac2d(pmid L,overline L) max{Z(p,L),ipoverline Lq}.
Exact mathematical passage
\boxed{ C\Bcal_L(p) = \Bcal_{\ell(p)}(q). }
\mathrm{Beta}Use in source · 4 passages
Search aliases: Beta mathrm
Use in source · atlas-pages.tex, line 305
vpage14Conjugacy tracks exponentsBETA MODEL / POSTERIOR AND EVIDENCETraditional Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-θ). Multiplication adds exponents, so the posterior belongs to Beta(4,4) with density 140θ^3(1-θ)^3. vfigbeta platecaptionThe dashed prior is asymmetr…
Exact mathematical passage
\mathrm{Beta}(2,3)
\operatorname{Beta}Use in source · 3 passages
Search aliases: Beta operatorname
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1071
[Beta--Binomial conjugate packet] A Beta(a,b) prior and s successes in n trials produce Beta(a+s,b+n-s). The prior exponents are additive packets: the likelihood increments them by (s,n-s). In the numerical example, (a,b)=(2,3), (s,n)=(7,10), so the posterior is Beta(9,6) with mean 0.6.
Exact mathematical passage
\operatorname{Beta}(a,b)
\mathbb CUse in source · 1 passage
Search aliases: C mathbb
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1654
For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. I…
Exact mathematical passage
\mathbb C^d
C^{1-\alpha}Use in source · 1 passage
Search aliases: C^1-α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1264
For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D^α. The zero cases are immediate.
Exact mathematical passage
\begin{aligned} \norm{D_\alpha} &\le \frac{\sin(\pi\alpha)}{\pi} \left( 2\norm C\int_0^r t^{\alpha-1}\,dt + \norm D\int_r^\infty t^{\alpha-2}\,dt \right)\\ &= K_\alpha\norm C^{1-\alpha}\norm D^\alpha. \end{aligned}
C^4Use in source · 1 passage
Search aliases: C^4
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4241
[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.
Exact mathematical passage
u\in C^4[0,1]
\mathbb C^dUse in source · 1 passage
Search aliases: C^d mathbb
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1654
For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. I…
Exact mathematical passage
\mathbb C^d
C^TUse in source · 1 passage
Search aliases: C^T
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5643
[Coarse Feynman Kac transfer] Assume CQ=overline QC, and let J=C^T lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so VJ=Joverline V. Then e^t(Q^T-V)J = J e^t(overline Q^T-overline V) for every t≥0.
Exact mathematical passage
J=C^T
C_0Use in source · 3 passages
Search aliases: C_0
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2436
Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.
Exact mathematical passage
X_0\xleftarrow{C_0}X_1\xleftarrow{C_1}X_2\xleftarrow{}\cdots
C_1Use in source · 1 passage
Search aliases: C_1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2436
Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.
Exact mathematical passage
X_0\xleftarrow{C_0}X_1\xleftarrow{C_1}X_2\xleftarrow{}\cdots
C_{ai}Use in source · 1 passage
Search aliases: C_ai
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5348
The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement. FloatBarrier Reversible coarse dynamics. Let Q be a finite Markov generator acting on column probability vectors, let (B_a) be a partition into nonempty blocks, and let C_ai= 1_i∈ B_a. Strong lumpability means CQ=overline Q C for a coarse Markov generator. For observables, the backward generator is Q^T; weighted L^2 in…
Exact mathematical passage
C_{ai}=\one_{i\in B_a}
C_{j+1,m}Use in source · 2 passages
Search aliases: C_j+1,m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5252
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
\boxed{ C_{\ell,m}T_m-T_\ell C_{\ell,m} = \sum_{j=\ell}^{m-1} C_{\ell,j}E_jC_{j+1,m}. }
C_{j,j}Use in source · 1 passage
Search aliases: C_j,j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5245
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
C_{j,j}=I
C_{j,m}Use in source · 1 passage
Search aliases: C_j,m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5264
Expand each summand as C_ℓ,j+1T_j+1C_j+1,m - C_ℓ,jT_jC_j,m. The intermediate terms cancel. The first surviving term is C_ℓ,mT_m, and the last is -T_ℓ C_ℓ,m.
Exact mathematical passage
C_{\ell,j+1}T_{j+1}C_{j+1,m} - C_{\ell,j}T_jC_{j,m}.
C_mUse in source · 1 passage
Search aliases: C_m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2522
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,n}=C_mC_{m+1}\cdots C_{n-1}, \qquad R_{n,m}=R_{n-1}\cdots R_m.
C_{m+1}Use in source · 1 passage
Search aliases: C_m+1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2522
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,n}=C_mC_{m+1}\cdots C_{n-1}, \qquad R_{n,m}=R_{n-1}\cdots R_m.
C_{m,m}Use in source · 1 passage
Search aliases: C_m,m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2528
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,m}=I_{X_m}, \qquad R_{m,m}=I_{X_m}.
C_{m-1}Use in source · 1 passage
Search aliases: C_m-1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5245
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
C_{\ell,m}=C_\ell\cdots C_{m-1}
C_nUse in source · 4 passages
Search aliases: C_n
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2543
[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
E_n(t)=C_nT_{n+1}(t)-T_n(t)C_n.
c_nUse in source · 2 passages
Search aliases: c_n
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3286
A mean zero function has no h_0 component, so every surviving Hermite multiplier is at most e^-t. For a polynomial f- E_γ f=∑_n≥1c_nh_n. Then Var_γ(f)=∑_n≥1absc_n^2 ≤ ∑_n≥1nabsc_n^2 = ip- L ff. Apply [source reference].
Exact mathematical passage
f-\Exp_\gamma f=\sum_{n\geq1}c_nh_n.
C_{n-1}Use in source · 1 passage
Search aliases: C_n-1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2522
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,n}=C_mC_{m+1}\cdots C_{n-1}, \qquad R_{n,m}=R_{n-1}\cdots R_m.
C_{\alpha i}Use in source · 1 passage
Search aliases: C_α i alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2847
Let {B_1,…,B_m} be a partition of the fine state set. Its aggregation matrix is C_α i= 1_{i∈ B_α}.
Exact mathematical passage
C_{\alpha i}=\one_{\{i\in B_\alpha\}}.
C_\etaUse in source · 4 passages
Search aliases: C_η eta
Use in source · atlas-pages.tex, line 943
vpage41An operator substitution transports the defect tooFUNCTIONAL CALCULUS / AN EXACT RATIONAL TESTTraditional Part II: Theorem E3.4; compare Traditional Part I, Theorem 15.1, and V--25. Use the operators from V--25, but perturb the rectangular comparison map: A=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, B=mat2 · 1 0 · 2, C_η=mat1 · 0 · η 0 · 1 · 0. The generator defect is Δ=C_η A-BC_η, with only one nonzero entry, Δ_13=7η (matrix indices begin at one). vfigext_substitution platecaptionThe perturbation…
Exact mathematical passage
A=\mat{2&1&0\\0&2&0\\0&0&9},\quad B=\mat{2&1\\0&2},\quad C_\eta=\mat{1&0&\eta\\0&1&0}.
C_\piUse in source · 8 passages
Search aliases: C_π pi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 840
Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero.
Exact mathematical passage
(C_\pi p)_a=P_a=\sum_{i\in I_a}p_i.
C_\ellUse in source · 14 passages
Search aliases: C_ℓ ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1328
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces; no operator sum across unrelated spaces…
Exact mathematical passage
D_\ell=C_\ell A_{\ell+1}-A_\ell C_\ell
C_{\ell,j}Use in source · 2 passages
Search aliases: C_ℓ,j ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5252
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
\boxed{ C_{\ell,m}T_m-T_\ell C_{\ell,m} = \sum_{j=\ell}^{m-1} C_{\ell,j}E_jC_{j+1,m}. }
C_{\ell,j+1}Use in source · 1 passage
Search aliases: C_ℓ,j+1 ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5264
Expand each summand as C_ℓ,j+1T_j+1C_j+1,m - C_ℓ,jT_jC_j,m. The intermediate terms cancel. The first surviving term is C_ℓ,mT_m, and the last is -T_ℓ C_ℓ,m.
Exact mathematical passage
C_{\ell,j+1}T_{j+1}C_{j+1,m} - C_{\ell,j}T_jC_{j,m}.
C_{\ell,m}Use in source · 5 passages
Search aliases: C_ℓ,m ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5245
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
C_{\ell,m}=C_\ell\cdots C_{m-1}
\cBUse in source · 13 passages
Search aliases: cB B mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 730
Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.
Exact mathematical passage
\boxed{ \cB_\lambda(p) = \frac{\lambda\odot p}{Z_\lambda(p)}. }
\cB_{\bar\lambda}Use in source · 5 passages
Search aliases: cB_barλ B_barλ cB bar lambda mathcal bar lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 880
[Bayesian refinement criterion] Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive. The identity C_π B_λ(p) = B_barλ(C_π p) holds for every fine prior p if and only if there exists one scalar c>0 such that λ_i=c barλ_a (i∈ I_a). Equivalently, the likelihood is constant on each coarse packet, with the coarse values determined up to the globally invisible Bayesian scaling c.
Exact mathematical passage
\boxed{ C_\pi\cB_\lambda(p) = \cB_{\bar\lambda}(C_\pi p)}
\cB_{\bar\lambda+\delta}Use in source · 1 passage
Search aliases: cB_barλ+δ B_barλ+δ cB bar lambda delta mathcal bar lambda delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 954
The effective-likelihood theorem reduces the defect to B_barλ+δ(P)- B_barλ(P). Since Z_r=Z_barλ+P^ Tδ, direct subtraction gives frac1Z_r (diag(P)-bar bP^ T)δ. Also, (diag(P)-bar bP^ T)barλ =Podotbarλ-bar bZ_barλ=0. The first norm bound is the likelihood perturbation theorem applied on the coarse simplex. For a zero-mass packet, both sides of the following estimate vanish. For P_a>0, P_a|δ_a| = P_a|∑_i∈ I_afracp_iP_a (λ_i-barλ_a)| ≤ ∑_i∈ I_ap_i|λ_i-barλ_a|, and summing over a proves the second in…
Exact mathematical passage
\cB_{\bar\lambda+\delta}(P)-\cB_{\bar\lambda}(P).
\cB_{r(p)}Use in source · 2 passages
Search aliases: cB_r(p) B_r(p) cB mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 854
[Exact effective-likelihood factorization] Whenever the fine evidence is positive, C_π B_λ(p) = B_r(p)(C_π p). Thus the only fine information needed for the coarse posterior, beyond the coarse prior masses, is the vector of conditional packet evidences r(p).
Exact mathematical passage
\boxed{ C_\pi\cB_\lambda(p) = \cB_{r(p)}(C_\pi p).}
\cB_\lambdaUse in source · 10 passages
Search aliases: cB_λ B_λ cB lambda mathcal lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 730
Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.
Exact mathematical passage
\boxed{ \cB_\lambda(p) = \frac{\lambda\odot p}{Z_\lambda(p)}. }
\cB_{\lambda^{(r)}}Use in source · 1 passage
Search aliases: cB_λ^(r) B_λ^(r) cB lambda mathcal lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1092
Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).
Exact mathematical passage
\widehat p_r=Kp_{r-1}, \qquad p_r=\cB_{\lambda^{(r)}}(\widehat p_r).
\cB_{\lambda\odot\mu}Use in source · 1 passage
Search aliases: cB_λodotμ B_λodotμ cB lambda odot mu mathcal lambda odot mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 741
[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).
Exact mathematical passage
\boxed{ \cB_\mu(\cB_\lambda(p)) = \cB_{\lambda\odot\mu}(p). }
\cB_\muUse in source · 2 passages
Search aliases: cB_μ B_μ cB mu mathcal mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 741
[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).
Exact mathematical passage
\boxed{ \cB_\mu(\cB_\lambda(p)) = \cB_{\lambda\odot\mu}(p). }
\cKUse in source · 3 passages
Search aliases: cK K mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1532
For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.
Exact mathematical passage
\dot x_P(t) =A_{PP}x_P(t)+A_{PQ}e^{tA_{QQ}}x_Q(0) +\int_0^t\cK(t-r)x_P(r)\,dr,
\cosUse in source · 4 passages
Search aliases: cos
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2695
[Summable defects do not force a semigroup limit] Let X_0= R and let X_n= R^2 for every n≥1. Set C_0(x,y)=x, R_0x=(x,0), and use identity connecting maps and reconstructions above level one. Let T_0(t)=I and let every higher level carry the rotation group T_n(t) = cos t · -sin t sin t · cos t . Only the first adjacent defect is nonzero, so every defect sum is finite. Yet for n≥1, U_0,n(t)=C_0T_n(t)R_0=cos t. The limiting family is not a semigroup because in general cos(s+t)≠cos scos t. The failu…
Exact mathematical passage
T_n(t) = \begin{pmatrix} \cos t&-\sin t\\ \sin t&\cos t \end{pmatrix}.
\CovUse in source · 1 passage
Search aliases: Cov operatorname
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5516
Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.
Exact mathematical passage
\frac{\dd}{\dd t}\EE_{p(t)}[h] = \Cov_{p(t)}(h,g).
\Cov_{p(t)}Use in source · 1 passage
Search aliases: Cov_p(t) Cov operatorname
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5516
Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.
Exact mathematical passage
\frac{\dd}{\dd t}\EE_{p(t)}[h] = \Cov_{p(t)}(h,g).
\mathfrak DUse in source · 8 passages
Search aliases: D mathfrak
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 456
Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.
Exact mathematical passage
\mathfrak D(t)=Ce^{tA_f}-e^{tA_c}C.
dUse in source · 59 passages
Search aliases: d
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 462
[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.
Exact mathematical passage
\boxed{ Ce^{tA_f}-e^{tA_c}C = \int_0^t e^{(t-r)A_c}\Delta e^{rA_f}\,dr. }
d^2Use in source · 1 passage
Search aliases: d^2
Use in source · atlas-pages.tex, line 832
t t=log2, they are Z=7/3 and p=(1,2,4)/7. vfigext_cumulants platecaptionThe score values stay fixed. Their probabilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores. headTwo derivatives have different statistical meanings The first derivative of log evidence is the current mean score, and the second is its variance: fracddtlog Z(t)= E_p(t)u, fracd^2dt^2log Z(t)=Var_p(t)(u)≥0. At q=2 the exact values are E u=frac107, E u^2=frac187, V…
Exact mathematical passage
\frac{d}{dt}\log Z(t)=\EE_{p(t)}u, \qquad \frac{d^2}{dt^2}\log Z(t)=\Var_{p(t)}(u)\ge0.
D^\alphaUse in source · 1 passage
Search aliases: D^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1264
For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D^α. The zero cases are immediate.
Exact mathematical passage
\begin{aligned} \norm{D_\alpha} &\le \frac{\sin(\pi\alpha)}{\pi} \left( 2\norm C\int_0^r t^{\alpha-1}\,dt + \norm D\int_r^\infty t^{\alpha-2}\,dt \right)\\ &= K_\alpha\norm C^{1-\alpha}\norm D^\alpha. \end{aligned}
D_0Use in source · 5 passages
Search aliases: D_0
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 690
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
D_0=P_0, \qquad D_j=P_j-P_{j-1}.
D_1Use in source · 5 passages
Search aliases: D_1
Use in source · atlas-pages.tex, line 172
vpage8Where the variance goesMARTINGALE PACKETS / THREE EXACT PIECESTraditional Part I: Theorem 6.2. Nested weighted projections and their differences. Keep the weights and observable from pagerefV7. Let E_0f be the constant global mean, let E_1=E average within A and B, and let E_2=I retain every value. Define D_0=E_0, D_1=E_1-E_0, D_2=I-E_1. Then D_0f is the mean, D_1f describes the contrast between block means, and D_2f is the fluctuation left inside each block. vfigvariance platecaptionThe s…
Exact mathematical passage
D_0=E_0,\qquad D_1=E_1-E_0,\qquad D_2=I-E_1.
D_2Use in source · 4 passages
Search aliases: D_2
Use in source · atlas-pages.tex, line 172
vpage8Where the variance goesMARTINGALE PACKETS / THREE EXACT PIECESTraditional Part I: Theorem 6.2. Nested weighted projections and their differences. Keep the weights and observable from pagerefV7. Let E_0f be the constant global mean, let E_1=E average within A and B, and let E_2=I retain every value. Define D_0=E_0, D_1=E_1-E_0, D_2=I-E_1. Then D_0f is the mean, D_1f describes the contrast between block means, and D_2f is the fluctuation left inside each block. vfigvariance platecaptionThe s…
Exact mathematical passage
D_0=E_0,\qquad D_1=E_1-E_0,\qquad D_2=I-E_1.
D_iUse in source · 3 passages
Search aliases: D_i
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 696
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
D_iD_j=\delta_{ij}D_j, \qquad P_r=\sum_{j=0}^rD_j.
D_jUse in source · 9 passages
Search aliases: D_j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 690
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
D_0=P_0, \qquad D_j=P_j-P_{j-1}.
d_jUse in source · 2 passages
Search aliases: d_j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5766
If B_j and C_j have an intermediate space of dimension d_j, use a block diagonal matrix D with blocks λ_j I_d_j, or the required Jordan blocks when poles repeat. Assemble the matrices B_j and C_j into the output and input maps.
Exact mathematical passage
d_j
d_pUse in source · 3 passages
Search aliases: d_p
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3408
[Likelihood perturbation] Let λ,μ≥0 have positive evidences under a fixed prior p. Set d_p(λ,μ) = ∑_iabsλ_i-μ_ip_i. Then norm B_λ(p)- B_μ(p)_1 ≤ frac2d_p(λ,μ) max{Z_λ(p),Z_μ(p)}.
Exact mathematical passage
d_p(\lambda,\mu) = \sum_i\abs{\lambda_i-\mu_i}p_i.
D_rUse in source · 5 passages
Search aliases: D_r
Use in source · atlas-pages.tex, line 289
unchanged. Predicting afterward gives (3/4,1/4) instead. The order of operations expresses which time the observation concerns. vfigfilter_order platecaptionBoth routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem. headKeep an unnormalized path alongside the filter For a sequence of observations, let D_r=diag(λ^(r)). The forward recursion is widehat p_r=Kp_r-1, p_r=D_rwidehat p_r/Z_r. Its unnormalized companion i…
Exact mathematical passage
D_r=\diag(\lambda^{(r)})
D_VUse in source · 12 passages
Search aliases: D_V
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4346
Let K be column stochastic. Let V be a real potential on the states and let D_V=diag(V). For functions written as columns, the backward Markov operator is K^T.
Exact mathematical passage
D_V=\diag(V).
D_wUse in source · 5 passages
Search aliases: D_w
Use in source · atlas-pages.tex, line 301
Here D_rK means an observation after prediction. In the Feynman--Kac example on pagerefV20, departure-state weighting has forward operator K D_w. Both conventions are legitimate once their meanings are fixed, but they cannot be substituted for one another. takeawayWrite the chronological sequence before multiplying matrices. A product can be normalized correctly and still answer the wrong time-ordered question.
Exact mathematical passage
K D_w
D_\alphaUse in source · 8 passages
Search aliases: D_α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1193
Let A≥0 act on a fine Hilbert space, let B≥0 act on a coarse Hilbert space, and let C be a bounded coarsening map. Set D=CA-BC, D_α=CA^α-B^α C, 0<α<1.
Exact mathematical passage
D=CA-BC, \qquad D_\alpha=CA^\alpha-B^\alpha C, \qquad 0<\alpha<1.
D_\ellUse in source · 3 passages
Search aliases: D_ℓ ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1328
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces; no operator sum across unrelated spaces…
Exact mathematical passage
D_\ell=C_\ell A_{\ell+1}-A_\ell C_\ell
\ddUse in source · 50 passages
Search aliases: dd d mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2092
[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.
Exact mathematical passage
\frac{\dd}{\dd t}T_{\beta\alpha}(t) = \sum_{\gamma\in I} A_{\beta\gamma}T_{\gamma\alpha}(t) = \sum_{\gamma\in I} T_{\beta\gamma}(t)A_{\gamma\alpha}.
\ddotsUse in source · 2 passages
Search aliases: ddots
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1367
For n interior points with spacing h=1/(n+1), define L_h=frac1h^2 2 · -1 · · -1 · 2 · ddots · · ddots · ddots · -1 · · -1 · 2 .
Exact mathematical passage
L_h=\frac1{h^2} \begin{pmatrix} 2&-1&&\\ -1&2&\ddots&\\ &\ddots&\ddots&-1\\ &&-1&2 \end{pmatrix}.
\detUse in source · 1 passage
Search aliases: det
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4488
[Schur Feshbach resolvent] Assume zI-A_QQ is invertible. Then zI-A is invertible if and only if F_P(z) is invertible. In that case, P(zI-A)^-1P = F_P(z)^-1. In finite dimension, det(zI-A) = det(zI-A_QQ)det F_P(z).
Exact mathematical passage
\det(zI-A) = \det(zI-A_{QQ})\det F_P(z).
\DfunUse in source · 6 passages
Search aliases: Dfun D mathsf
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5910
This section receives the operator-substitution result deliberately moved out of Paper IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=log D_N once and define S_N^(r)(z)=exp(-r g(z)) for complex r. The bounded Banach-space result below uses classical holomorphic functional calculus; it …
Exact mathematical passage
\Dfun_N(z)=\sum_{m=0}^{\infty}\frac{(-1)^m z^{Nm}}{(Nm)!}.
\Dfun_NUse in source · 5 passages
Search aliases: Dfun_N D_N Dfun mathsf
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5910
This section receives the operator-substitution result deliberately moved out of Paper IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=log D_N once and define S_N^(r)(z)=exp(-r g(z)) for complex r. The bounded Banach-space result below uses classical holomorphic functional calculus; it …
Exact mathematical passage
\Dfun_N(z)=\sum_{m=0}^{\infty}\frac{(-1)^m z^{Nm}}{(Nm)!}.
\Dfun_N^{-r}Use in source · 1 passage
Search aliases: Dfun_N^-r D_N^-r Dfun mathsf
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5927
[Operator reciprocal family] Let A be bounded and suppose Spec(A) lies in a simply connected region on which D_N has no zeros. Fix a holomorphic branch g=log D_N there. Then S_N^(r)(A) = exp(-r g(A)) = ( D_N^-r)(A) is well defined and S_N^(r)(A) S_N^(s)(A) = S_N^(r+s)(A). If (P_i) is a finite complete family of bounded idempotents, then P_i f(A)P_j = frac12π i ∫_Γ f(z)P_i(zI-A)^-1P_j dz for every holomorphic f on a neighborhood of the spectrum.
Exact mathematical passage
\Sfun_N^{(r)}(A) = \exp\bigl(-r g(A)\bigr) = \bigl(\Dfun_N^{-r}\bigr)(A)
\diagUse in source · 11 passages
Search aliases: diag operatorname
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 926
[Bayesian information-loss packet] Assume nonnegative fine and declared coarse likelihoods, with both Z_r>0 and Z_barλ>0. The coarsening defect is C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood; the full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together. Moreover, C_π B_λ(p)- B_barλ(P) …
Exact mathematical passage
\boxed{ C_\pi\cB_\lambda(p) - \cB_{\bar\lambda}(P) = \frac{1}{Z_r} \left(\diag(P)-\bar bP^{\mathsf T}\right) \delta(p).}
\operatorname{diag}Use in source · 4 passages
Search aliases: diag operatorname
Use in source · atlas-pages.tex, line 35
B_λ · diag(λ)p/(λ^ Tp). · Nonnegative likelihood and positive total evidence.
Exact mathematical passage
\operatorname{diag}(\lambda)p/(\lambda^{\mathsf T}p)
\dimUse in source · 2 passages
Search aliases: dim
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4586
[Criterion for exact local closure] Assume x_Q(0)=0 and d=dim X_Q<∞. The following conditions are equivalent. item For every retained initial state, P e^tAP = e^tA_PPP for all t≥0. item Σ_P(z)=0 for every sufficiently large z. item K(t)=0 for every t≥0. item If d=dim X_Q, then A_PQA_QQ^kA_QP=0 for 0≤ k<d.
Exact mathematical passage
d=\dim X_Q<\infty
\operatorname{Dom}Use in source · 1 passage
Search aliases: Dom operatorname
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5334
[Coordinate generators] Assume the setting of [source reference]. If x=(x_j) satisfies x_j∈Dom(A_j) and the sequence (A_jx_j) belongs to X_∞, then x lies in the generator domain of T_∞ and A_∞ x=(A_jx_j)_j≥0.
Exact mathematical passage
x_j\in\operatorname{Dom}(A_j)
\dot pUse in source · 4 passages
Search aliases: dot p dot
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1057
The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.
Exact mathematical passage
\dot p_i=g_ip_i-p_i\frac{\sum_jg_je^{tg_j}p_j(0)}{\sum_je^{tg_j}p_j(0)}.
\dot p_0Use in source · 2 passages
Search aliases: dot p_0 dot
Use in source · atlas-pages.tex, line 277
Differentiation of the normalized flow yields the centered score equation dot p_i=(g_i-⟨ g,p⟩)p_i. For the displayed model, dot p_0=p_0(1-p_0). The negative average-score term is what keeps the total derivative zero. Omitting the derivative of the normalizer would not preserve mass.
Exact mathematical passage
\dot p_0=p_0(1-p_0)
\dot p_iUse in source · 2 passages
Search aliases: dot p_i dot
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1057
The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.
Exact mathematical passage
\dot p_i=g_ip_i-p_i\frac{\sum_jg_je^{tg_j}p_j(0)}{\sum_je^{tg_j}p_j(0)}.
\dot uUse in source · 1 passage
Search aliases: dot u dot
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4403
Conditioning over the first small time interval gives the backward equation dot u=(Q^T-D_V)u, u(0)=f. The finite system has a unique solution, namely the displayed matrix exponential. The expectation bound follows from exp(-∫_0^tV(X_s) d s) ≤ e^-v_0t. This finite argument is the matrix form of the Feynman Kac principle originating in [source reference].
Exact mathematical passage
\dot u=(Q^T-D_V)u, \qquad u(0)=f.
\dot xUse in source · 9 passages
Search aliases: dot x dot
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1526
For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.
Exact mathematical passage
\dot x=Ax
\dot x_PUse in source · 3 passages
Search aliases: dot x_P dot
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1532
For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.
Exact mathematical passage
\dot x_P(t) =A_{PP}x_P(t)+A_{PQ}e^{tA_{QQ}}x_Q(0) +\int_0^t\cK(t-r)x_P(r)\,dr,
\dot yUse in source · 2 passages
Search aliases: dot y dot
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4801
[Finite auxiliary state realization] Every kernel of the form K(t)=B e^tDC can be represented exactly by adjoining the finite auxiliary state dot y=Dy+Cx_P, dot x_P=A_PPx_P+By, y(0)=0. A nonzero auxiliary initial state adds the explicit forcing B e^tDy(0). The self energy of the augmented system is B(zI-D)^-1C.
Exact mathematical passage
\dot y=Dy+Cx_P, \qquad \dot x_P=A_{PP}x_P+By, \qquad y(0)=0.
\dot y_1Use in source · 1 passage
Search aliases: dot y_1 dot
Use in source · atlas-pages.tex, line 895
vpage39Two auxiliary states realize a repeated poleFINITE REALIZATION / AN EXACT DIFFERENTIAL MODELTraditional Part II: Corollaries E15.4 and E15.5. This is a supplied rational realization. The kernel K(t)=te^-2t from V--23 has transform 1/(z+2)^2. It is realized by two auxiliary states: dot y_1 · =-2y_1+y_2, · y_1(0) · =0, dot y_2 · =-2y_2+x, · y_2(0) · =0, dot x · =a x+y_1. · With D=mat-2 · 1
Exact mathematical passage
\begin{aligned} \dot y_1&=-2y_1+y_2,& y_1(0)&=0,\\ \dot y_2&=-2y_2+x,& y_2(0)&=0,\\ \dot x&=a x+y_1.& \end{aligned}
\dot y_2Use in source · 1 passage
Search aliases: dot y_2 dot
Use in source · atlas-pages.tex, line 895
vpage39Two auxiliary states realize a repeated poleFINITE REALIZATION / AN EXACT DIFFERENTIAL MODELTraditional Part II: Corollaries E15.4 and E15.5. This is a supplied rational realization. The kernel K(t)=te^-2t from V--23 has transform 1/(z+2)^2. It is realized by two auxiliary states: dot y_1 · =-2y_1+y_2, · y_1(0) · =0, dot y_2 · =-2y_2+x, · y_2(0) · =0, dot x · =a x+y_1. · With D=mat-2 · 1
Exact mathematical passage
\begin{aligned} \dot y_1&=-2y_1+y_2,& y_1(0)&=0,\\ \dot y_2&=-2y_2+x,& y_2(0)&=0,\\ \dot x&=a x+y_1.& \end{aligned}
\dot{\widetilde x}Use in source · 1 passage
Search aliases: dotwidetilde x dot widetilde
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5681
entiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II. [Stability of memory compression] Let x and widetilde x solve dot x(t) =Ax(t)+∫_0^t K(t-s)x(s) d s+f(t) and dotwidetilde x(t) =Awidetilde x(t) +∫_0^twidetilde K(t-s)widetilde x(s) d s +f(t) with the same initial value. Fix T>0, and assume sup_0≤ t≤ Tnormwidetilde x(t)≤ M. Set a=norm A, k=norm K_L^1(0,T), δ=norm K-widetil…
Exact mathematical passage
\dot{\widetilde x}(t) =A\widetilde x(t) +\int_0^t\widetilde\Kcal(t-s)\widetilde x(s)\dd s +f(t)
\downarrowUse in source · 2 passages
Search aliases: downarrow
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2476
ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x_ℓ-x_ℓ=0 for every (x_ℓ)_ℓ∈ X_∞.
Exact mathematical passage
\lim_{t\downarrow0} \sup_\ell \norm{T_\ell(t)x_\ell-x_\ell}=0
EUse in source · 27 passages
Search aliases: E
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1655
For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. I…
Exact mathematical passage
E_\mu=\ker(A-\mu I)^d
E^{\mathsf T}Use in source · 2 passages
Search aliases: E^ T mathsf
Use in source · atlas-pages.tex, line 160
)^ T becomes Ef=(2,2,26/7,26/7)^ T. vfigweighted_projection platecaptionThe bars are function values, not state probabilities. Their total need not be one. The block masses 3/10 and 7/10 determine the denominators in the averaging formula. headWhich inner product makes this an orthogonal projection? Put W=diag(π). Direct multiplication checks E^2=E, WE=E^ TW, Eones=ones. The middle equation is weighted self-adjointness. Ordinary symmetry fails: E_01=2/3 whereas E_10=1/3. Testing E=E^ T would the…
Exact mathematical passage
E^2=E,\qquad WE=E^{\mathsf T}W,\qquad E\ones=\ones.
\e^{(a+k)T}Use in source · 1 passage
Search aliases: e^(a+k)T e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5700
e x solve dot x(t) =Ax(t)+∫_0^t K(t-s)x(s) d s+f(t) and dotwidetilde x(t) =Awidetilde x(t) +∫_0^twidetilde K(t-s)widetilde x(s) d s +f(t) with the same initial value. Fix T>0, and assume sup_0≤ t≤ Tnormwidetilde x(t)≤ M. Set a=norm A, k=norm K_L^1(0,T), δ=norm K-widetilde K_L^1(0,T). Then sup_0≤ t≤ T normx(t)-widetilde x(t) ≤ MTδ e^(a+k)T.
Exact mathematical passage
\boxed{ \sup_{0\leq t\leq T} \norm{x(t)-\widetilde x(t)} \leq MT\delta\e^{(a+k)T}. }
\e^{(a+k)t}Use in source · 1 passage
Search aliases: e^(a+k)t e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5719
Let e=x-widetilde x. Integrating the difference equation and changing the order of integration gives norme(t) ≤ (a+k)∫_0^tnorme(r) d r +Mtδ. Since t≤ T, Gronwall's inequality gives norme(t) ≤ MTδ e^(a+k)t. Take the supremum over [0,T].
Exact mathematical passage
\norm{e(t)} \leq MT\delta\e^{(a+k)t}.
e^{(s+t)A}Use in source · 2 passages
Search aliases: e^(s+t)A
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 334
[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.
Exact mathematical passage
\boxed{e^{(s+t)A}=e^{sA}e^{tA}.}
e^{(s+t)g}Use in source · 1 passage
Search aliases: e^(s+t)g
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1056
The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.
Exact mathematical passage
e^{sg}e^{tg}=e^{(s+t)g}
e^{(s+t)Q}Use in source · 1 passage
Search aliases: e^(s+t)Q
Use in source · atlas-pages.tex, line 90
Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.
Exact mathematical passage
P_b e^{(s+t)Q}P_a=\sum_{c\in\{\mathrm A,\mathrm B\}}(P_b e^{sQ}P_c)(P_c e^{tQ}P_a).
\e^{(s_m-s_{m-1})D}Use in source · 1 passage
Search aliases: e^(s_m-s_m-1)D e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2235
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
e^{(t-r)A_c}Use in source · 3 passages
Search aliases: e^(t-r)A_c
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 462
[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.
Exact mathematical passage
\boxed{ Ce^{tA_f}-e^{tA_c}C = \int_0^t e^{(t-r)A_c}\Delta e^{rA_f}\,dr. }
e^{(t-r)A_{QQ}}Use in source · 1 passage
Search aliases: e^(t-r)A_QQ
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1527
For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.
Exact mathematical passage
x_Q(t)=e^{tA_{QQ}}x_Q(0) +\int_0^te^{(t-r)A_{QQ}}A_{QP}x_P(r)\,dr.
e^{(t-r)D}Use in source · 1 passage
Search aliases: e^(t-r)D
Use in source · atlas-pages.tex, line 907
0 · -2, B=(1,0), and C=(0,1)^ T, the feedback kernel is Be^tDC=te^-2t. vfigext_auxiliary platecaptionAn input first enters y_2, transfers to y_1, and then returns to x. The two equal decay rates produce a repeated pole through the intervening link. headElimination recovers the convolution, including its preparation Variation of constants gives y(t)=e^tDy(0)+∫_0^t e^(t-r)DCx(r) dr, so the retained equation is dot x(t)=a x(t)+Be^tDy(0) +∫_0^t (t-r)e^-2(t-r)x(r) dr. The convolution-only version is …
Exact mathematical passage
y(t)=e^{tD}y(0)+\int_0^t e^{(t-r)D}Cx(r)\,dr,
\e^{(t-s)(QAQ)|_{QX}}Use in source · 1 passage
Search aliases: e^(t-s)(QAQ)|_QX e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2785
For x∈ PX, write p(t)=PT(t)x, q(t)=QT(t)x. The hidden component satisfies q'(t)=QAQq(t)+QAPp(t), q(0)=0. Variation of constants gives q(t)= ∫_0^t e^(t-s)(QAQ)|_QX QAPp(s) d s. Since normp(s)≤norm PnormT(s)norm x and x=Px, the first estimate follows. The contraction case is immediate.
Exact mathematical passage
q(t)= \int_0^t \e^{(t-s)(QAQ)|_{QX}} QAPp(s) \dd s.
\e^{(t-s)A_c}Use in source · 3 passages
Search aliases: e^(t-s)A_c e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2289
[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\mathfrak D(t) = \int_0^t \e^{(t-s)A_c}\Delta\e^{sA_f} \dd s.
\e^{(t-s)A_{QQ}}Use in source · 1 passage
Search aliases: e^(t-s)A_QQ e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4518
For the homogeneous equation dot x=Ax, variation of constants gives x_Q(t) = e^tA_QQx_Q(0) + ∫_0^t e^(t-s)A_QQA_QPx_P(s) d s. Substitution into the retained equation gives dot x_P(t) = · A_PPx_P(t) + A_PQ e^tA_QQx_Q(0)
Exact mathematical passage
x_Q(t) = \e^{tA_{QQ}}x_Q(0) + \int_0^t \e^{(t-s)A_{QQ}}A_{QP}x_P(s) \dd s.
\e^{(t-s)D}Use in source · 1 passage
Search aliases: e^(t-s)D e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2253
Write the variation of constants identity e^tA = e^tD + ∫_0^t e^(t-s)DN e^sA d s. Iterating the identity gives the displayed series. The norm of the term of order m is at most e^tnorm Dfrac(tnorm N)^mm!. The majorant is summable. Inserting the packet resolution around every factor N leaves only products whose consecutive packet labels are joined by nonzero transfer blocks.
Exact mathematical passage
\e^{tA} = \e^{tD} + \int_0^t\e^{(t-s)D}N\e^{sA}\dd s.
\e^{(t-s_m)D}Use in source · 1 passage
Search aliases: e^(t-s_m)D e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2235
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
e^{(\lambda-z)t}Use in source · 1 passage
Search aliases: e^(λ-z)t lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1558
Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace; equivalently, apply the decomposition to left multiplication by B on the matrix algebra and evaluate the exponential at I. The strict spectral inequality makes every such term absolutely integrable on (0,∞) and forces it to tend to zero. H…
Exact mathematical passage
t^k e^{(\lambda-z)t}
e^{(\mu-z)t}Use in source · 1 passage
Search aliases: e^(μ-z)t mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1930
General nonnormal memory (13.2). The actual damped exponential is decomposed into finitely many terms of the form t^k e^(μ-z)t. Integrability and convergence to zero follow from Reμ<Rez. The formal proof uses generalized eigenspaces of left multiplication on the matrix algebra, so nilpotent Jordan parts are included. Both inverse identities follow by integrating the two exponential derivative formulas. A continuous linear sandwich map gives the literal rectangular matrix memory integral and the …
Exact mathematical passage
t^k e^{(\mu-z)t}
\e^{-(s+t)}Use in source · 1 passage
Search aliases: e^-(s+t) e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3156
Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p contraction. The joint Gaussian pair ( X, e^-tX+sqrt1- e^-2tY ) is symmetric under interchange, which proves self adjointness.
Exact mathematical passage
\e^{-(s+t)}x + \sqrt{1-\e^{-2(s+t)}}Z
e^{-(s-r)B^\alpha}Use in source · 2 passages
Search aliases: e^-(s-r)B^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1296
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ Ce^{-sA^\alpha}-e^{-sB^\alpha}C = -\int_0^s e^{-(s-r)B^\alpha}D_\alpha e^{-rA^\alpha}\,dr. }
e^{-(t-r)}Use in source · 2 passages
Search aliases: e^-(t-r)
Use in source · atlas-pages.tex, line 493
vpage22Hidden state returns as memoryEXACT RETAINED DYNAMICS / NOT A SCALAR SEMIGROUPTraditional Part I: Theorem 13.2. The retained-state memory identity, including hidden initial data. Write the same system as x'=-x+y, y'=x-y. Solving the second equation first gives y(t)=e^-ty(0)+∫_0^t e^-(t-r)x(r) dr. Substitution is exact: x'(t)=-x(t)+e^-ty(0)+∫_0^t e^-(t-r)x(r) dr. The hidden initial state produces a separate forcing term. It must be retained even when the memory kernel is known. vfigmemory …
Exact mathematical passage
y(t)=e^{-t}y(0)+\int_0^t e^{-(t-r)}x(r)\,dr.
e^{-(z+2)t}Use in source · 1 passage
Search aliases: e^-(z+2)t
Use in source · atlas-pages.tex, line 529
t. The nilpotent link carries an input from the second hidden coordinate to the first. vfigjordan platecaptionThe memory kernel starts at zero, becomes positive, and peaks at t=1/2. A rule that retained the eigenvalue -2 but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps. headThe transfer function sees pole order The exact transform is U(zI-J)^-1V=frac1(z+2)^2 =∫_0^∞ t e^-(z+2)t dt, Rez>-2. The repeated pole and the polynomial factor t are the same info…
Exact mathematical passage
U(zI-J)^{-1}V=\frac1{(z+2)^2} =\int_0^\infty t e^{-(z+2)t}\,dt, \qquad \operatorname{Re}z>-2.
\e^{-2(s+t)}Use in source · 1 passage
Search aliases: e^-2(s+t) e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3156
Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p contraction. The joint Gaussian pair ( X, e^-tX+sqrt1- e^-2tY ) is symmetric under interchange, which proves self adjointness.
Exact mathematical passage
\e^{-(s+t)}x + \sqrt{1-\e^{-2(s+t)}}Z
e^{-2(s+t)}Use in source · 1 passage
Search aliases: e^-2(s+t)
Use in source · atlas-pages.tex, line 66
A point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory. headTwo checks explain the mechanism The constant mode is unchanged, while each elapsed interval multiplies the contrast by e^-2t. Consequently two intervals multiply their contrast factors: e^-2se^-2t=e^-2(s+t), T(s)T(t)=T(s+t). For t≥0, the entries are nonnegative and every column sums to one. Diffe…
Exact mathematical passage
e^{-2s}e^{-2t}=e^{-2(s+t)},\qquad T(s)T(t)=T(s+t).
e^{-2(t-r)}Use in source · 2 passages
Search aliases: e^-2(t-r)
Use in source · atlas-pages.tex, line 142
The continuous counterpart is equally explicit. With fine generator -1, coarse generator -2, and C=1, the generator defect is 1 and e^-t-e^-2t=∫_0^t e^-2(t-r)e^-r dr. Its resolvent version is 1/(z+1)-1/(z+2)=1/((z+1)(z+2)) for z>0. takeawayA small defect does not disappear. The identity specifies how it accumulates, and the norm assumptions turn that identity into a finite-horizon certificate.
Exact mathematical passage
e^{-t}-e^{-2t}=\int_0^t e^{-2(t-r)}e^{-r}\,dr.
e^{-2jt}Use in source · 1 passage
Search aliases: e^-2jt
Use in source · atlas-pages.tex, line 330
itive self-adjoint finite model and its spectral powers. Return to the four-state generator on pagerefV4 and put L=-Q. It is real symmetric and nonnegative, with orthogonal projectors P_0,P_1,P_2,P_3 and eigenvalues 0,2,4,6. Its square root is L^1/2=sqrt2P_1+2P_2+sqrt6P_3, (L^1/2)^2=L. Both heat evolutions retain these modes. Ordinary diffusion multiplies mode j by e^-2jt; square-root diffusion multiplies it by e^-sqrt2j t. The constant mode is unchanged in both. vfigfractional platecaptionThe h…
Exact mathematical passage
e^{-2jt}
\e^{-2nt}Use in source · 1 passage
Search aliases: e^-2nt e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3204
[Hermite packet reconstruction] Let P_n be the orthogonal projection onto C h_n. Then I=∑_n=0^∞ P_n in the strong operator topology and O_t = ∑_n=0^∞ e^-ntP_n. For f∈ L^2(γ), norm O_t f_2^2 = ∑_n=0^∞ e^-2ntnormP_nf_2^2.
Exact mathematical passage
\norm{\Ocal_t f}_2^2 = \sum_{n=0}^\infty \e^{-2nt}\norm{P_nf}_2^2.
\e^{-2s}Use in source · 1 passage
Search aliases: e^-2s e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3147
Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p contraction. The joint Gaussian pair ( X, e^-tX+sqrt1- e^-2tY ) is symmetric under interchange, which proves self adjointness.
Exact mathematical passage
\e^{-s} \left( \e^{-t}x+\sqrt{1-\e^{-2t}}X \right) + \sqrt{1-\e^{-2s}}Y
e^{-2s}Use in source · 2 passages
Search aliases: e^-2s
Use in source · atlas-pages.tex, line 66
A point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory. headTwo checks explain the mechanism The constant mode is unchanged, while each elapsed interval multiplies the contrast by e^-2t. Consequently two intervals multiply their contrast factors: e^-2se^-2t=e^-2(s+t), T(s)T(t)=T(s+t). For t≥0, the entries are nonnegative and every column sums to one. Diffe…
Exact mathematical passage
e^{-2s}e^{-2t}=e^{-2(s+t)},\qquad T(s)T(t)=T(s+t).
\e^{-2t}Use in source · 5 passages
Search aliases: e^-2t e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3120
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
(\Ocal_t f)(x) = \int_\RR f\left( \e^{-t}x+\sqrt{1-\e^{-2t}}\,y \right) \gamma(\dd y).
e^{-2t}Use in source · 22 passages
Search aliases: e^-2t
Use in source · atlas-pages.tex, line 59
1 · -1. The total-mass direction (1,1) has eigenvalue zero. The contrast (1,-1) has eigenvalue -2. Resolving those two directions gives the full transfer, not only its first-order approximation: T(t)=e^tG=frac121+e^-2t · 1-e^-2t 1-e^-2t · 1+e^-2t. vfigclock platecaptionA point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory. headTwo checks explain the mecha…
Exact mathematical passage
T(t)=e^{tG}=\frac12\begin{pmatrix}1+e^{-2t}&1-e^{-2t}\\1-e^{-2t}&1+e^{-2t}\end{pmatrix}.
e^{-3t}Use in source · 1 passage
Search aliases: e^-3t
Use in source · atlas-pages.tex, line 850
vpage37Keep blocks by their certified contributionMEMORY COMPRESSION / AN OPTIMIZATION WITH A DEFINED COSTTraditional Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks; block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified integrated weights are w_1=frac14, w_2=frac32+frac22^2=2, w…
Exact mathematical passage
\mathcal K(t)=\tfrac14e^{-t}+(3+2t)e^{-2t}+\tfrac13e^{-3t}.
e^{-4t}Use in source · 1 passage
Search aliases: e^-4t
Use in source · atlas-pages.tex, line 762
=-nH_n for these three polynomials. Their squared norms are n! and different Hermite degrees are orthogonal. vfigext_hermite platecaptionFor f=H_1+H_2, the two bars are its separate squared-energy contributions. At t=log2, their sum is 3/8, below the spectral-gap bound 3/4. headReconstruction preserves the modes; time changes their weights The semigroup sends f=H_1+H_2 to T(t)f=e^-tH_1+e^-2tH_2, T(t)f _2^2=e^-2t+2e^-4t. Its mean is zero. The Gaussian gap bounds this by e^-2t f _2^2=3e^-2t. It ne…
Exact mathematical passage
T(t)f=e^{-t}H_1+e^{-2t}H_2,\qquad \|T(t)f\|_2^2=e^{-2t}+2e^{-4t}.
e^{-6t}Use in source · 1 passage
Search aliases: e^-6t
Use in source · fractional.tex, line 48
) (12.144,0.650009) (12.21,0.650008) (12.276,0.650007) (12.342,0.650006) (12.408,0.650005) (12.474,0.650004) (12.54,0.650004) (12.606,0.650003) (12.672,0.650003) (12.738,0.650002) (12.804,0.650002) (12.87,0.650002) (12.936,0.650002) (13.002,0.650001) (13.068,0.650001) (13.134,0.650001) (13.2,0.650001); node[font=small,align=center,font=scriptsize] at (10,4.45) solid: e^-tsqrt6; dashed: e^-6t;
Exact mathematical passage
e^{-6t}
e^{-8t}Use in source · 1 passage
Search aliases: e^-8t
Use in source · atlas-pages.tex, line 95
For later discrete calculations define K=I+Q/8. This matrix is stochastic, but it is not e^Q/8. It is the step kernel in the uniformization identity e^tQ=e^-8t∑_n≥0(8t)^nK^n/n!. takeawayChanging packet coordinates does not change the evolution. Replacing continuous time by one discrete step does.
Exact mathematical passage
e^{tQ}=e^{-8t}\sum_{n\ge0}(8t)^nK^n/n!
e^{-a_nt}Use in source · 1 passage
Search aliases: e^-a_nt
Use in source · atlas-pages.tex, line 712
vpage31A tail bound states what refinement buysSUMMABLE DEFECTS / A SCALAR COMPLETIONTraditional Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps. Take X_n= R, C_n=R_n=1, and a_n=1-2^-n, A_n=-a_n, T_n(t)=e^-a_nt (n≥0). Each semigroup is contractive for t≥0. The exact adjacent generator-defect magnitude is |A_n+1-A_n|=2^-(n+1), ∑_j=n^∞|A_j+1-A_j|=2^-n. vfigext_tails platecaptionThe bars show the certified tail 2^-n at horizon T=1. They are bounds on a whole time…
Exact mathematical passage
a_n=1-2^{-n},\qquad A_n=-a_n,\qquad T_n(t)=e^{-a_nt}\quad(n\ge0).
\e^{-at}Use in source · 1 passage
Search aliases: e^-at e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2837
Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.
Exact mathematical passage
\e^{tQ}=\e^{-at}\e^{t(Q+aI)}
e^{-at}Use in source · 1 passage
Search aliases: e^-at
Use in source · atlas-pages.tex, line 723
0. The exact adjacent generator-defect magnitude is |A_n+1-A_n|=2^-(n+1), ∑_j=n^∞|A_j+1-A_j|=2^-n. vfigext_tails platecaptionThe bars show the certified tail 2^-n at horizon T=1. They are bounds on a whole time interval, not errors sampled at the right endpoint. headThe estimate follows before any graph is drawn For 0≤ t≤ T, the derivative of a↦ e^-at has absolute value at most T when a≥0. Thus sup_0≤ t≤ T|T_n(t)-e^-t|≤ T2^-n. The limit is e^-t, which does have the semigroup law. Here every reco…
Exact mathematical passage
a\mapsto e^{-at}
\e^{-bt}Use in source · 1 passage
Search aliases: e^-bt e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5869
d · -b , a,b>0, and retain the first coordinate. The memory kernel and self energy are K(t)=cd e^-bt, Σ(z)=fraccdz+b. The retained coordinate has an exact time local closure for every retained initial value precisely when cd=0. When cd≠0, one auxiliary scalar state realizes the memory exactly.
Exact mathematical passage
\Kcal(t)=cd\e^{-bt}, \qquad \Sigma(z)=\frac{cd}{z+b}.
\e^{-nt}Use in source · 4 passages
Search aliases: e^-nt e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3139
[Ornstein Uhlenbeck semigroup] The operators ( O_t)_t≥0 form a positive Markov contraction semigroup on L^p(γ) for every 1≤ p≤∞. On L^2(γ), the semigroup is self adjoint and O_t h_n= e^-nth_n. The family (h_n)_n≥0 is an orthonormal basis of L^2(γ).
Exact mathematical passage
\Ocal_t h_n=\e^{-nt}h_n.
e^{-nt}Use in source · 1 passage
Search aliases: e^-nt
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 716
[Gaussian chaos realization] For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally: T_tH_n=e^-ntH_n. Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.
Exact mathematical passage
T_tH_n=e^{-nt}H_n.
e^{-r}Use in source · 1 passage
Search aliases: e^-r
Use in source · atlas-pages.tex, line 142
The continuous counterpart is equally explicit. With fine generator -1, coarse generator -2, and C=1, the generator defect is 1 and e^-t-e^-2t=∫_0^t e^-2(t-r)e^-r dr. Its resolvent version is 1/(z+1)-1/(z+2)=1/((z+1)(z+2)) for z>0. takeawayA small defect does not disappear. The identity specifies how it accumulates, and the norm assumptions turn that identity into a finite-horizon certificate.
Exact mathematical passage
e^{-t}-e^{-2t}=\int_0^t e^{-2(t-r)}e^{-r}\,dr.
e^{-rA^\alpha}Use in source · 2 passages
Search aliases: e^-rA^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1296
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ Ce^{-sA^\alpha}-e^{-sB^\alpha}C = -\int_0^s e^{-(s-r)B^\alpha}D_\alpha e^{-rA^\alpha}\,dr. }
\e^{-s}Use in source · 1 passage
Search aliases: e^-s e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3147
Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p contraction. The joint Gaussian pair ( X, e^-tX+sqrt1- e^-2tY ) is symmetric under interchange, which proves self adjointness.
Exact mathematical passage
\e^{-s} \left( \e^{-t}x+\sqrt{1-\e^{-2t}}X \right) + \sqrt{1-\e^{-2s}}Y
e^{-s}Use in source · 1 passage
Search aliases: e^-s
Use in source · atlas-pages.tex, line 931
/2 and f_2(t)=e^-2t. vfigext_risks platecaptionAt t=log2, surviving mass is 3/8. Cause 1 has accumulated probability 1/4 and cause 2 has accumulated probability 3/8; the three pieces sum to one. headCondition on survival before restarting the clock At this time the transient masses are (1/4,1/8), so the conditional distribution of survivors is (2/3,1/3). Consequently fracS(t+s)S(t)=frac23e^-s+frac13e^-2s, E[τ-tmidτ>t]=frac23+frac16=frac56. The original mean lifetime was tfrac12·1+tfrac12·tfrac12…
Exact mathematical passage
\frac{S(t+s)}{S(t)}=\frac23e^{-s}+\frac13e^{-2s}, \qquad \EE[\tau-t\mid\tau>t]=\frac23+\frac16=\frac56.
e^{-sA^\alpha}Use in source · 3 passages
Search aliases: e^-sA^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1296
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ Ce^{-sA^\alpha}-e^{-sB^\alpha}C = -\int_0^s e^{-(s-r)B^\alpha}D_\alpha e^{-rA^\alpha}\,dr. }
e^{-sB^\alpha}Use in source · 3 passages
Search aliases: e^-sB^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1296
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ Ce^{-sA^\alpha}-e^{-sB^\alpha}C = -\int_0^s e^{-(s-r)B^\alpha}D_\alpha e^{-rA^\alpha}\,dr. }
\e^{-sL}Use in source · 2 passages
Search aliases: e^-sL e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4083
nstandardfact[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\e^{-tL^\alpha} = \int_0^\infty \e^{-sL}\eta_{\alpha,t}(\dd s)
e^{-sL}Use in source · 2 passages
Search aliases: e^-sL
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1166
[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.
Exact mathematical passage
\boxed{ e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)}e^{-sL}\,ds. }
e^{-sm^\alpha}Use in source · 2 passages
Search aliases: e^-sm^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1310
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ \norm{Ce^{-sA^\alpha}-e^{-sB^\alpha}C} \le s e^{-sm^\alpha}\alpha m^{\alpha-1}\norm D. }
e^{-\sqrt{2j}\,t}Use in source · 1 passage
Search aliases: e^-sqrt2j t sqrt
Use in source · atlas-pages.tex, line 330
wers. Return to the four-state generator on pagerefV4 and put L=-Q. It is real symmetric and nonnegative, with orthogonal projectors P_0,P_1,P_2,P_3 and eigenvalues 0,2,4,6. Its square root is L^1/2=sqrt2P_1+2P_2+sqrt6P_3, (L^1/2)^2=L. Both heat evolutions retain these modes. Ordinary diffusion multiplies mode j by e^-2jt; square-root diffusion multiplies it by e^-sqrt2j t. The constant mode is unchanged in both. vfigfractional platecaptionThe highest mode decays at rates 6 and sqrt6. The graph …
Exact mathematical passage
e^{-\sqrt{2j}\,t}
\e^{-st}Use in source · 1 passage
Search aliases: e^-st e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4996
The Laplace transform follows by integrating the density in [source reference]. Stability of A gives ∫_0^∞ e^-st e^tA d t = (sI-A)^-1. At s=0, a^T(-A)^-1p_0 = 1^Tp_0=1. For the moments, ∫_0^∞ t^m e^tA d t = m!(-A)^-m-1. Multiplying by a^T= 1^T(-A) gives the result. These are the standard matrix formulas for phase type laws [source reference].
Exact mathematical passage
\int_0^\infty \e^{-st}\e^{tA}\dd t = (sI-A)^{-1}.
e^{-st}Use in source · 1 passage
Search aliases: e^-st
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5770
For the survival consequences, A is a finite Metzler matrix with nonpositive column sums and spectrum in the open left half-plane, p_0 is a probability vector, S(t)= 1^T e^tAp_0, and f(t)=-S'(t). The one-phase model has λ>0. The survival Laplace identity of Part II also holds for every complex s with Res≥0: stability gives absolute convergence of ∫_0^∞ e^-ste^tA dt=(sI-A)^-1, and multiplying by the exit row and p_0 gives the same transform. [Phase type rationality] The Laplace transform of every…
Exact mathematical passage
\int_0^\infty e^{-st}e^{tA}\,dt=(sI-A)^{-1}
\e^{-s\lambda}Use in source · 1 passage
Search aliases: e^-sλ e lambda mathrm lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4077
[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\int_0^\infty \e^{-s\lambda}\eta_{\alpha,t}(\dd s) = \e^{-t\lambda^\alpha}
\e^{-s\tau}Use in source · 1 passage
Search aliases: e^-sτ e tau mathrm tau
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4982
[Laplace transform and moments] Assume the spectrum of A lies in the open left half plane. Then absorption occurs with probability one. For s≥0, E[ e^-sτ] = a^T(sI-A)^-1p_0. For every integer m≥1, E[τ^m] = m! 1^T(-A)^-mp_0.
Exact mathematical passage
\Exp[\e^{-s\tau}] = a^T(sI-A)^{-1}p_0.
\e^{-t}Use in source · 7 passages
Search aliases: e^-t e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3120
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
(\Ocal_t f)(x) = \int_\RR f\left( \e^{-t}x+\sqrt{1-\e^{-2t}}\,y \right) \gamma(\dd y).
e^{-t}Use in source · 15 passages
Search aliases: e^-t
Use in source · atlas-pages.tex, line 142
The continuous counterpart is equally explicit. With fine generator -1, coarse generator -2, and C=1, the generator defect is 1 and e^-t-e^-2t=∫_0^t e^-2(t-r)e^-r dr. Its resolvent version is 1/(z+1)-1/(z+2)=1/((z+1)(z+2)) for z>0. takeawayA small defect does not disappear. The identity specifies how it accumulates, and the norm assumptions turn that identity into a finite-horizon certificate.
Exact mathematical passage
e^{-t}-e^{-2t}=\int_0^t e^{-2(t-r)}e^{-r}\,dr.
\e^{-t(-\Lcal)^\alpha}Use in source · 1 passage
Search aliases: e^-t(-Lcal)^α e^-t(- L)^α e Lcal alpha mathrm mathcal alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3306
[Fractional Hermite transfer] Let - L be the nonnegative Ornstein Uhlenbeck number operator with - L h_n=nh_n. For every α>0, e^-t(- L)^αf = ∑_n=0^∞ e^-tn^αP_nf. This is a contraction semigroup on L^2(γ).
Exact mathematical passage
\e^{-t(-\Lcal)^\alpha}f = \sum_{n=0}^\infty \e^{-tn^\alpha}P_nf.
\e^{-t(k\pi)^{2\alpha}}Use in source · 1 passage
Search aliases: e^-t(kπ)^2α e pi alpha mathrm pi alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5593
set λ_k,h=4h^-2sin^2(kπ h/2). The retained explicit coefficient below is valid for every discrete mode. [Fractional low mode dispersion] Let 0<α≤1. Then 0 ≤ (kπ)^2α-λ_k,h^α ≤ fracα(4k^2)^α-1(kπ)^412h^2. For every t≥0, abs e^-tλ_k,h^α - e^-t(kπ)^2α ≤ fractα(4k^2)^α-1(kπ)^412h^2.
Exact mathematical passage
\abs{ \e^{-t\lambda_{k,h}^\alpha} - \e^{-t(k\pi)^{2\alpha}} } \leq \frac{t\alpha(4k^2)^{\alpha-1}(k\pi)^4}{12}h^2.
\e^{-t^2/(4s)}Use in source · 1 passage
Search aliases: e^-t^2/(4s) e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4101
[Poisson subordination] For t>0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s) e^-sL d s. The integral converges in operator norm.
Exact mathematical passage
\e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2} \e^{-t^2/(4s)} \e^{-sL} \dd s.
e^{-t^2/(4s)}Use in source · 2 passages
Search aliases: e^-t^2/(4s)
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1166
[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.
Exact mathematical passage
\boxed{ e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)}e^{-sL}\,ds. }
\e^{-t^2/(4s)-\lambda s}Use in source · 1 passage
Search aliases: e^-t^2/(4s)-λ s e lambda mathrm lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4116
On a spectral packet with eigenvalue λ, the formula reduces to e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s)-λ s d s. This scalar identity follows from the Gaussian integral. Recombine the finitely many spectral packets.
Exact mathematical passage
\e^{-t\sqrt\lambda} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}\e^{-t^2/(4s)-\lambda s}\dd s.
e^{-t^2/(4s)-\lambda s}Use in source · 1 passage
Search aliases: e^-t^2/(4s)-λ s lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1180
By the spectral theorem it suffices to prove the scalar identity e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)-λ s ds for λ≥0. This is the classical Gaussian integral after the substitution u=t/(2sqrt s). Applying the identity to every eigenvalue and recombining the spectral projectors proves the operator formula.
Exact mathematical passage
e^{-t\sqrt\lambda} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)-\lambda s}\,ds
\e^{-ta}Use in source · 1 passage
Search aliases: e^-ta e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5613
proof The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.
Exact mathematical passage
\abs{\e^{-ta}-\e^{-tb}} \leq t\abs{a-b}
\e^{-tb}Use in source · 1 passage
Search aliases: e^-tb e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5613
proof The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.
Exact mathematical passage
\abs{\e^{-ta}-\e^{-tb}} \leq t\abs{a-b}
\e^{-tD_V/n}Use in source · 1 passage
Search aliases: e^-tD_V/n e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4419
[Alternating propagation and killing] For every t≥0, e^t(Q^T-D_V) = lim_n→∞ ( e^-tD_V/n e^tQ^T/n )^n in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.
Exact mathematical passage
\e^{t(Q^T-D_V)} = \lim_{n\to\infty} \left( \e^{-tD_V/n}\e^{tQ^T/n} \right)^n
e^{-tL}Use in source · 3 passages
Search aliases: e^-tL
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 303
Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usually written with e^tA; positive diffusion and elliptic operators are usually written with e^-tL.
Exact mathematical passage
H(t)=e^{-tL}.
\e^{-tL^\alpha}Use in source · 4 passages
Search aliases: e^-tL^α e alpha mathrm alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2002
For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional semigroup defect. This closes a finite version of fractional naturality under explicit spectral hypotheses.
Exact mathematical passage
\e^{-tL^\alpha}
e^{-tL^\alpha}Use in source · 2 passages
Search aliases: e^-tL^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1129
Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.
Exact mathematical passage
T_\alpha(t)=e^{-tL^\alpha} =\sum_je^{-t\lambda_j^\alpha}P_j.
\e^{-tL_c^\alpha}Use in source · 2 passages
Search aliases: e^-tL_c^α e alpha mathrm alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3990
[Exact fractional naturality] Let L_f and L_c be finite dimensional positive self adjoint operators and let C satisfy CL_f=L_cC. Then, for every α>0 and t≥0, CL_f^α=L_c^α C and C e^-tL_f^α = e^-tL_c^αC.
Exact mathematical passage
C\e^{-tL_f^\alpha} = \e^{-tL_c^\alpha}C.
\e^{-tL_f^\alpha}Use in source · 2 passages
Search aliases: e^-tL_f^α e alpha mathrm alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3990
[Exact fractional naturality] Let L_f and L_c be finite dimensional positive self adjoint operators and let C satisfy CL_f=L_cC. Then, for every α>0 and t≥0, CL_f^α=L_c^α C and C e^-tL_f^α = e^-tL_c^αC.
Exact mathematical passage
C\e^{-tL_f^\alpha} = \e^{-tL_c^\alpha}C.
\e^{-tn^\alpha}Use in source · 1 passage
Search aliases: e^-tn^α e alpha mathrm alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3306
[Fractional Hermite transfer] Let - L be the nonnegative Ornstein Uhlenbeck number operator with - L h_n=nh_n. For every α>0, e^-t(- L)^αf = ∑_n=0^∞ e^-tn^αP_nf. This is a contraction semigroup on L^2(γ).
Exact mathematical passage
\e^{-t(-\Lcal)^\alpha}f = \sum_{n=0}^\infty \e^{-tn^\alpha}P_nf.
\e^{-t\sqrt L}Use in source · 1 passage
Search aliases: e^-tsqrt L e sqrt mathrm sqrt
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4101
[Poisson subordination] For t>0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s) e^-sL d s. The integral converges in operator norm.
Exact mathematical passage
\e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2} \e^{-t^2/(4s)} \e^{-sL} \dd s.
e^{-t\sqrt L}Use in source · 2 passages
Search aliases: e^-tsqrt L sqrt
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1166
[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.
Exact mathematical passage
\boxed{ e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)}e^{-sL}\,ds. }
e^{-t\sqrt6}Use in source · 1 passage
Search aliases: e^-tsqrt6 sqrt
Use in source · fractional.tex, line 48
650013) (12.078,0.650011) (12.144,0.650009) (12.21,0.650008) (12.276,0.650007) (12.342,0.650006) (12.408,0.650005) (12.474,0.650004) (12.54,0.650004) (12.606,0.650003) (12.672,0.650003) (12.738,0.650002) (12.804,0.650002) (12.87,0.650002) (12.936,0.650002) (13.002,0.650001) (13.068,0.650001) (13.134,0.650001) (13.2,0.650001); node[font=small,align=center,font=scriptsize] at (10,4.45) solid: e^-tsqrt6; dashed: e^-6t;
Exact mathematical passage
e^{-t\sqrt6}
\e^{-t\sqrt\lambda}Use in source · 1 passage
Search aliases: e^-tsqrtλ e sqrt lambda mathrm sqrt lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4116
On a spectral packet with eigenvalue λ, the formula reduces to e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s)-λ s d s. This scalar identity follows from the Gaussian integral. Recombine the finitely many spectral packets.
Exact mathematical passage
\e^{-t\sqrt\lambda} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}\e^{-t^2/(4s)-\lambda s}\dd s.
e^{-t\sqrt\lambda}Use in source · 2 passages
Search aliases: e^-tsqrtλ sqrt lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1180
By the spectral theorem it suffices to prove the scalar identity e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)-λ s ds for λ≥0. This is the classical Gaussian integral after the substitution u=t/(2sqrt s). Applying the identity to every eigenvalue and recombining the spectral projectors proves the operator formula.
Exact mathematical passage
e^{-t\sqrt\lambda} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)-\lambda s}\,ds
\e^{-tx^\alpha}Use in source · 1 passage
Search aliases: e^-tx^α e alpha mathrm alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3998
The union of the two finite spectra is a finite set. There is a polynomial that agrees with x^α on that set, and another polynomial that agrees with e^-tx^α. Exact intertwining passes to every polynomial by induction.
Exact mathematical passage
\e^{-tx^\alpha}
\e^{-t\lambda^\alpha}Use in source · 2 passages
Search aliases: e^-tλ^α e lambda alpha mathrm lambda alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4077
[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\int_0^\infty \e^{-s\lambda}\eta_{\alpha,t}(\dd s) = \e^{-t\lambda^\alpha}
\e^{-t\lambda_j^\alpha}Use in source · 1 passage
Search aliases: e^-tλ_j^α e lambda alpha mathrm lambda alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3929
Let H be a finite dimensional Hilbert space and let L=L^*≥0. Write its spectral resolution as L=∑_jλ_jP_j. For α>0, define L^α = ∑_jλ_j^α P_j and T_α(t) = e^-tL^α = ∑_j e^-tλ_j^αP_j.
Exact mathematical passage
T_\alpha(t) = \e^{-tL^\alpha} = \sum_j\e^{-t\lambda_j^\alpha}P_j.
e^{-t\lambda_j^\alpha}Use in source · 1 passage
Search aliases: e^-tλ_j^α lambda alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1129
Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.
Exact mathematical passage
T_\alpha(t)=e^{-tL^\alpha} =\sum_je^{-t\lambda_j^\alpha}P_j.
\e^{-t\lambda_k}Use in source · 1 passage
Search aliases: e^-tλ_k e lambda mathrm lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4312
[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.
Exact mathematical passage
0\leq \e^{-t\lambda_{k,h}}-\e^{-t\lambda_k} \leq \frac{t(k\pi)^4}{12}h^2\e^{-t\lambda_{k,h}}.
\e^{-t\lambda_{k,h}}Use in source · 1 passage
Search aliases: e^-tλ_k,h e lambda mathrm lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4312
[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.
Exact mathematical passage
0\leq \e^{-t\lambda_{k,h}}-\e^{-t\lambda_k} \leq \frac{t(k\pi)^4}{12}h^2\e^{-t\lambda_{k,h}}.
\e^{-t\lambda_{k,h}^\alpha}Use in source · 2 passages
Search aliases: e^-tλ_k,h^α e lambda alpha mathrm lambda alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4325
[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.
Exact mathematical passage
\e^{-t\lambda_{k,h}^\alpha} = \e^{-t\lambda_k^\alpha}+O(th^2)
\e^{-t\lambda_k^\alpha}Use in source · 1 passage
Search aliases: e^-tλ_k^α e lambda alpha mathrm lambda alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4325
[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.
Exact mathematical passage
\e^{-t\lambda_{k,h}^\alpha} = \e^{-t\lambda_k^\alpha}+O(th^2)
e^{-u^2}Use in source · 1 passage
Search aliases: e^-u^2
Use in source · atlas-pages.tex, line 385
al time; s is the ordinary diffusion time being mixed. They are not interchangeable variables. vfigrandom_clock platecaptionThe density η_1 peaks at s=1/6 but has a long right tail. The drawing ends at s=2; it does not claim that the missing tail has zero mass. headCheck the normalizer before interpreting a density Under s=t^2/(4u^2), reversing the integration limits gives η_t(s) |ds|=frac2sqrtπe^-u^2 du. Its integral over u>0 is one, so η_t is a probability density. On each eigenmode of L, the …
Exact mathematical passage
\eta_t(s)\,|ds|=\frac2{\sqrt\pi}e^{-u^2}\,du.
\e^{-V}Use in source · 1 passage
Search aliases: e^-V e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4353
[Discrete Feynman Kac formula] Define M_V=diag( e^-V)K^T. For terminal data f, (M_V^nf)_i = E_i[ exp( -∑_r=0^n-1V(X_r) ) f(X_n) ].
Exact mathematical passage
M_V=\diag(\e^{-V})K^T.
e^{-V}Use in source · 2 passages
Search aliases: e^-V
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1442
Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.
Exact mathematical passage
M_V=\diag(e^{-V})K^T.
\e^{-V(i)}Use in source · 1 passage
Search aliases: e^-V(i) e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4371
For one step, (M_Vf)_i = e^-V(i)∑_jK_jif_j. Iterating expands the matrix product over all paths. The product of transition entries is the path probability, and the diagonal factors give the exponential potential weight.
Exact mathematical passage
(M_Vf)_i = \e^{-V(i)}\sum_jK_{ji}f_j.
e^{-V(i)}Use in source · 2 passages
Search aliases: e^-V(i)
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1465
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
(M_Vf)_i=e^{-V(i)}\sum_jK_{j i}f_j,
e^{-V(i_r)}Use in source · 1 passage
Search aliases: e^-V(i_r)
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1468
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
\prod_re^{-V(i_r)}
\e^{-v_0t}Use in source · 2 passages
Search aliases: e^-v_0t e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4394
[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.
Exact mathematical passage
\norm{\e^{t(Q^T-D_V)}}_{\infty\to\infty} \leq \e^{-v_0t}.
\e^{-zt}Use in source · 10 passages
Search aliases: e^-zt e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2117
[Laplace resolvent and packet transform] Let A∈ B(X), let T(t)= e^tA, and let Rez>norm A. Then the Bochner integral converges in operator norm and (zI-A)^-1 = ∫_0^∞ e^-ztT(t) d t. For every pair of packets, P_β(zI-A)^-1P_α = ∫_0^∞ e^-ztT_βα(t) d t.
Exact mathematical passage
(zI-A)^{-1} = \int_0^\infty \e^{-zt}T(t)\dd t.
e^{-zt}Use in source · 2 passages
Search aliases: e^-zt
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1546
[Memory/self-energy correspondence] For finite complex block matrices, assume Rez>Reλ for every λ∈σ(A_QQ). Then the following improper integral is absolutely convergent and ∫_0^∞ e^-zt K(t) dt =A_PQ(zI-A_QQ)^-1A_QP =Σ_P(z).
Exact mathematical passage
\boxed{ \int_0^\infty e^{-zt}\cK(t)\,dt =A_{PQ}(zI-A_{QQ})^{-1}A_{QP} =\Sigma_P(z). }
\e^{-\gamma t}Use in source · 2 passages
Search aliases: e^-γ t e gamma mathrm gamma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4557
[Memory and self energy] If z lies to the right of the spectrum of A_QQ, then ∫_0^∞ e^-zt K(t) d t = Σ_P(z). If, for constants M≥0 and γ>0, norm e^tA_QQ ≤ M e^-γ t, then norm K(t) ≤ MnormA_PQnormA_QP e^-γ t and norm K_L^1(0,∞) ≤ fracMγ normA_PQnormA_QP.
Exact mathematical passage
\norm{\e^{tA_{QQ}}} \leq M\e^{-\gamma t},
\e^{-\delta t}Use in source · 2 passages
Search aliases: e^-δ t e delta mathrm delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3838
[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).
Exact mathematical passage
1-\sum_{i\in M}p_i(t) \leq \frac{1-p_M(0)}{p_M(0)}\e^{-\delta t},
e^{-\lambda s}Use in source · 1 passage
Search aliases: e^-λ s lambda
Use in source · atlas-pages.tex, line 389
ends at s=2; it does not claim that the missing tail has zero mass. headCheck the normalizer before interpreting a density Under s=t^2/(4u^2), reversing the integration limits gives η_t(s) |ds|=frac2sqrtπe^-u^2 du. Its integral over u>0 is one, so η_t is a probability density. On each eigenmode of L, the operator formula reduces to the scalar Laplace identity ∫_0^∞η_t(s)e^-λ s ds=e^-tsqrtλ. The manuscript supplies that scalar identity and its spectral assembly. The appendix separately checks the…
Exact mathematical passage
\int_0^\infty\eta_t(s)e^{-\lambda s}\,ds=e^{-t\sqrt\lambda}.
\e^{-\lambda t}Use in source · 3 passages
Search aliases: e^-λ t e lambda mathrm lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5781
[One phase model] If A=(-λ) and p_0=1, then S(t)= e^-λ t, f(t)=λ e^-λ t, h(t)=λ.
Exact mathematical passage
S(t)=\e^{-\lambda t}, \qquad f(t)=\lambda\e^{-\lambda t}, \qquad h(t)=\lambda.
e^{-\lambda t}Use in source · 1 passage
Search aliases: e^-λ t lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 551
Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.
Exact mathematical passage
e^{tQ}=e^{-\lambda t}e^{tB}
\e^{-\mu t}Use in source · 2 passages
Search aliases: e^-μ t e mu mathrm mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5890
0, where λ≠μ are positive. The first phase moves to the second at rate λ, and the second is absorbed at rate μ. Direct calculation gives S(t)= fracμ e^-λ t-λ e^-μ tμ-λ, f(t)= fracλμμ-λ ( e^-λ t- e^-μ t). The ratio f(t)/S(t) is generally not constant. A time local Markov model on two transient phases can therefore produce a nonconstant hazard after the phase coordinate is hidden.
Exact mathematical passage
S(t)= \frac{\mu\e^{-\lambda t}-\lambda\e^{-\mu t}}{\mu-\lambda},
e^{1/4}Use in source · 1 passage
Search aliases: e^1/4
Use in source · atlas-pages.tex, line 888
und is deliberately shown separately from the trajectory. headTrack every constant before applying the estimate Here M_T=1, B=sup widetilde x =1, and L= K _L^1(0,T)=T, ε= K-widetilde K _L^1(0,T)=T. The theorem yields sup_0≤ t≤ T x(t)-widetilde x(t) =fracT^22≤ T^2e^T^2. At T=1/2, the exact error is 1/8 and the certificate is e^1/4/4. Nilpotence makes this example polynomial; it does not remove the time horizon from the general theorem.
Exact mathematical passage
e^{1/4}/4
E^2Use in source · 1 passage
Search aliases: E^2
Use in source · atlas-pages.tex, line 160
)^ T becomes Ef=(2,2,26/7,26/7)^ T. vfigweighted_projection platecaptionThe bars are function values, not state probabilities. Their total need not be one. The block masses 3/10 and 7/10 determine the denominators in the averaging formula. headWhich inner product makes this an orthogonal projection? Put W=diag(π). Direct multiplication checks E^2=E, WE=E^ TW, Eones=ones. The middle equation is weighted self-adjointness. Ordinary symmetry fails: E_01=2/3 whereas E_10=1/3. Testing E=E^ T would the…
Exact mathematical passage
E^2=E,\qquad WE=E^{\mathsf T}W,\qquad E\ones=\ones.
\e^{at}Use in source · 2 passages
Search aliases: e^at e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5808
c · b and let the two coordinate axes be the packet spaces. If a≠ b, then e^tA = e^at · 0 displaystyle cfrac e^at- e^bta-b · e^bt . The lower left block is produced by one transfer from the first packet to the second. Its expansion begins with ct, in agreement with [source reference]. When a=b, the same block is ct e^at.
Exact mathematical passage
\e^{tA} = \begin{pmatrix} \e^{at}&0\\ \displaystyle c\frac{\e^{at}-\e^{bt}}{a-b}&\e^{bt} \end{pmatrix}.
\e^{bt}Use in source · 1 passage
Search aliases: e^bt e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5808
c · b and let the two coordinate axes be the packet spaces. If a≠ b, then e^tA = e^at · 0 displaystyle cfrac e^at- e^bta-b · e^bt . The lower left block is produced by one transfer from the first packet to the second. Its expansion begins with ct, in agreement with [source reference]. When a=b, the same block is ct e^at.
Exact mathematical passage
\e^{tA} = \begin{pmatrix} \e^{at}&0\\ \displaystyle c\frac{\e^{at}-\e^{bt}}{a-b}&\e^{bt} \end{pmatrix}.
e^{Q/8}Use in source · 1 passage
Search aliases: e^Q/8
Use in source · atlas-pages.tex, line 95
For later discrete calculations define K=I+Q/8. This matrix is stochastic, but it is not e^Q/8. It is the step kernel in the uniformization identity e^tQ=e^-8t∑_n≥0(8t)^nK^n/n!. takeawayChanging packet coordinates does not change the evolution. Replacing continuous time by one discrete step does.
Exact mathematical passage
e^{Q/8}
e^{rA_f}Use in source · 3 passages
Search aliases: e^rA_f
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 462
[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.
Exact mathematical passage
\boxed{ Ce^{tA_f}-e^{tA_c}C = \int_0^t e^{(t-r)A_c}\Delta e^{rA_f}\,dr. }
\e^{\operatorname{Re}\mu_jt}Use in source · 1 passage
Search aliases: e^Reμ_jt e operatorname mu mathrm operatorname mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4714
esolvent expansion is (zI-A_QQ)^-1Q_j = ∑_k=0^ν_j-1 fracN_j^kQ_j(z-μ_j)^k+1. Multiplication by the coupling blocks gives the self energy formula for every z outside the hidden spectrum. Equivalently, on a right half plane it is the Laplace transform identity ∫_0^∞ e^-zt e^μ_jt fract^kk! d t = frac1(z-μ_j)^k+1. Under stability, ∫_0^∞ e^Reμ_jt fract^kk! d t = frac1(-Reμ_j)^k+1. Sum the omitted terms.
Exact mathematical passage
\int_0^\infty \e^{\operatorname{Re}\mu_jt} \frac{t^k}{k!} \dd t = \frac{1}{(-\operatorname{Re}\mu_j)^{k+1}}.
\e^{s_1D}Use in source · 1 passage
Search aliases: e^s_1D e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2235
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
\e^{sA}Use in source · 3 passages
Search aliases: e^sA e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2253
Write the variation of constants identity e^tA = e^tD + ∫_0^t e^(t-s)DN e^sA d s. Iterating the identity gives the displayed series. The norm of the term of order m is at most e^tnorm Dfrac(tnorm N)^mm!. The majorant is summable. Inserting the packet resolution around every factor N leaves only products whose consecutive packet labels are joined by nonzero transfer blocks.
Exact mathematical passage
\e^{tA} = \e^{tD} + \int_0^t\e^{(t-s)D}N\e^{sA}\dd s.
e^{sA}Use in source · 3 passages
Search aliases: e^sA
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 334
[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.
Exact mathematical passage
\boxed{e^{(s+t)A}=e^{sA}e^{tA}.}
\e^{sA_f}Use in source · 3 passages
Search aliases: e^sA_f e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2289
[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\mathfrak D(t) = \int_0^t \e^{(t-s)A_c}\Delta\e^{sA_f} \dd s.
\e^{sg}Use in source · 1 passage
Search aliases: e^sg e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3795
The semigroup law follows from multiplication of the likelihoods e^sg and e^tg. Differentiating the normalized formula gives the replicator equation. A coordinate that starts at zero remains zero, while a positive coordinate remains positive. Dividing two explicit coordinate formulas gives the odds identity.
Exact mathematical passage
\e^{sg}
e^{sg}Use in source · 1 passage
Search aliases: e^sg
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1056
The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.
Exact mathematical passage
e^{sg}e^{tg}=e^{(s+t)g}
e^{sQ}Use in source · 1 passage
Search aliases: e^sQ
Use in source · atlas-pages.tex, line 90
Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.
Exact mathematical passage
P_b e^{(s+t)Q}P_a=\sum_{c\in\{\mathrm A,\mathrm B\}}(P_b e^{sQ}P_c)(P_c e^{tQ}P_a).
e^tUse in source · 6 passages
Search aliases: e^t
Use in source · atlas-pages.tex, line 259
vpage12A nonlinear semigroup of oddsBAYESIAN FLOW / COMPOSITION WITHOUT LINEARITYTraditional Part I: Theorems 7.1 and 9.1. Likelihood composition and the replicator field. Let the score vector be g=(1,0) and start at (1/2,1/2). The likelihood at time t is (e^t,1), so the first posterior coordinate is p_0(t)=frace^t1+e^t, p_1(t)=frac11+e^t. The odds p_0/p_1 grow exponentially, while the probabilities remain inside the simplex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionTh…
Exact mathematical passage
(e^t,1)
\e^{t(B+C)}Use in source · 1 passage
Search aliases: e^t(B+C) e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4441
Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.
Exact mathematical passage
\e^{t(B+C)}
e^{t(B-zI)}Use in source · 1 passage
Search aliases: e^t(B-zI)
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1556
Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace; equivalently, apply the decomposition to left multiplication by B on the matrix algebra and evaluate the exponential at I. The strict spectral inequality makes every such term absolutely integrable on (0,∞) and forces it to tend to zero. H…
Exact mathematical passage
T_z(t)=e^{-zt}e^{tB}=e^{t(B-zI)}
\e^{t(g_i-g_j)}Use in source · 1 passage
Search aliases: e^t(g_i-g_j) e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3786
[Bayesian flow and replicator equation] The maps Φ_t form a nonlinear semigroup: Φ_s+t=Φ_s∘Φ_t. Every coordinate satisfies frac d d tp_i(t) = (g_i-ipgp(t))p_i(t). The support of p(t) is independent of t, and for any two coordinates in the support, fracp_i(t)p_j(t) = e^t(g_i-g_j) fracp_i(0)p_j(0).
Exact mathematical passage
\frac{p_i(t)}{p_j(t)} = \e^{t(g_i-g_j)} \frac{p_i(0)}{p_j(0)}.
\e^{t(\overline Q^T-\overline V)}Use in source · 1 passage
Search aliases: e^t(overline Q^T-overline V) e overline overline mathrm overline overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5648
[Coarse Feynman Kac transfer] Assume CQ=overline QC, and let J=C^T lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so VJ=Joverline V. Then e^t(Q^T-V)J = J e^t(overline Q^T-overline V) for every t≥0.
Exact mathematical passage
\boxed{ \e^{t(Q^T-V)}J = J\e^{t(\overline Q^T-\overline V)} }
\e^{t(Q+aI)}Use in source · 1 passage
Search aliases: e^t(Q+aI) e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2837
Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.
Exact mathematical passage
\e^{tQ}=\e^{-at}\e^{t(Q+aI)}
\e^{t(Q-D_V)}Use in source · 1 passage
Search aliases: e^t(Q-D_V) e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4942
Average [source reference] with terminal function f= 1 against the initial distribution. Transposition gives p_0^T e^t(Q^T-D_V) 1 = 1^T e^t(Q-D_V)p_0. Since 1^TQ=0, differentiation gives S_V'(t) = - 1^TD_V e^tAp_0.
Exact mathematical passage
p_0^T\e^{t(Q^T-D_V)}\one = \one^T\e^{t(Q-D_V)}p_0.
\e^{t(Q^T-D_V)}Use in source · 5 passages
Search aliases: e^t(Q^T-D_V) e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4381
[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.
Exact mathematical passage
\left( \e^{t(Q^T-D_V)}f \right)_i = \Exp_i\left[ \exp\left( -\int_0^tV(X_s)\dd s \right) f(X_t) \right].
\e^{t(Q^T-V)}Use in source · 2 passages
Search aliases: e^t(Q^T-V) e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5625
Let Q and overline Q be finite Markov generators in the column convention. All times in this section satisfy t≥0. [Potential perturbation bound] Let V and W be nonnegative diagonal matrices. In the supremum norm, norm e^t(Q^T-V) - e^t(Q^T-W) _∞→∞ ≤ tnormV-W_∞→∞.
Exact mathematical passage
\boxed{ \norm{ \e^{t(Q^T-V)} - \e^{t(Q^T-W)} }_{\infty\to\infty} \leq t\norm{V-W}_{\infty\to\infty}. }
\e^{t(Q^T-W)}Use in source · 1 passage
Search aliases: e^t(Q^T-W) e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5625
Let Q and overline Q be finite Markov generators in the column convention. All times in this section satisfy t≥0. [Potential perturbation bound] Let V and W be nonnegative diagonal matrices. In the supremum norm, norm e^t(Q^T-V) - e^t(Q^T-W) _∞→∞ ≤ tnormV-W_∞→∞.
Exact mathematical passage
\boxed{ \norm{ \e^{t(Q^T-V)} - \e^{t(Q^T-W)} }_{\infty\to\infty} \leq t\norm{V-W}_{\infty\to\infty}. }
\e^{t(QAQ)|_{QX}}Use in source · 1 passage
Search aliases: e^t(QAQ)|_QX e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2749
[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.
Exact mathematical passage
\norm{\e^{t(QAQ)|_{QX}}} \leq M_Q\e^{\omega_Qt}.
e^{T^2}Use in source · 2 passages
Search aliases: e^T^2
Use in source · atlas-pages.tex, line 884
trajectory platecaptionThe first-coordinate error is exactly t^2/2. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory. headTrack every constant before applying the estimate Here M_T=1, B=sup widetilde x =1, and L= K _L^1(0,T)=T, ε= K-widetilde K _L^1(0,T)=T. The theorem yields sup_0≤ t≤ T x(t)-widetilde x(t) =fracT^22≤ T^2e^T^2. At T=1/2, the exact error is 1/8 and the certificate is e^1/4/4. Nilpotence makes this example po…
Exact mathematical passage
\sup_{0\le t\le T}\|x(t)-\widetilde x(t)\| =\frac{T^2}{2}\le T^2e^{T^2}.
e^{tA}Use in source · 17 passages
Search aliases: e^tA
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 299
Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usually written with e^tA; positive diffusion and elliptic operators are usually written with e^-tL.
Exact mathematical passage
T(t)=e^{tA},\qquad t\ge 0,
\e^{tA_c}Use in source · 4 passages
Search aliases: e^tA_c e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2283
Let X_f and X_c be complex Banach spaces. Let A_f∈ B(X_f), A_c∈ B(X_c), and C∈ B(X_f,X_c). Define Δ=CA_f-A_cC and D(t)=C e^tA_f- e^tA_cC.
Exact mathematical passage
\mathfrak D(t)=C\e^{tA_f}-\e^{tA_c}C.
e^{tA_c}Use in source · 4 passages
Search aliases: e^tA_c
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 456
Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.
Exact mathematical passage
\mathfrak D(t)=Ce^{tA_f}-e^{tA_c}C.
\e^{tA_f}Use in source · 4 passages
Search aliases: e^tA_f e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2283
Let X_f and X_c be complex Banach spaces. Let A_f∈ B(X_f), A_c∈ B(X_c), and C∈ B(X_f,X_c). Define Δ=CA_f-A_cC and D(t)=C e^tA_f- e^tA_cC.
Exact mathematical passage
\mathfrak D(t)=C\e^{tA_f}-\e^{tA_c}C.
e^{tA_f}Use in source · 4 passages
Search aliases: e^tA_f
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 456
Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.
Exact mathematical passage
\mathfrak D(t)=Ce^{tA_f}-e^{tA_c}C.
\e^{tA_n}Use in source · 1 passage
Search aliases: e^tA_n e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2800
[A usable closure condition for refinement towers] Assume the contractive summability hypothesis of [source reference] holds on every bounded time interval. Suppose every P_m,n is an orthogonal projection on a Hilbert space and T_n(t)= e^tA_n is contractive. If, for every fixed m, norm(I-P_m,n)A_nP_m,n ⟶0, then the limiting family U_m is a semigroup.
Exact mathematical passage
T_n(t)=\e^{tA_n}
\e^{tA_{PP}}Use in source · 1 passage
Search aliases: e^tA_PP e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4589
[Criterion for exact local closure] Assume x_Q(0)=0 and d=dim X_Q<∞. The following conditions are equivalent. item For every retained initial state, P e^tAP = e^tA_PPP for all t≥0. item Σ_P(z)=0 for every sufficiently large z. item K(t)=0 for every t≥0. item If d=dim X_Q, then A_PQA_QQ^kA_QP=0 for 0≤ k<d.
Exact mathematical passage
P\e^{tA}P = \e^{tA_{PP}}P
\e^{tA_{QQ}}Use in source · 5 passages
Search aliases: e^tA_QQ e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4518
For the homogeneous equation dot x=Ax, variation of constants gives x_Q(t) = e^tA_QQx_Q(0) + ∫_0^t e^(t-s)A_QQA_QPx_P(s) d s. Substitution into the retained equation gives dot x_P(t) = · A_PPx_P(t) + A_PQ e^tA_QQx_Q(0)
Exact mathematical passage
x_Q(t) = \e^{tA_{QQ}}x_Q(0) + \int_0^t \e^{(t-s)A_{QQ}}A_{QP}x_P(s) \dd s.
e^{tA_{QQ}}Use in source · 3 passages
Search aliases: e^tA_QQ
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1527
For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.
Exact mathematical passage
x_Q(t)=e^{tA_{QQ}}x_Q(0) +\int_0^te^{(t-r)A_{QQ}}A_{QP}x_P(r)\,dr.
\e^{tA_{\alpha\alpha}}Use in source · 1 passage
Search aliases: e^tA_αα e alpha alpha mathrm alpha alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2189
thm:reducing_packets The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.
Exact mathematical passage
T(t)|_{P_\alpha X} = \e^{tA_{\alpha\alpha}}.
\e^{tA_\ell}Use in source · 3 passages
Search aliases: e^tA_ℓ e ell mathrm ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2483
If T_ℓ(t)= e^tA_ℓ, the generators are bounded, and sup_ℓnormA_ℓ<∞, C_ℓ A_ℓ+1=A_ℓ C_ℓ, then the continuity condition holds and A_∞(x_ℓ)_ℓ=(A_ℓ x_ℓ)_ℓ is the bounded generator of T_∞.
Exact mathematical passage
T_\ell(t)=\e^{tA_\ell}
e^{tA_\ell}Use in source · 1 passage
Search aliases: e^tA_ℓ ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1766
[Uniform refinement control] For a nested family of physically or analytically meaningful discretizations, there exist norms and constants independent of resolution such that normC_ℓ e^tA_ℓ+1-e^tA_ℓC_ℓ ≤ M t normC_ℓ A_ℓ+1-A_ℓ C_ℓ on every bounded time interval. Summable generator defects would then produce a coherent inverse-limit evolution.
Exact mathematical passage
\norm{C_\ell e^{tA_{\ell+1}}-e^{tA_\ell}C_\ell} \le M\,t\,\norm{C_\ell A_{\ell+1}-A_\ell C_\ell}
e^{tA_{\ell+1}}Use in source · 1 passage
Search aliases: e^tA_ℓ+1 ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1766
[Uniform refinement control] For a nested family of physically or analytically meaningful discretizations, there exist norms and constants independent of resolution such that normC_ℓ e^tA_ℓ+1-e^tA_ℓC_ℓ ≤ M t normC_ℓ A_ℓ+1-A_ℓ C_ℓ on every bounded time interval. Summable generator defects would then produce a coherent inverse-limit evolution.
Exact mathematical passage
\norm{C_\ell e^{tA_{\ell+1}}-e^{tA_\ell}C_\ell} \le M\,t\,\norm{C_\ell A_{\ell+1}-A_\ell C_\ell}
\e^{tB}Use in source · 3 passages
Search aliases: e^tB e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4579
pace, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus e^-zt e^tB is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for R=∫_0^∞ e^-zt e^tB d t, so R=(zI-B)^-1. This is the…
Exact mathematical passage
\norm{\e^{tB}}\leq M_\beta\exp(\norm B+\max(0,-\beta))\e^{\beta t}.
e^{tB}Use in source · 4 passages
Search aliases: e^tB
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 550
Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.
Exact mathematical passage
e^{tB}
\e^{tB/n}Use in source · 1 passage
Search aliases: e^tB/n e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4432
Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.
Exact mathematical passage
\e^{tB/n}\e^{tC/n} = I+\frac{t}{n}(B+C)+O(n^{-2})
e^{t\bar Q}Use in source · 1 passage
Search aliases: e^tbar Q bar
Use in source · atlas-pages.tex, line 119
2 · -2. The Duhamel identity therefore gives Ce^tQ=e^tbar QC. This is a second consequence of the corresponding generator identity, not an inference that K is a fixed-time sample of Q.
Exact mathematical passage
Ce^{tQ}=e^{t\bar Q}C
\e^{tC/n}Use in source · 1 passage
Search aliases: e^tC/n e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4432
Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.
Exact mathematical passage
\e^{tB/n}\e^{tC/n} = I+\frac{t}{n}(B+C)+O(n^{-2})
\e^{tD}Use in source · 5 passages
Search aliases: e^tD e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2235
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
e^{tD}Use in source · 5 passages
Search aliases: e^tD
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5731
A finite auxiliary realization of a matrix kernel means K(t)=B e^tDC, with finite matrices of compatible sizes. Laplace transforms are taken in a right half-plane of absolute convergence. The rational characterization concerns continuous kernels of exponential order, so uniqueness of the ordinary Laplace transform applies. [Rational memory realization] A matrix valued memory transform has a finite auxiliary realization if and only if it is a strictly proper rational matrix function. More precise…
Exact mathematical passage
\Kcal(t)=B e^{tD}C
\e^{tG}Use in source · 1 passage
Search aliases: e^tG e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3847
Factor e^tG from the normalizing constant. The total weight outside M is at most (1-p_M(0)) e^-δ t, while the weight on M is p_M(0).
Exact mathematical passage
\e^{tG}
\e^{tg}Use in source · 1 passage
Search aliases: e^tg e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3795
The semigroup law follows from multiplication of the likelihoods e^sg and e^tg. Differentiating the normalized formula gives the replicator equation. A coordinate that starts at zero remains zero, while a positive coordinate remains positive. Dividing two explicit coordinate formulas gives the odds identity.
Exact mathematical passage
\e^{tg}
e^{tG}Use in source · 1 passage
Search aliases: e^tG
Use in source · atlas-pages.tex, line 59
1 · -1. The total-mass direction (1,1) has eigenvalue zero. The contrast (1,-1) has eigenvalue -2. Resolving those two directions gives the full transfer, not only its first-order approximation: T(t)=e^tG=frac121+e^-2t · 1-e^-2t 1-e^-2t · 1+e^-2t. vfigclock platecaptionA point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory. headTwo checks explain the mecha…
Exact mathematical passage
T(t)=e^{tG}=\frac12\begin{pmatrix}1+e^{-2t}&1-e^{-2t}\\1-e^{-2t}&1+e^{-2t}\end{pmatrix}.
e^{tg}Use in source · 1 passage
Search aliases: e^tg
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1056
The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.
Exact mathematical passage
e^{sg}e^{tg}=e^{(s+t)g}
\e^{tg_i}Use in source · 2 passages
Search aliases: e^tg_i e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3767
Fix a score vector g∈ R^n. For t≥0, define Φ_t(p)_i = frac e^tg_ip_i ∑_j e^tg_jp_j.
Exact mathematical passage
\Phi_t(p)_i = \frac{\e^{tg_i}p_i} {\sum_j\e^{tg_j}p_j}.
e^{tg_i}Use in source · 3 passages
Search aliases: e^tg_i
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1029
Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.
Exact mathematical passage
\lambda_i(t)=e^{tg_i},
\e^{tg_j}Use in source · 1 passage
Search aliases: e^tg_j e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3767
Fix a score vector g∈ R^n. For t≥0, define Φ_t(p)_i = frac e^tg_ip_i ∑_j e^tg_jp_j.
Exact mathematical passage
\Phi_t(p)_i = \frac{\e^{tg_i}p_i} {\sum_j\e^{tg_j}p_j}.
e^{tg_j}Use in source · 3 passages
Search aliases: e^tg_j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1033
Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.
Exact mathematical passage
\Phi_t(p)_i = \frac{e^{tg_i}p_i}{\sum_je^{tg_j}p_j}.
e^{tJ}Use in source · 2 passages
Search aliases: e^tJ
Use in source · atlas-pages.tex, line 520
0 · 0, N^2=0. Choose the entrance column V=(0,1)^ T and exit row U=(1,0). The hidden exponential is not e^-2tI: e^tJ=e^-2t(I+tN), K_rm mem(t)=Ue^tJV=t e^-2t. The nilpotent link carries an input from the second hidden coordinate to the first. vfigjordan platecaptionThe memory kernel starts at zero, becomes positive, and peaks at t=1/2. A rule that retained the eigenvalue -2 but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps. headThe transfer function see…
Exact mathematical passage
e^{tJ}=e^{-2t}(I+tN), \qquad K_{\rm mem}(t)=Ue^{tJ}V=t e^{-2t}.
e^{tL}Use in source · 1 passage
Search aliases: e^tL
Use in source · atlas-pages.tex, line 36
L · A nonnegative self-adjoint operator. · Its evolution is e^-tL, not e^tL.
Exact mathematical passage
e^{tL}
\e^{tM_A}Use in source · 1 passage
Search aliases: e^tM_A e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2500
For bounded generators with common bound M_A, sup_ℓ norm( e^tA_ℓ-I)x_ℓ ≤ ( e^tM_A-1)sup_ℓnormx_ℓ. This tends to zero with t. Exact generator intertwining makes (A_ℓ x_ℓ)_ℓ compatible, and coordinatewise differentiation identifies the generator.
Exact mathematical passage
\sup_\ell \norm{(\e^{tA_\ell}-I)x_\ell} \leq (\e^{tM_A}-1)\sup_\ell\norm{x_\ell}.
\e^{t\norm A}Use in source · 1 passage
Search aliases: e^tnorm A e norm mathrm norm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2132
The estimate normT(t)≤ e^tnorm A gives absolute convergence of the operator valued integral. Let R(z)=∫_0^∞ e^-ztT(t) d t. Since T'(t)=AT(t)=T(t)A, integration by parts gives (zI-A)R(z) = -∫_0^∞ frac d d t ( e^-ztT(t)) d t =I. The same calculation on the other side gives R(z)(zI-A)=I . Multiplication by P_β and P_α may be passed under the norm convergent integral, which gives the packet formula.
Exact mathematical passage
\norm{T(t)}\leq\e^{t\norm A}
\e^{t\norm D}Use in source · 1 passage
Search aliases: e^tnorm D e norm mathrm norm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2261
Write the variation of constants identity e^tA = e^tD + ∫_0^t e^(t-s)DN e^sA d s. Iterating the identity gives the displayed series. The norm of the term of order m is at most e^tnorm Dfrac(tnorm N)^mm!. The majorant is summable. Inserting the packet resolution around every factor N leaves only products whose consecutive packet labels are joined by nonzero transfer blocks.
Exact mathematical passage
\e^{t\norm D}\frac{(t\norm N)^m}{m!}.
\e^{t\overline Q}Use in source · 1 passage
Search aliases: e^toverline Q e overline mathrm overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2944
[Continuous time lumping] A coarse Markov generator overline Q satisfying CQ=overline QC exists if and only if the aggregated rates ∑_r∈ B_αQ_ri depend only on the source block of i. In that case C e^tQ= e^toverline QC for every t≥0.
Exact mathematical passage
C\e^{tQ}=\e^{t\overline Q}C
\e^{tQ}Use in source · 5 passages
Search aliases: e^tQ e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2026
Markov chain · K^n or e^tQ · equality of aggregated transition masses
Exact mathematical passage
\e^{tQ}
e^{tQ}Use in source · 10 passages
Search aliases: e^tQ
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 541
Use column probability vectors. A finite continuous-time Markov generator Q has nonnegative off-diagonal entries and column sums zero. The stochastic semigroup is K(t)=e^tQ.
Exact mathematical passage
K(t)=e^{tQ}.
\e^{tQ^T/n}Use in source · 1 passage
Search aliases: e^tQ^T/n e mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4419
[Alternating propagation and killing] For every t≥0, e^t(Q^T-D_V) = lim_n→∞ ( e^-tD_V/n e^tQ^T/n )^n in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.
Exact mathematical passage
\e^{t(Q^T-D_V)} = \lim_{n\to\infty} \left( \e^{-tD_V/n}\e^{tQ^T/n} \right)^n
e^{tu}Use in source · 1 passage
Search aliases: e^tu
Use in source · atlas-pages.tex, line 822
vpage36The score rises because it has varianceBAYESIAN FLOW / EVIDENCE CUMULANTSTraditional Part II: Theorems E10.1 and E10.2. Exponential tilting of a fixed finite prior. Start with equal prior masses on scores u=(0,1,2) and likelihood e^tu. Put q=e^t. The evidence and tilted distribution are Z(t)=frac1+q+q^23, p(t)=frac(1,q,q^2)1+q+q^2. At t=log2, they are Z=7/3 and p=(1,2,4)/7. vfigext_cumulants platecaptionThe score values stay fixed. Their probabilities change. The rising mean score is a pr…
Exact mathematical passage
e^{tu}
\e^{\beta t}Use in source · 1 passage
Search aliases: e^β t e beta mathrm beta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4579
pace, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus e^-zt e^tB is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for R=∫_0^∞ e^-zt e^tB d t, so R=(zI-B)^-1. This is the…
Exact mathematical passage
\norm{\e^{tB}}\leq M_\beta\exp(\norm B+\max(0,-\beta))\e^{\beta t}.
\e^{\lambda_jt}Use in source · 1 passage
Search aliases: e^λ_jt e lambda mathrm lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5759
[Exponential sum memory] If K(t)=∑_j=1^mB_j e^λ_jtC_j, then K has a finite auxiliary realization with one auxiliary block for each exponential. Repeated poles produce polynomial factors in t and are realized by Jordan blocks.
Exact mathematical passage
\Kcal(t)=\sum_{j=1}^{m}B_j\e^{\lambda_jt}C_j,
\e^{\mu_jt}Use in source · 4 passages
Search aliases: e^μ_jt e mu mathrm mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4657
xtended:thm:modal_memory Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the complete pri…
Exact mathematical passage
\Kcal(t) = \sum_{j=1}^r \e^{\mu_jt} \sum_{k=0}^{\nu_j-1} \frac{t^k}{k!}B_{j,k}
\e^{\omega t}Use in source · 2 passages
Search aliases: e^ω t e omega mathrm omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2466
[Exact projective limit semigroup] For each ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x_ℓ-x_ℓ=0 for every (x_ℓ)_ℓ∈ X_∞.
Exact mathematical passage
\sup_\ell\norm{T_\ell(t)}\leq M\e^{\omega t}.
e^{\omega t}Use in source · 1 passage
Search aliases: e^ω t omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5179
section renewcommandthefigureRarabicfigure renewcommandtheHfigureR.arabicfigure renewcommandthetableRarabictable renewcommandtheHtableR.arabictable FloatBarrier Transfer paths and product-topology limits. Let T(t) be an arbitrary strongly continuous semigroup on a complex Banach space X, with generator A, and suppose normT(t)≤ M e^ω t for M≥1 and ω∈ R. Let (P_α)_α∈ I_0 be a finite family of bounded pairwise orthogonal idempotents summing to I, and put T_βα(t)=P_β T(t)P_α. The generator need not …
Exact mathematical passage
\norm{T(t)}\leq M e^{\omega t}
\e^{\omega_c t}Use in source · 1 passage
Search aliases: e^ω_c t e omega mathrm omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2297
[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{\e^{tA_f}}\leq M_f\e^{\omega_f t}, \qquad \norm{\e^{tA_c}}\leq M_c\e^{\omega_c t},
e^{\omega_c t}Use in source · 1 passage
Search aliases: e^ω_c t omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 486
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{e^{tA_f}}\le M_f e^{\omega_f t}, \qquad \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
e^{\omega_c(t-r)}Use in source · 1 passage
Search aliases: e^ω_c(t-r) omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 492
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{\mathfrak D(t)} \le M_fM_c\norm{\Delta} \int_0^t e^{\omega_c(t-r)}e^{\omega_fr}\,dr.
\e^{\omega_c(t-s)+\omega_fs}Use in source · 1 passage
Search aliases: e^ω_c(t-s)+ω_fs e omega omega mathrm omega omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2303
[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{\mathfrak D(t)} \leq M_fM_c\norm\Delta \int_0^t \e^{\omega_c(t-s)+\omega_fs} \dd s.
\e^{\omega_f t}Use in source · 1 passage
Search aliases: e^ω_f t e omega mathrm omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2297
[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{\e^{tA_f}}\leq M_f\e^{\omega_f t}, \qquad \norm{\e^{tA_c}}\leq M_c\e^{\omega_c t},
e^{\omega_f t}Use in source · 1 passage
Search aliases: e^ω_f t omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 486
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{e^{tA_f}}\le M_f e^{\omega_f t}, \qquad \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
e^{\omega_fr}Use in source · 1 passage
Search aliases: e^ω_fr omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 492
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{\mathfrak D(t)} \le M_fM_c\norm{\Delta} \int_0^t e^{\omega_c(t-r)}e^{\omega_fr}\,dr.
\e^{\omega_Q(t-s)+\omega s}Use in source · 1 passage
Search aliases: e^ω_Q(t-s)+ω s e omega omega mathrm omega omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2754
[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.
Exact mathematical passage
\norm{QT(t)P} \leq M M_Q\norm P^2\norm{QAP} \int_0^t \e^{\omega_Q(t-s)+\omega s} \dd s.
\e^{\omega_Qt}Use in source · 1 passage
Search aliases: e^ω_Qt e omega mathrm omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2749
[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.
Exact mathematical passage
\norm{\e^{t(QAQ)|_{QX}}} \leq M_Q\e^{\omega_Qt}.
E_0Use in source · 3 passages
Search aliases: E_0
Use in source · atlas-pages.tex, line 171
vpage8Where the variance goesMARTINGALE PACKETS / THREE EXACT PIECESTraditional Part I: Theorem 6.2. Nested weighted projections and their differences. Keep the weights and observable from pagerefV7. Let E_0f be the constant global mean, let E_1=E average within A and B, and let E_2=I retain every value. Define D_0=E_0, D_1=E_1-E_0, D_2=I-E_1. Then D_0f is the mean, D_1f describes the contrast between block means, and D_2f is the fluctuation left inside each block. vfigvariance platecaptionThe s…
Exact mathematical passage
E_0f
e_0Use in source · 12 passages
Search aliases: e_0
Use in source · atlas-pages.tex, line 42
bottomrule headOne convention test For K_2=3/4 · 1/4 1/4 · 3/4, p=e_0, f=(2,1)^ T, we obtain K_2p=(3/4,1/4)^ T and f^ TK_2p=7/4=(K_2^ Tf)_0. With departure weights w_i=e^-V(i), the backward Feynman--Kac operator is diag(w)K_2^ T. Its forward dual is K_2diag(w). Interchanging these factors changes when the weight is applied. takeawayAlways name the object being transported. The transpose and the position of the likelihood or potential are mathematical choices, not typographic details.
Exact mathematical passage
K_2=\begin{pmatrix}3/4&1/4\\1/4&3/4\end{pmatrix}, \qquad p=e_0,\qquad f=(2,1)^{\mathsf T},
E_{01}Use in source · 1 passage
Search aliases: E_01
Use in source · atlas-pages.tex, line 163
state probabilities. Their total need not be one. The block masses 3/10 and 7/10 determine the denominators in the averaging formula. headWhich inner product makes this an orthogonal projection? Put W=diag(π). Direct multiplication checks E^2=E, WE=E^ TW, Eones=ones. The middle equation is weighted self-adjointness. Ordinary symmetry fails: E_01=2/3 whereas E_10=1/3. Testing E=E^ T would therefore reject a correct conditional-expectation projector.
Exact mathematical passage
E_{01}=2/3
E_1Use in source · 3 passages
Search aliases: E_1
Use in source · atlas-pages.tex, line 171
vpage8Where the variance goesMARTINGALE PACKETS / THREE EXACT PIECESTraditional Part I: Theorem 6.2. Nested weighted projections and their differences. Keep the weights and observable from pagerefV7. Let E_0f be the constant global mean, let E_1=E average within A and B, and let E_2=I retain every value. Define D_0=E_0, D_1=E_1-E_0, D_2=I-E_1. Then D_0f is the mean, D_1f describes the contrast between block means, and D_2f is the fluctuation left inside each block. vfigvariance platecaptionThe s…
Exact mathematical passage
E_1=E
e_1Use in source · 4 passages
Search aliases: e_1
Use in source · atlas-pages.tex, line 112
ds for B. vfiglumpability platecaptionThe fine columns are visibly different. Their sums over each target packet agree within each source packet. That equality is the reason a two-state description works. headA one-step calculation controls every discrete horizon The displayed columns give CK=bar KC. Multiplication then yields CK^n=bar K^nC (n≥0). For example, e_0 and e_1 have the same coarse initial vector (1,0)^ T. Their fine evolutions differ, but their coarse vectors agree at every step. No …
Exact mathematical passage
e_1
E_{10}Use in source · 1 passage
Search aliases: E_10
Use in source · atlas-pages.tex, line 163
Their total need not be one. The block masses 3/10 and 7/10 determine the denominators in the averaging formula. headWhich inner product makes this an orthogonal projection? Put W=diag(π). Direct multiplication checks E^2=E, WE=E^ TW, Eones=ones. The middle equation is weighted self-adjointness. Ordinary symmetry fails: E_01=2/3 whereas E_10=1/3. Testing E=E^ T would therefore reject a correct conditional-expectation projector.
Exact mathematical passage
E_{10}=1/3
E_2Use in source · 1 passage
Search aliases: E_2
Use in source · atlas-pages.tex, line 171
vpage8Where the variance goesMARTINGALE PACKETS / THREE EXACT PIECESTraditional Part I: Theorem 6.2. Nested weighted projections and their differences. Keep the weights and observable from pagerefV7. Let E_0f be the constant global mean, let E_1=E average within A and B, and let E_2=I retain every value. Define D_0=E_0, D_1=E_1-E_0, D_2=I-E_1. Then D_0f is the mean, D_1f describes the contrast between block means, and D_2f is the fluctuation left inside each block. vfigvariance platecaptionThe s…
Exact mathematical passage
E_2=I
e_2Use in source · 2 passages
Search aliases: e_2
Use in source · atlas-pages.tex, line 417
A discrete unit coordinate vector and a discretized unit point mass use different normalizations. For f=e_2, the result is the second inverse column. For f=e_2/h, it is the unscaled tent sampled at the nodes. Either convention can be useful, but it must be specified before comparing answers.
Exact mathematical passage
f=e_2
e_iUse in source · 3 passages
Search aliases: e_i
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2896
[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.
Exact mathematical passage
e_i-e_j
E_jUse in source · 3 passages
Search aliases: E_j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2560
fine the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
\sup_{0\leq t\leq T} \norm{U_m(t)-U_{m,n}(t)} \leq \sum_{j=n}^{\infty}w_{m,j} \sup_{0\leq t\leq T}\norm{E_j(t)}.
e_jUse in source · 3 passages
Search aliases: e_j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2896
[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.
Exact mathematical passage
e_i-e_j
E_nUse in source · 3 passages
Search aliases: E_n
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2543
[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
E_n(t)=C_nT_{n+1}(t)-T_n(t)C_n.
E_\muUse in source · 2 passages
Search aliases: E_μ mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1655
For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. I…
Exact mathematical passage
E_\mu=\ker(A-\mu I)^d
E_\piUse in source · 1 passage
Search aliases: E_π pi
Use in source · atlas-pages.tex, line 34
E_π · Conditional expectation of observables. · A weighted average, self-adjoint for the π inner product.
Exact mathematical passage
E_\pi
E_\ellUse in source · 2 passages
Search aliases: E_ℓ ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2599
[Contractive summable defect completion] Assume normC_ℓ≤1, normR_ℓ≤1, normT_ℓ(t)≤1 for every ℓ and t≥0. If ∑_ℓ=0^∞ sup_0≤ t≤ TnormE_ℓ(t)<∞, then the conclusion of [source reference] holds, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_ℓ=n^∞ sup_0≤ t≤ TnormE_ℓ(t).
Exact mathematical passage
\sum_{\ell=0}^{\infty} \sup_{0\leq t\leq T}\norm{E_\ell(t)}<\infty,
\EEUse in source · 8 passages
Search aliases: EE E mathbb
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5516
Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.
Exact mathematical passage
\frac{\dd}{\dd t}\EE_{p(t)}[h] = \Cov_{p(t)}(h,g).
\EE_0Use in source · 1 passage
Search aliases: EE_0 E_0 EE mathbb
Use in source · atlas-pages.tex, line 448
1 · 3, start at state 0, use weights w=(1,1/2), and terminal observable f=(2,1). For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate E_0[∏_r=0^2w(X_r) f(X_3)]. Each path contributes its transition probability times its three departure weights times its terminal value. vfigpaths platecaptionEvery possible path is included. The probability column sums to one. The contribution column sums to 311/256, agreeing exactly with the transfer calculati…
Exact mathematical passage
\EE_0\left[\prod_{r=0}^{2}w(X_r)\,f(X_3)\right].
\EE_{p(t)}Use in source · 3 passages
Search aliases: EE_p(t) E_p(t) EE mathbb
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5516
Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.
Exact mathematical passage
\frac{\dd}{\dd t}\EE_{p(t)}[h] = \Cov_{p(t)}(h,g).
\operatorname{End}Use in source · 1 passage
Search aliases: End operatorname
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 298
Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usually written with e^tA; positive diffusion and elliptic operators are usually written with e^-tL.
Exact mathematical passage
A\in\operatorname{End}(X)
\ExpUse in source · 9 passages
Search aliases: Exp E mathbb
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 579
[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.
Exact mathematical passage
\Exp[X_n]=0, \qquad \Var(X_n)=n.
\expUse in source · 14 passages
Search aliases: exp
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1452
[Discrete Feynman--Kac path formula] For an initial state i, (u_n)_i = E_i[ exp(-∑_r=0^n-1V(X_r))f(X_n) ].
Exact mathematical passage
\boxed{ (u_n)_i = \Exp_i\left[ \exp\left(-\sum_{r=0}^{n-1}V(X_r)\right)f(X_n) \right]. }
\Exp_iUse in source · 3 passages
Search aliases: Exp_i E_i Exp mathbb
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1452
[Discrete Feynman--Kac path formula] For an initial state i, (u_n)_i = E_i[ exp(-∑_r=0^n-1V(X_r))f(X_n) ].
Exact mathematical passage
\boxed{ (u_n)_i = \Exp_i\left[ \exp\left(-\sum_{r=0}^{n-1}V(X_r)\right)f(X_n) \right]. }
\Exp_{p_0}Use in source · 1 passage
Search aliases: Exp_p_0 E_p_0 Exp mathbb
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4921
[State dependent killing as survival] Let Q be a finite Markov generator, let V≥0, and set A=Q-D_V. For an initial distribution p_0, define S_V(t)= 1^T e^tAp_0. Then S_V(t) = E_p_0 [ exp( -∫_0^tV(X_s) d s ) ]. Also, -S_V'(t) = 1^TD_V e^tAp_0. The matrix A=Q-D_V has nonnegative off diagonal entries and nonpositive column sums, so it is the subgenerator of the killed, possibly nontransient chain. Thus the potential in the finite Feynman Kac formula is a state dependent killing rate in the forward …
Exact mathematical passage
S_V(t) = \Exp_{p_0} \left[ \exp\left( -\int_0^tV(X_s)\dd s \right) \right].
\Exp_\gammaUse in source · 1 passage
Search aliases: Exp_γ E_γ Exp gamma mathbb gamma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3286
A mean zero function has no h_0 component, so every surviving Hermite multiplier is at most e^-t. For a polynomial f- E_γ f=∑_n≥1c_nh_n. Then Var_γ(f)=∑_n≥1absc_n^2 ≤ ∑_n≥1nabsc_n^2 = ip- L ff. Apply [source reference].
Exact mathematical passage
f-\Exp_\gamma f=\sum_{n\geq1}c_nh_n.
\Exp_\piUse in source · 1 passage
Search aliases: Exp_π E_π Exp pi mathbb pi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3091
[Law of total variance] For a real valued f, let P_0f= E_π[f] 1. Then Var_π(f) = Var_π(P_ Qf) + normf-P_ Qf_π^2.
Exact mathematical passage
P_0f=\Exp_\pi[f]\one
F'Use in source · 6 passages
Search aliases: F'
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 478
Set F(r)=e^(t-r)A_cCe^rA_f. Then F'(r)=e^(t-r)A_c(CA_f-A_cC)e^rA_f. Integrating from 0 to t gives F(t)-F(0)=Ce^tA_f-e^tA_cC and proves the formula.
Exact mathematical passage
F'(r)=e^{(t-r)A_c}(CA_f-A_cC)e^{rA_f}.
f'Use in source · 8 passages
Search aliases: f'
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3218
[Generator and Dirichlet form] On the polynomial subspace, the infinitesimal generator of ( O_t) is the differential operator L f=f''-xf'. It satisfies L h_n=-nh_n. For every polynomial f, ip- L ff_L^2(γ) = ∫_ Rabsf'(x)^2γ( d x).
Exact mathematical passage
\Lcal f=f''-xf'.
f''Use in source · 4 passages
Search aliases: f''
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3218
[Generator and Dirichlet form] On the polynomial subspace, the infinitesimal generator of ( O_t) is the differential operator L f=f''-xf'. It satisfies L h_n=-nh_n. For every polynomial f, ip- L ff_L^2(γ) = ∫_ Rabsf'(x)^2γ( d x).
Exact mathematical passage
\Lcal f=f''-xf'.
f^{\mathsf T}Use in source · 2 passages
Search aliases: f^ T mathsf
Use in source · atlas-pages.tex, line 24
vpage2A dictionary with orientationsREAD BEFORE MULTIPLYING / CONVENTIONSTraditional Part I: Sections 2, 4, 6, 7, and 12. Column probabilities and backward observables. An apparent disagreement between two transfer formulas often comes from acting on different objects. Throughout this guide, probabilities are columns and observables are functions written as columns. Their pairing is f^ Tp. The transpose distinguishes the forward action on distributions from the backward action on observations. s…
Exact mathematical passage
f^{\mathsf T}p
F^{(k)}Use in source · 2 passages
Search aliases: F^(k)
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1657
For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. I…
Exact mathematical passage
\left.F(A)\right|_{E_\mu} =\left.\sum_{k=0}^{d-1}\frac{F^{(k)}(\mu)}{k!}(A-\mu I)^k\right|_{E_\mu}.
f^hUse in source · 3 passages
Search aliases: f^h
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4200
[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.
Exact mathematical passage
L_hu^h=f^h,
f^h_iUse in source · 1 passage
Search aliases: f^h_i
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4242
[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.
Exact mathematical passage
f^h_i=f(x_i)
f_1Use in source · 1 passage
Search aliases: f_1
Use in source · atlas-pages.tex, line 926
ausesCOMPETING RISKS / RESIDUAL LIFETraditional Part II: Proposition E16.4 and Theorem E16.5. This example is stable and absorbs surely. Two transient states have no internal transitions. Their exit rates are 1 and 2, their initial masses are both 1/2, and the two states exit by distinct causes. Thus A=diag(-1,-2), S(t)=tfrac12e^-t+tfrac12e^-2t, with cause densities f_1(t)=e^-t/2 and f_2(t)=e^-2t. vfigext_risks platecaptionAt t=log2, surviving mass is 3/8. Cause 1 has accumulated probability 1/4…
Exact mathematical passage
f_1(t)=e^{-t}/2
f_2Use in source · 2 passages
Search aliases: f_2
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3271
[Gaussian spectral gap] If f∈ L^2(γ) has mean zero, then norm O_tf_2 ≤ e^-tnorm f_2. For every polynomial f, Var_γ(f) ≤ ∫_ Rabsf'(x)^2γ( d x).
Exact mathematical passage
\norm{\Ocal_tf}_2 \leq \e^{-t}\norm f_2.
f_j^hUse in source · 1 passage
Search aliases: f_j^h
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4204
[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.
Exact mathematical passage
u_i^h = h\sum_{j=1}^nG(x_i,x_j)f_j^h.
F_PUse in source · 8 passages
Search aliases: F_P
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1496
[Schur--Feshbach projected resolvent] Assume both zI-A and zI-A_QQ are invertible. Define F_P(z) =zI-A_PP-A_PQ(zI-A_QQ)^-1A_QP. Then .P(zI-A)^-1|_Ran P=F_P(z)^-1.
Exact mathematical passage
\boxed{ F_P(z) =zI-A_{PP}-A_{PQ}(zI-A_{QQ})^{-1}A_{QP}. }
f_rUse in source · 4 passages
Search aliases: f_r
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5044
[Competing risk packets] Suppose the exit row splits into nonnegative cause rows a^T = ∑_r=1^q a_r^T. Define f_r(t)=a_r^T e^tAp_0. Then f(t)=∑_r=1^qf_r(t). The cause specific hazard is h_r(t)=fracf_r(t)S(t), and h(t)=∑_r=1^qh_r(t). If the spectrum of A lies in the open left half-plane, the eventual probability of cause r is ∫_0^∞ f_r(t) d t = a_r^T(-A)^-1p_0.
Exact mathematical passage
f_r(t)=a_r^T\e^{tA}p_0.
F_VUse in source · 2 passages
Search aliases: F_V
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4446
[Positivity and composition] The operators F_V(t)= e^t(Q^T-D_V) form a positive semigroup. If f≥0, then F_V(t)f≥0.
Exact mathematical passage
F_V(t)=\e^{t(Q^T-D_V)}
f_\pi^2Use in source · 1 passage
Search aliases: f_π^2 pi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3077
P_0,P_1,…,P_r come from increasingly fine partitions, so that Ran P_0⊆Ran P_1⊆⋯⊆Ran P_r. Define D_0=P_0, D_j=P_j-P_j-1 (1≤ j≤ r). Then P_iP_j=P_min(i,j), D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j. Every D_j is an orthogonal projector, and normP_rf_π^2 = ∑_j=0^rnormD_jf_π^2 and norm f_π^2 = ∑_j=0^rnormD_jf_π^2 + normf-P_rf_π^2.
Exact mathematical passage
\norm f_\pi^2 = \sum_{j=0}^r\norm{D_jf}_\pi^2 + \norm{f-P_rf}_\pi^2.
GUse in source · 20 passages
Search aliases: G
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1358
On the interval (0,1) with Dirichlet boundary conditions, the Green kernel for -u''=f is G(x,y)=min(x,y)-xy. Thus the point source δ_y generates the solution kernel, and a general source is reconstructed by u(x)=∫_0^1G(x,y)f(y) dy.
Exact mathematical passage
\boxed{G(x,y)=\min(x,y)-xy.}
g_aUse in source · 1 passage
Search aliases: g_a
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5386
[Coarse Dirichlet form] Under the hypotheses of [source reference], let Jg be the fine function that is constant with value g_a on B_a. Then -ipJgQ^TJg_L^2(π) = -ipgoverline Q^Tg_L^2(overlineπ).
Exact mathematical passage
g_a
g_iUse in source · 12 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1029
Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.
Exact mathematical passage
\lambda_i(t)=e^{tg_i},
g_jUse in source · 5 passages
Search aliases: g_j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1033
Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.
Exact mathematical passage
\Phi_t(p)_i = \frac{e^{tg_i}p_i}{\sum_je^{tg_j}p_j}.
G_{\rm nodes}Use in source · 1 passage
Search aliases: G_rm nodes rm
Use in source · green.tex, line 51
node[font=small,align=center,font=small] at (10.43,1.2) 1; path[draw=black!30,fill=black!8] (10.86,0.77) rectangle (11.72,1.63); node[font=small,align=center,font=small] at (11.29,1.2) 2; path[draw=black!30,fill=black!8] (11.72,0.77) rectangle (12.58,1.63); node[font=small,align=center,font=small] at (12.15,1.2) 3; node[font=small,align=center,font=small] at (11.3,0.4) L_h^-1=hG_rm nodes;
Exact mathematical passage
L_h^{-1}=hG_{\rm nodes}
HUse in source · 15 passages
Search aliases: H
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 303
Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usually written with e^tA; positive diffusion and elliptic operators are usually written with e^-tL.
Exact mathematical passage
H(t)=e^{-tL}.
h^{-2}Use in source · 2 passages
Search aliases: h^-2
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5582
FloatBarrier Fractional modes and killed transfer. For the one-dimensional Dirichlet difference operator, let h=1/M with an integer M≥2, let 1≤ k≤ M-1, and set λ_k,h=4h^-2sin^2(kπ h/2). The retained explicit coefficient below is valid for every discrete mode. [Fractional low mode dispersion] Let 0<α≤1. Then 0 ≤ (kπ)^2α-λ_k,h^α ≤ fracα(4k^2)^α-1(kπ)^412h^2. For every t≥0, abs e^-tλ_k,h^α - e^-t(kπ)^2α ≤ fractα(4k^2)^α-1(kπ)^412h^2.
Exact mathematical passage
\lambda_{k,h}=4h^{-2}\sin^2(k\pi h/2)
h^2Use in source · 25 passages
Search aliases: h^2
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1367
For n interior points with spacing h=1/(n+1), define L_h=frac1h^2 2 · -1 · · -1 · 2 · ddots · · ddots · ddots · -1 · · -1 · 2 .
Exact mathematical passage
L_h=\frac1{h^2} \begin{pmatrix} 2&-1&&\\ -1&2&\ddots&\\ &\ddots&\ddots&-1\\ &&-1&2 \end{pmatrix}.
h_0Use in source · 1 passage
Search aliases: h_0
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3285
A mean zero function has no h_0 component, so every surviving Hermite multiplier is at most e^-t. For a polynomial f- E_γ f=∑_n≥1c_nh_n. Then Var_γ(f)=∑_n≥1absc_n^2 ≤ ∑_n≥1nabsc_n^2 = ip- L ff. Apply [source reference].
Exact mathematical passage
h_0
H_1Use in source · 5 passages
Search aliases: H_1
Use in source · atlas-pages.tex, line 753
vpage33Gaussian packets have separate clocksORNSTEIN--UHLENBECK / HERMITE ENERGYTraditional Part II: Theorem E7.1 and Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure. In L^2(γ) for the standard Gaussian law on R, use the probabilists' Hermite polynomials H_1(x)=x, H_2(x)=x^2-1, H_4(x)=x^4-6x^2+3. The Ornstein--Uhlenbeck generator is Lf=f''-xf'. Direct polynomial differentiation gives LH_n=-nH_n for these three polynomials. Their squared norms are n! and different Hermite degrees are …
Exact mathematical passage
H_1(x)=x,\qquad H_2(x)=x^2-1,\qquad H_4(x)=x^4-6x^2+3.
H_2Use in source · 4 passages
Search aliases: H_2
Use in source · atlas-pages.tex, line 753
vpage33Gaussian packets have separate clocksORNSTEIN--UHLENBECK / HERMITE ENERGYTraditional Part II: Theorem E7.1 and Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure. In L^2(γ) for the standard Gaussian law on R, use the probabilists' Hermite polynomials H_1(x)=x, H_2(x)=x^2-1, H_4(x)=x^4-6x^2+3. The Ornstein--Uhlenbeck generator is Lf=f''-xf'. Direct polynomial differentiation gives LH_n=-nH_n for these three polynomials. Their squared norms are n! and different Hermite degrees are …
Exact mathematical passage
H_1(x)=x,\qquad H_2(x)=x^2-1,\qquad H_4(x)=x^4-6x^2+3.
H_4Use in source · 2 passages
Search aliases: H_4
Use in source · atlas-pages.tex, line 753
vpage33Gaussian packets have separate clocksORNSTEIN--UHLENBECK / HERMITE ENERGYTraditional Part II: Theorem E7.1 and Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure. In L^2(γ) for the standard Gaussian law on R, use the probabilists' Hermite polynomials H_1(x)=x, H_2(x)=x^2-1, H_4(x)=x^4-6x^2+3. The Ornstein--Uhlenbeck generator is Lf=f''-xf'. Direct polynomial differentiation gives LH_n=-nH_n for these three polynomials. Their squared norms are n! and different Hermite degrees are …
Exact mathematical passage
H_1(x)=x,\qquad H_2(x)=x^2-1,\qquad H_4(x)=x^4-6x^2+3.
h_iUse in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5530
Differentiate ∑_ih_ip_i(t) and insert the replicator equation.
Exact mathematical passage
\sum_ih_ip_i(t)
H_mUse in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3186
· = ∑_n=0^∞ e^-ntH_n(x)fracu^nn!. Comparison of coefficients proves the eigenvalue formula. The same generating function shows ∫ H_mH_n dγ=n!δ_mn. The Hermite polynomials span all polynomials, and polynomials are dense in L^2(γ). Hence the normalized family is an orthonormal basis. See [source reference] for the general Gaussian Hilbert space setting.
Exact mathematical passage
\int H_mH_n\dd\gamma=n!\delta_{mn}.
H_nUse in source · 7 passages
Search aliases: H_n
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 716
[Gaussian chaos realization] For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally: T_tH_n=e^-ntH_n. Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.
Exact mathematical passage
T_tH_n=e^{-nt}H_n.
h_nUse in source · 8 passages
Search aliases: h_n
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3135
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
h_n=H_n/\sqrt{n!}
h_rUse in source · 2 passages
Search aliases: h_r
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5052
[Competing risk packets] Suppose the exit row splits into nonnegative cause rows a^T = ∑_r=1^q a_r^T. Define f_r(t)=a_r^T e^tAp_0. Then f(t)=∑_r=1^qf_r(t). The cause specific hazard is h_r(t)=fracf_r(t)S(t), and h(t)=∑_r=1^qh_r(t). If the spectrum of A lies in the open left half-plane, the eventual probability of cause r is ∫_0^∞ f_r(t) d t = a_r^T(-A)^-1p_0.
Exact mathematical passage
h_r(t)=\frac{f_r(t)}{S(t)},
i'Use in source · 4 passages
Search aliases: i'
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 898
Conversely, fix two fine states i,i'∈ I_a and one state j∈ I_b in another coarse packet. Vary the distribution of a fixed coarse mass inside I_a between i and i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the conditional average of λ_i and λ_i'. It can be independent of that variation only if λ_i=λ_i'. Hence λ is constant on every packet; write the value on I_…
Exact mathematical passage
i,i'\in I_a
I_0Use in source · 3 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5179
ureR.arabicfigure renewcommandthetableRarabictable renewcommandtheHtableR.arabictable FloatBarrier Transfer paths and product-topology limits. Let T(t) be an arbitrary strongly continuous semigroup on a complex Banach space X, with generator A, and suppose normT(t)≤ M e^ω t for M≥1 and ω∈ R. Let (P_α)_α∈ I_0 be a finite family of bounded pairwise orthogonal idempotents summing to I, and put T_βα(t)=P_β T(t)P_α. The generator need not be bounded. In the following resolvent statement the integral …
Exact mathematical passage
(P_\alpha)_{\alpha\in I_0}
i_0Use in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1468
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
(i=i_0,i_1,\ldots,i_n)
I_1Use in source · 1 passage
Search aliases: I_1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3367
Let I=I_1sqcup⋯sqcup I_m be a partition. Define packet evidences Z_α = ∑_i∈ I_αλ_ip_i. For Z_α>0, define m_α=fracZ_α∑_β Z_β, q_α,i=fracλ_ip_iZ_α (i∈ I_α). Packets with zero evidence receive posterior mass zero.
Exact mathematical passage
I=I_1\sqcup\cdots\sqcup I_m
i_1Use in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1468
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
(i=i_0,i_1,\ldots,i_n)
I_aUse in source · 11 passages
Search aliases: I_a
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 839
Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero.
Exact mathematical passage
I_a
I_bUse in source · 1 passage
Search aliases: I_b
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 898
Conversely, fix two fine states i,i'∈ I_a and one state j∈ I_b in another coarse packet. Vary the distribution of a fixed coarse mass inside I_a between i and i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the conditional average of λ_i and λ_i'. It can be independent of that variation only if λ_i=λ_i'. Hence λ is constant on every packet; write the value on I_…
Exact mathematical passage
j\in I_b
I_{d_j}Use in source · 1 passage
Search aliases: I_d_j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5766
If B_j and C_j have an intermediate space of dimension d_j, use a block diagonal matrix D with blocks λ_j I_d_j, or the required Jordan blocks when poles repeat. Assemble the matrices B_j and C_j into the output and input maps.
Exact mathematical passage
\lambda_j I_{d_j}
I_mUse in source · 1 passage
Search aliases: I_m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3367
Let I=I_1sqcup⋯sqcup I_m be a partition. Define packet evidences Z_α = ∑_i∈ I_αλ_ip_i. For Z_α>0, define m_α=fracZ_α∑_β Z_β, q_α,i=fracλ_ip_iZ_α (i∈ I_α). Packets with zero evidence receive posterior mass zero.
Exact mathematical passage
I=I_1\sqcup\cdots\sqcup I_m
i_nUse in source · 1 passage
Search aliases: i_n
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1468
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
(i=i_0,i_1,\ldots,i_n)
i_rUse in source · 2 passages
Search aliases: i_r
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1468
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
\prod_rK_{i_{r+1},i_r}
i_{r+1}Use in source · 1 passage
Search aliases: i_r+1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1468
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
\prod_rK_{i_{r+1},i_r}
I_XUse in source · 2 passages
Search aliases: I_X
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2046
Let X be a complex Banach space. A finite packet system is a family of bounded idempotents P=(P_α)_α∈ I such that P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I_X. No orthogonality is needed in this section.
Exact mathematical passage
P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I_X.
I_{X_m}Use in source · 1 passage
Search aliases: I_X_m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2528
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,m}=I_{X_m}, \qquad R_{m,m}=I_{X_m}.
I_{X_\ell}Use in source · 1 passage
Search aliases: I_X_ℓ ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2516
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
R_\ell:X_\ell\to X_{\ell+1}, \qquad C_\ell R_\ell=I_{X_\ell}.
\iiUse in source · 1 passage
Search aliases: ii i mathrm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2379
elextended:thm:holomorphic_defect Let Γ be the positively oriented rectifiable boundary of a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that Γ lies in the common resolvent set. Thus its winding number is one about every spectral point. If g is holomorphic on a neighborhood of the closure of the union, then Cg(A_f)-g(A_c)C = frac12π i ∫_Γ g(z)R_c(z)Δ R_f(z) d z. In particular, normCg(A_f)-g(A_c)C ≤ · fraclength(Γ)2π sup_z∈Γabsg(z) …
Exact mathematical passage
Cg(A_f)-g(A_c)C = \frac{1}{2\pi\ii} \int_\Gamma g(z)R_c(z)\Delta R_f(z) \dd z.
JUse in source · 25 passages
Search aliases: J
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1566
Let J=∫_0^∞ T_z(t) dt. Integrating both derivative identities T_z'=(B-zI)T_z=T_z(B-zI) and using T_z(0)=I gives (zI-B)J=J(zI-B)=I. Thus J=(zI-B)^-1. Continuous left and right multiplication carry the Bochner integral through the two rectangular transfer blocks, yielding the claim.
Exact mathematical passage
J=\int_0^\infty T_z(t)\,dt
\mathcal KUse in source · 2 passages
Search aliases: K mathcal
Use in source · atlas-pages.tex, line 38
K(t) · The memory kernel after hidden-state elimination. · It is not the transition matrix K.
Exact mathematical passage
\mathcal K(t)
kUse in source · 62 passages
Search aliases: k
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 574
[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.
Exact mathematical passage
\bbP(X_n=k)= 2^{-n}\binom{n}{(n+k)/2}
K^{\mathsf T}Use in source · 1 passage
Search aliases: K^ T mathsf
Use in source · atlas-pages.tex, line 31
K^ Tf · Expected next observable given the current state. · The same dynamics, with the opposite action on the dual object.
Exact mathematical passage
K^{\mathsf T}f
k^2Use in source · 3 passages
Search aliases: k^2
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5585
:04 For the one-dimensional Dirichlet difference operator, let h=1/M with an integer M≥2, let 1≤ k≤ M-1, and set λ_k,h=4h^-2sin^2(kπ h/2). The retained explicit coefficient below is valid for every discrete mode. [Fractional low mode dispersion] Let 0<α≤1. Then 0 ≤ (kπ)^2α-λ_k,h^α ≤ fracα(4k^2)^α-1(kπ)^412h^2. For every t≥0, abs e^-tλ_k,h^α - e^-t(kπ)^2α ≤ fractα(4k^2)^α-1(kπ)^412h^2.
Exact mathematical passage
0 \leq (k\pi)^{2\alpha}-\lambda_{k,h}^\alpha \leq \frac{\alpha(4k^2)^{\alpha-1}(k\pi)^4}{12}h^2.
K^jUse in source · 4 passages
Search aliases: K^j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 612
[Finite-horizon defect formula] For arbitrary Δ=CK-bar KC, CK^n-bar K^nC = ∑_j=0^n-1bar K^n-1-jΔ K^j. If K and bar K are contractions in the chosen norm, then normCK^n-bar K^nC≤ nnormΔ.
Exact mathematical passage
\boxed{ CK^n-\bar K^nC = \sum_{j=0}^{n-1}\bar K^{n-1-j}\Delta K^j. }
K^{j+1}Use in source · 2 passages
Search aliases: K^j+1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 627
The identity follows by telescoping: bar K^n-1-jCK^j+1-bar K^n-jCK^j =bar K^n-1-j(CK-bar KC)K^j. Summing from j=0 to n-1 leaves CK^n-bar K^nC. The norm bound follows termwise.
Exact mathematical passage
\bar K^{n-1-j}CK^{j+1}-\bar K^{n-j}CK^j =\bar K^{n-1-j}(CK-\bar KC)K^j.
K^nUse in source · 12 passages
Search aliases: K^n
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 597
[Exact lumpability] If Δ=0, then CK^n=bar K^nC for every n≥0. Thus fine evolution followed by coarse reconstruction equals coarse evolution from the beginning.
Exact mathematical passage
\boxed{CK^n=\bar K^nC}
K^{n+1}Use in source · 1 passage
Search aliases: K^n+1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 605
The result is immediate for n=0. If CK^n=bar K^nC, then CK^n+1=bar K^nCK=bar K^n+1C.
Exact mathematical passage
CK^{n+1}=\bar K^nCK=\bar K^{n+1}C.
K^TUse in source · 4 passages
Search aliases: K^T
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1442
Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.
Exact mathematical passage
M_V=\diag(e^{-V})K^T.
K_2Use in source · 9 passages
Search aliases: K_2
Use in source · atlas-pages.tex, line 42
bottomrule headOne convention test For K_2=3/4 · 1/4 1/4 · 3/4, p=e_0, f=(2,1)^ T, we obtain K_2p=(3/4,1/4)^ T and f^ TK_2p=7/4=(K_2^ Tf)_0. With departure weights w_i=e^-V(i), the backward Feynman--Kac operator is diag(w)K_2^ T. Its forward dual is K_2diag(w). Interchanging these factors changes when the weight is applied. takeawayAlways name the object being transported. The transpose and the position of the likelihood or potential are mathematical choices, not typographic details.
Exact mathematical passage
K_2=\begin{pmatrix}3/4&1/4\\1/4&3/4\end{pmatrix}, \qquad p=e_0,\qquad f=(2,1)^{\mathsf T},
K_2^{\mathsf T}Use in source · 4 passages
Search aliases: K_2^ T mathsf
Use in source · atlas-pages.tex, line 47
1/4 · 3/4, p=e_0, f=(2,1)^ T, we obtain K_2p=(3/4,1/4)^ T and f^ TK_2p=7/4=(K_2^ Tf)_0. With departure weights w_i=e^-V(i), the backward Feynman--Kac operator is diag(w)K_2^ T. Its forward dual is K_2diag(w). Interchanging these factors changes when the weight is applied. takeawayAlways name the object being transported. The transpose and the position of the likelihood or potential are mathematical choices, not typographic details.
Exact mathematical passage
f^{\mathsf T}K_2p=7/4=(K_2^{\mathsf T}f)_0.
K_cUse in source · 1 passage
Search aliases: K_c
Use in source · atlas-pages.tex, line 614
Two distinct equalities justify the coarse computation. First, CK=K_cC transports prediction. Second, the likelihood is constant on each packet, so CD_r=overline D_rC transports observation weighting. Their composition preserves both the posterior packet totals and the evidence factors.
Exact mathematical passage
CK=K_cC
K_{i_{r+1},i_r}Use in source · 1 passage
Search aliases: K_i_r+1,i_r
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1468
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
\prod_rK_{i_{r+1},i_r}
K_{ij}Use in source · 1 passage
Search aliases: K_ij
Use in source · atlas-pages.tex, line 30
K · One discrete stochastic step. · ones^ TK=ones^ T and K_ij≥0.
Exact mathematical passage
K_{ij}\ge0
K_{j i}Use in source · 1 passage
Search aliases: K_j i
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1465
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
(M_Vf)_i=e^{-V(i)}\sum_jK_{j i}f_j,
K_{ji}Use in source · 1 passage
Search aliases: K_ji
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4371
For one step, (M_Vf)_i = e^-V(i)∑_jK_jif_j. Iterating expands the matrix product over all paths. The product of transition entries is the path probability, and the diagonal factors give the exponential potential weight.
Exact mathematical passage
(M_Vf)_i = \e^{-V(i)}\sum_jK_{ji}f_j.
K_{L^1(0,T)}Use in source · 1 passage
Search aliases: K_L^1(0,T)
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4840
· + ∫_0^tS(t-s) ∫_0^sK(s-r)x(r) d r d s. Assume sup_0≤ t≤ Tnormwidetilde x(t)≤ B. Set L=norm K_L^1(0,T), ε=normK-widetilde K_L^1(0,T). Then sup_0≤ t≤ T normx(t)-widetilde x(t) ≤ M_TBTε exp(M_TLT).
Exact mathematical passage
L=\norm K_{L^1(0,T)}, \qquad \varepsilon=\norm{K-\widetilde K}_{L^1(0,T)}.
K_{ri}Use in source · 3 passages
Search aliases: K_ri
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2897
[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.
Exact mathematical passage
\Ob_C(K)(e_i-e_j) = \left( \sum_{r\in B_\alpha}(K_{ri}-K_{rj}) \right)_{\alpha=1}^m.
K_{rj}Use in source · 2 passages
Search aliases: K_rj
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2897
[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.
Exact mathematical passage
\Ob_C(K)(e_i-e_j) = \left( \sum_{r\in B_\alpha}(K_{ri}-K_{rj}) \right)_{\alpha=1}^m.
K_{\rm mem}Use in source · 2 passages
Search aliases: K_rm mem rm
Use in source · atlas-pages.tex, line 520
0 · 0, N^2=0. Choose the entrance column V=(0,1)^ T and exit row U=(1,0). The hidden exponential is not e^-2tI: e^tJ=e^-2t(I+tN), K_rm mem(t)=Ue^tJV=t e^-2t. The nilpotent link carries an input from the second hidden coordinate to the first. vfigjordan platecaptionThe memory kernel starts at zero, becomes positive, and peaks at t=1/2. A rule that retained the eigenvalue -2 but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps. headThe transfer function see…
Exact mathematical passage
e^{tJ}=e^{-2t}(I+tN), \qquad K_{\rm mem}(t)=Ue^{tJ}V=t e^{-2t}.
K_\alphaUse in source · 2 passages
Search aliases: K_α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1212
[Fractional intertwining theorem] For bounded nonnegative self-adjoint A and B, D_α = fracsin(πα)π ∫_0^∞ t^α(tI+B)^-1D(tI+A)^-1 dt. The improper integral converges in operator norm. Moreover, normD_α ≤ K_αnormC^1-αnormD^α, K_α= frac2^1-αsin(πα) πα(1-α). If A,B≥ mI for some m>0, then the stronger linear estimate holds: normD_α ≤ α m^α-1normD. In particular, exact generator intertwining CA=BC implies exact fractional intertwining CA^α=B^α C.
Exact mathematical passage
\boxed{ \norm{D_\alpha} \le K_\alpha\norm{C}^{1-\alpha}\norm{D}^{\alpha}, \qquad K_\alpha= \frac{2^{1-\alpha}\sin(\pi\alpha)} {\pi\alpha(1-\alpha)}. }
K_\varepsilonUse in source · 9 passages
Search aliases: K_ε varepsilon
Use in source · atlas-pages.tex, line 125
vpage6A broken lump and its error budgetDEFECTS / A GUARANTEE WITHOUT A FITTraditional Part I: Theorems 3.1, 3.3, and 5.2. Continuous and discrete defect identities. Modify only column zero of K: move 1/16 of its mass from target state zero to target state two. The perturbed matrix K_ε remains stochastic, but its aggregate column now differs from column one. Relative to the unchanged coarse kernel, Δ=CK_ε-bar KC =-1/16 · 0 · 0 · 0
Exact mathematical passage
K_\varepsilon
K_\varepsilon^jUse in source · 1 passage
Search aliases: K_ε^j varepsilon
Use in source · atlas-pages.tex, line 135
1/16 · 0 · 0 · 0, Δ _1=1/8. vfigdefect_budget platecaptionThe solid values are computed with rational matrix powers. The dashed certificate uses only stochasticity and the exact one-step defect. It need not be sharp. headTelescoping tells us where every error originates For n≥0, CK_ε^n-bar K^nC =∑_j=0^n-1bar K^n-1-jΔ K_ε^j. Each summand propagates one defect between an earlier fine evolution and a later coarse evolution. Column-stochastic matrices have induced ℓ^1 norm one, so the norm is at mos…
Exact mathematical passage
CK_\varepsilon^n-\bar K^nC =\sum_{j=0}^{n-1}\bar K^{n-1-j}\Delta K_\varepsilon^j.
K_\varepsilon^nUse in source · 1 passage
Search aliases: K_ε^n varepsilon
Use in source · atlas-pages.tex, line 135
1/16 · 0 · 0 · 0, Δ _1=1/8. vfigdefect_budget platecaptionThe solid values are computed with rational matrix powers. The dashed certificate uses only stochasticity and the exact one-step defect. It need not be sharp. headTelescoping tells us where every error originates For n≥0, CK_ε^n-bar K^nC =∑_j=0^n-1bar K^n-1-jΔ K_ε^j. Each summand propagates one defect between an earlier fine evolution and a later coarse evolution. Column-stochastic matrices have induced ℓ^1 norm one, so the norm is at mos…
Exact mathematical passage
CK_\varepsilon^n-\bar K^nC =\sum_{j=0}^{n-1}\bar K^{n-1-j}\Delta K_\varepsilon^j.
\KcalUse in source · 22 passages
Search aliases: Kcal K mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4528
For the homogeneous equation dot x=Ax, variation of constants gives x_Q(t) = e^tA_QQx_Q(0) + ∫_0^t e^(t-s)A_QQA_QPx_P(s) d s. Substitution into the retained equation gives dot x_P(t) = · A_PPx_P(t) + A_PQ e^tA_QQx_Q(0) · + ∫_0^t K(t-s)x_P(s) d s, where K(t) = A_PQ e^tA_QQA_QP.
Exact mathematical passage
\begin{aligned} \dot x_P(t) ={}& A_{PP}x_P(t) + A_{PQ}\e^{tA_{QQ}}x_Q(0)\\ &+ \int_0^t \Kcal(t-s)x_P(s) \dd s, \end{aligned}
\Kcal_JUse in source · 4 passages
Search aliases: Kcal_J K_J Kcal mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4673
he least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the complete primary blocks indexed by J, then norm K- K_J_L^1(0,∞) ≤ ∑_j∉ J ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1.
Exact mathematical passage
\Kcal_J
LUse in source · 132 passages
Search aliases: L
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 302
Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usually written with e^tA; positive diffusion and elliptic operators are usually written with e^-tL.
Exact mathematical passage
L
\mathcal LUse in source · 9 passages
Search aliases: L mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3543
[Coarsest fixed sufficient partition] Let L be a nonempty family of strictly positive likelihood vectors. Define i∼_ Lj Longleftrightarrow λ_i=λ_j hboxfor every λ∈ L. Let Q_ L be the partition into equivalence classes. A partition Q gives a fixed coarse sufficient description in the sense of [source reference] for every λ∈ L if and only if every block of Q is contained in a block of Q_ L. Hence Q_ L is the unique coarsest partition with this property.
Exact mathematical passage
\mathcal L
lUse in source · 1 passage
Search aliases: l
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1799
The accompanying Python package implements the formulas and finite models used in the paper. projector_transfer_semigroups The computational organization mirrors the paper: c|l packets.py · projectors, blocks, reconstruction, block composition semigroup.py · matrix semigroups, defects, Feshbach operators, memory markov.py · random walks, exact lumping, survival bayes.py · normalized updates, packet evidence, conjugacy, flows, HMM conditional.py · conditional expectation and Gaussian chaos fracti…
Exact mathematical passage
\begin{array}{c|l} \texttt{packets.py} & \text{projectors, blocks, reconstruction, block composition}\\ \texttt{semigroup.py} & \text{matrix semigroups, defects, Feshbach operators, memory}\\ \texttt{markov.py} & \text{random walks, exact lumping, survival}\\ \texttt{bayes.py} & \text{normalized updates, packet evidence, conjugacy, flows, HMM}\\ \texttt{conditional.py} & \text{conditional expectation and Gaussian chaos}\\ \texttt{fractional.py} & \text{spectral powers, subordination, and fractional refinement certificates}\\ \texttt{pde.py} & \text{Dirichlet Laplacians, Green kernels, eigenvalue error}\\ \texttt{feynman\_kac.py} & \text{weighted transfer and exhaustive path enumeration}. \end{array}
L^*Use in source · 2 passages
Search aliases: L^*
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1120
Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.
Exact mathematical passage
L=L^*\ge0
L^1Use in source · 16 passages
Search aliases: L^1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 823
Add and subtract μodot p/Z_λ and use the triangle inequality: normfracλodot pZ_λ-fracμodot pZ_μ_1 ≤ fracnorm(λ-μ)odot p_1Z_λ + normμodot p_1|frac1Z_λ-frac1Z_μ|. The second term equals |Z_λ-Z_μ|/Z_λ and is bounded by the same weighted L^1 difference divided by Z_λ. Interchanging λ and μ gives the minimum-evidence form.
Exact mathematical passage
L^1
L^{1/2}Use in source · 4 passages
Search aliases: L^1/2
Use in source · atlas-pages.tex, line 326
vpage15The modes stay; their clocks changeFRACTIONAL DIFFUSION / NEW COUPLINGSTraditional Part I: Theorem 10.1. A positive self-adjoint finite model and its spectral powers. Return to the four-state generator on pagerefV4 and put L=-Q. It is real symmetric and nonnegative, with orthogonal projectors P_0,P_1,P_2,P_3 and eigenvalues 0,2,4,6. Its square root is L^1/2=sqrt2P_1+2P_2+sqrt6P_3, (L^1/2)^2=L. Both heat evolutions retain these modes. Ordinary diffusion multiplies mode j by e^-2jt; square-…
Exact mathematical passage
L^{1/2}=\sqrt2P_1+2P_2+\sqrt6P_3, \qquad (L^{1/2})^2=L.
L^2Use in source · 13 passages
Search aliases: L^2
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3113
[General conditional expectations] On a probability space, conditional expectation onto a sub sigma algebra is the orthogonal projector in L^2. Nested sigma algebras satisfy the same tower law. Under the usual hypotheses, martingale convergence gives a limit in L^2 and almost surely. Standard treatments include [source reference].
Exact mathematical passage
L^2
L^pUse in source · 2 passages
Search aliases: L^p
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3138
[Ornstein Uhlenbeck semigroup] The operators ( O_t)_t≥0 form a positive Markov contraction semigroup on L^p(γ) for every 1≤ p≤∞. On L^2(γ), the semigroup is self adjoint and O_t h_n= e^-nth_n. The family (h_n)_n≥0 is an orthonormal basis of L^2(γ).
Exact mathematical passage
L^p(\gamma)
L^\alphaUse in source · 15 passages
Search aliases: L^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1125
Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.
Exact mathematical passage
\boxed{L^\alpha=\sum_j\lambda_j^\alpha P_j.}
L^{\alpha+\beta}Use in source · 2 passages
Search aliases: L^α+β alpha beta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1136
[Fractional packet calculus] For α,β>0, L^α L^β=L^α+β, and for s,t≥0, T_α(s)T_α(t)=T_α(s+t). Every spectral packet P_j reduces L^α and T_α(t).
Exact mathematical passage
L^\alpha L^\beta=L^{\alpha+\beta},
L^\betaUse in source · 3 passages
Search aliases: L^β beta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1136
[Fractional packet calculus] For α,β>0, L^α L^β=L^α+β, and for s,t≥0, T_α(s)T_α(t)=T_α(s+t). Every spectral packet P_j reduces L^α and T_α(t).
Exact mathematical passage
L^\alpha L^\beta=L^{\alpha+\beta},
L_{03}Use in source · 1 passage
Search aliases: L_03
Use in source · atlas-pages.tex, line 334
by e^-2jt; square-root diffusion multiplies it by e^-sqrt2j t. The constant mode is unchanged in both. vfigfractional platecaptionThe highest mode decays at rates 6 and sqrt6. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit. headFractional does not mean entrywise Although L_03=0, its spectral square root satisfies (L^1/2)_03=fracsqrt6-sqrt2-24<0. Thus -L^1/2 permits a direct transition between state…
Exact mathematical passage
L_{03}=0
L_cUse in source · 7 passages
Search aliases: L_c
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2003
For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional semigroup defect. This closes a finite version of fractional naturality under explicit spectral hypotheses.
Exact mathematical passage
\norm{CL_f^\alpha-L_c^\alpha C} \leq \alpha m^{\alpha-1} \norm{CL_f-L_cC}, \qquad 0<\alpha<1.
L_c^\alphaUse in source · 7 passages
Search aliases: L_c^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2003
For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional semigroup defect. This closes a finite version of fractional naturality under explicit spectral hypotheses.
Exact mathematical passage
\norm{CL_f^\alpha-L_c^\alpha C} \leq \alpha m^{\alpha-1} \norm{CL_f-L_cC}, \qquad 0<\alpha<1.
L_fUse in source · 7 passages
Search aliases: L_f
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2003
For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional semigroup defect. This closes a finite version of fractional naturality under explicit spectral hypotheses.
Exact mathematical passage
\norm{CL_f^\alpha-L_c^\alpha C} \leq \alpha m^{\alpha-1} \norm{CL_f-L_cC}, \qquad 0<\alpha<1.
L_f^\alphaUse in source · 7 passages
Search aliases: L_f^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2003
For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional semigroup defect. This closes a finite version of fractional naturality under explicit spectral hypotheses.
Exact mathematical passage
\norm{CL_f^\alpha-L_c^\alpha C} \leq \alpha m^{\alpha-1} \norm{CL_f-L_cC}, \qquad 0<\alpha<1.
L_hUse in source · 9 passages
Search aliases: L_h
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1367
For n interior points with spacing h=1/(n+1), define L_h=frac1h^2 2 · -1 · · -1 · 2 · ddots · · ddots · ddots · -1 · · -1 · 2 .
Exact mathematical passage
L_h=\frac1{h^2} \begin{pmatrix} 2&-1&&\\ -1&2&\ddots&\\ &\ddots&\ddots&-1\\ &&-1&2 \end{pmatrix}.
L_h^{-1}Use in source · 11 passages
Search aliases: L_h^-1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1379
[Exact discrete Green reconstruction at grid nodes] Let x_i=ih. Then frac1h(L_h^-1)_ij =G(x_i,x_j). Thus the scaled discrete inverse reproduces the continuum Dirichlet Green kernel exactly at the grid nodes.
Exact mathematical passage
\boxed{ \frac1h(L_h^{-1})_{ij} =G(x_i,x_j). }
L_iUse in source · 4 passages
Search aliases: L_i
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5404
tly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
\overline L_a = \sum_{i\in B_a}L_ir_i^{(a)}.
L_yUse in source · 5 passages
Search aliases: L_y
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5534
For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
L_y,\overline L_y
L_zUse in source · 2 passages
Search aliases: L_z
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4497
Set L_z= I · -A_PQ(zI-A_QQ)^-1 0 · I and S_z= F_P(z) · 0
Exact mathematical passage
L_z= \begin{pmatrix} I&-A_{PQ}(zI-A_{QQ})^{-1}\\ 0&I \end{pmatrix}
\LcalUse in source · 7 passages
Search aliases: Lcal L mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3218
[Generator and Dirichlet form] On the polynomial subspace, the infinitesimal generator of ( O_t) is the differential operator L f=f''-xf'. It satisfies L h_n=-nh_n. For every polynomial f, ip- L ff_L^2(γ) = ∫_ Rabsf'(x)^2γ( d x).
Exact mathematical passage
\Lcal f=f''-xf'.
\operatorname{length}Use in source · 1 passage
Search aliases: length operatorname
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2388
a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that Γ lies in the common resolvent set. Thus its winding number is one about every spectral point. If g is holomorphic on a neighborhood of the closure of the union, then Cg(A_f)-g(A_c)C = frac12π i ∫_Γ g(z)R_c(z)Δ R_f(z) d z. In particular, normCg(A_f)-g(A_c)C ≤ · fraclength(Γ)2π sup_z∈Γabsg(z) sup_z∈ΓnormR_c(z) sup_z∈ΓnormR_f(z) normΔ.
Exact mathematical passage
\begin{aligned} \norm{Cg(A_f)-g(A_c)C} \leq{}& \frac{\operatorname{length}(\Gamma)}{2\pi} \sup_{z\in\Gamma}\abs{g(z)} \sup_{z\in\Gamma}\norm{R_c(z)} \sup_{z\in\Gamma}\norm{R_f(z)} \norm\Delta. \end{aligned}
\limUse in source · 4 passages
Search aliases: lim
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1609
[Semigroup survival identities] For t≥0, the finite-time exit density and, wherever S(t)>0, the hazard are f(t)=a^Te^tAp_0=-S'(t), h(t)=fracf(t)S(t). Consequently, S is nonincreasing and S(t)=exp(-∫_0^th(r) dr) whenever S(t)>0. The density may be defective: its total mass is 1-lim_t→∞S(t). It is a probability density when absorption is eventual, so that this limit is zero.
Exact mathematical passage
1-\lim_{t\to\infty}S(t)
\lim_{n\to\infty}Use in source · 1 passage
Search aliases: lim_n→∞ lim to infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4419
[Alternating propagation and killing] For every t≥0, e^t(Q^T-D_V) = lim_n→∞ ( e^-tD_V/n e^tQ^T/n )^n in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.
Exact mathematical passage
\e^{t(Q^T-D_V)} = \lim_{n\to\infty} \left( \e^{-tD_V/n}\e^{tQ^T/n} \right)^n
\lim_{t\downarrow0}Use in source · 1 passage
Search aliases: lim_tdownarrow0 lim downarrow
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2476
ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x_ℓ-x_ℓ=0 for every (x_ℓ)_ℓ∈ X_∞.
Exact mathematical passage
\lim_{t\downarrow0} \sup_\ell \norm{T_\ell(t)x_\ell-x_\ell}=0
\lim_{t\to0}Use in source · 1 passage
Search aliases: lim_t→0 lim to
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2331
[Infinitesimal recovery of the defect] The generator defect is the first order semigroup defect: lim_t→0 frac D(t)t = Δ in operator norm. Consequently, C e^tA_f= e^tA_cC hboxfor every t if and only if CA_f=A_cC.
Exact mathematical passage
\lim_{t\to0} \frac{\mathfrak D(t)}{t} = \Delta
\lim_{t\to\infty}Use in source · 1 passage
Search aliases: lim_t→∞ lim to infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1609
[Semigroup survival identities] For t≥0, the finite-time exit density and, wherever S(t)>0, the hazard are f(t)=a^Te^tAp_0=-S'(t), h(t)=fracf(t)S(t). Consequently, S is nonincreasing and S(t)=exp(-∫_0^th(r) dr) whenever S(t)>0. The density may be defective: its total mass is 1-lim_t→∞S(t). It is a probability density when absorption is eventual, so that this limit is zero.
Exact mathematical passage
1-\lim_{t\to\infty}S(t)
\LongleftrightarrowUse in source · 3 passages
Search aliases: Longleftrightarrow
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 989
[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its evidence is positive. If all likelihoods are strictly positive and there are at least two profile classes, this is also the unique coarsest nontrivia…
Exact mathematical passage
i\sim j \quad\Longleftrightarrow\quad \lambda_i^{(y)}=\lambda_j^{(y)} \text{ for every }y\in Y.
m^\alphaUse in source · 2 passages
Search aliases: m^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1310
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ \norm{Ce^{-sA^\alpha}-e^{-sB^\alpha}C} \le s e^{-sm^\alpha}\alpha m^{\alpha-1}\norm D. }
m^{\alpha-1}Use in source · 9 passages
Search aliases: m^α-1 alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1224
D_α = fracsin(πα)π ∫_0^∞ t^α(tI+B)^-1D(tI+A)^-1 dt. The improper integral converges in operator norm. Moreover, normD_α ≤ K_αnormC^1-αnormD^α, K_α= frac2^1-αsin(πα) πα(1-α). If A,B≥ mI for some m>0, then the stronger linear estimate holds: normD_α ≤ α m^α-1normD. In particular, exact generator intertwining CA=BC implies exact fractional intertwining CA^α=B^α C.
Exact mathematical passage
\boxed{ \norm{D_\alpha} \le \alpha m^{\alpha-1}\norm{D}. }
m_{\mathrm A}Use in source · 2 passages
Search aliases: m_ A mathrm
Use in source · atlas-pages.tex, line 205
vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditional shapes are m_ A=5/7, q_ A=(4/5,1/5,0,0)^ T, m_ B=2/7, q_ B=(0,0,1/2,1/2)^ T. Thus m_ Aq_ A+m_ Bq_ B is exactly the displayed posterior. Keeping o…
Exact mathematical passage
m_{\mathrm A}=5/7,\quad q_{\mathrm A}=(4/5,1/5,0,0)^{\mathsf T}, \qquad m_{\mathrm B}=2/7,\quad q_{\mathrm B}=(0,0,1/2,1/2)^{\mathsf T}.
m_{\mathrm B}Use in source · 2 passages
Search aliases: m_ B mathrm
Use in source · atlas-pages.tex, line 205
vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditional shapes are m_ A=5/7, q_ A=(4/5,1/5,0,0)^ T, m_ B=2/7, q_ B=(0,0,1/2,1/2)^ T. Thus m_ Aq_ A+m_ Bq_ B is exactly the displayed posterior. Keeping o…
Exact mathematical passage
m_{\mathrm A}=5/7,\quad q_{\mathrm A}=(4/5,1/5,0,0)^{\mathsf T}, \qquad m_{\mathrm B}=2/7,\quad q_{\mathrm B}=(0,0,1/2,1/2)^{\mathsf T}.
m_0Use in source · 4 passages
Search aliases: m_0
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5231
atement, now assume A is bounded as in Part II. A directed edge from α to β means that P_β A P_α is nonzero, with α≠β; a packet always has the length-zero path to itself. [Minimum transfer order] Suppose every directed packet path from α to β uses at least m_0 off diagonal transfers. Then P_β e^tAP_α =O(t^m_0) as tdownarrow0. If there is no directed path from α to β, the block vanishes for all t.
Exact mathematical passage
m_0
M_AUse in source · 2 passages
Search aliases: M_A
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2499
For bounded generators with common bound M_A, sup_ℓ norm( e^tA_ℓ-I)x_ℓ ≤ ( e^tM_A-1)sup_ℓnormx_ℓ. This tends to zero with t. Exact generator intertwining makes (A_ℓ x_ℓ)_ℓ compatible, and coordinatewise differentiation identifies the generator.
Exact mathematical passage
M_A
M_cUse in source · 4 passages
Search aliases: M_c
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 486
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{e^{tA_f}}\le M_f e^{\omega_f t}, \qquad \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
M_fUse in source · 4 passages
Search aliases: M_f
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 486
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{e^{tA_f}}\le M_f e^{\omega_f t}, \qquad \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
M_QUse in source · 2 passages
Search aliases: M_Q
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2749
[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.
Exact mathematical passage
\norm{\e^{t(QAQ)|_{QX}}} \leq M_Q\e^{\omega_Qt}.
M_TUse in source · 5 passages
Search aliases: M_T
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4820
[Volterra stability under kernel compression] Let S(t) be a strongly continuous semigroup on a Banach space X, fix T>0, and assume normS(t)≤ M_T (0≤ t≤ T). Let K,widetilde K∈ L^1((0,T), B(X)), let g∈ L^1((0,T),X), and let x,widetilde x∈ C([0,T],X). Suppose these are mild solutions with the same initial state and forcing, except that K is replaced by widetilde K: x(t) = · S(t)x_0+ ∫_0^tS(t-s)g(s) d s
Exact mathematical passage
\norm{S(t)}\leq M_T \quad(0\leq t\leq T).
M_VUse in source · 4 passages
Search aliases: M_V
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1442
Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.
Exact mathematical passage
M_V=\diag(e^{-V})K^T.
M_V^nUse in source · 2 passages
Search aliases: M_V^n
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1446
Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.
Exact mathematical passage
u_n=M_V^nf.
m_\alphaUse in source · 8 passages
Search aliases: m_α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 762
Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.
Exact mathematical passage
Z_\alpha=\sum_{i\in I_\alpha}\lambda_ip_i, \qquad m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta}, \qquad q_{\alpha,i}=\begin{cases} \lambda_ip_i/Z_\alpha,&Z_\alpha>0,\ i\in I_\alpha,\\ 0,&\text{otherwise}. \end{cases}
\max_{1\leq i\leq n}Use in source · 1 passage
Search aliases: max_1≤ i≤ n max leq leq
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4222
[Discrete maximum bound] The inverse L_h^-1 has nonnegative entries and normL_h^-1_∞→∞ = max_1≤ i≤ n fracx_i(1-x_i)2 ≤ frac18.
Exact mathematical passage
\norm{L_h^{-1}}_{\infty\to\infty} = \max_{1\leq i\leq n} \frac{x_i(1-x_i)}{2} \leq \frac18.
\max_iUse in source · 3 passages
Search aliases: max_i max
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 550
Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.
Exact mathematical passage
\lambda\ge\max_i(-Q_{ii})
\max_{i\in B_a}Use in source · 1 passage
Search aliases: max_i∈ B_a max in
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5490
[Oscillation bound] Suppose overline L_a is any representative likelihood for block B_a, and set ω_a = max_i∈ B_aabsL_i-overline L_a. Then d(pmid L,overline L) ≤ ∑_aq_aω_a. Thus small within block likelihood oscillation gives a small coarse posterior defect whenever the evidence is bounded away from zero.
Exact mathematical passage
\omega_a = \max_{i\in B_a}\abs{L_i-\overline L_a}.
\max_{i\in I_\alpha}Use in source · 1 passage
Search aliases: max_i∈ I_α max in alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3719
[Information created by a Bayesian update] Let q= B_λ(p) with positive likelihood and positive prior. Then D(qVert p) = ∑_iq_ilogλ_i - log Z_λ(p). Inside a packet, D(q_αVert r_α) = ∑_i∈ I_α q_α,i logfracλ_ioverlineλ_α(p). Hence D(q_αVert r_α) ≤ log fracmax_i∈ I_αλ_i min_i∈ I_αλ_i.
Exact mathematical passage
D(q_\alpha\Vert r_\alpha) \leq \log \frac{\max_{i\in I_\alpha}\lambda_i} {\min_{i\in I_\alpha}\lambda_i}.
\midUse in source · 7 passages
Search aliases: mid
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5462
Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0; choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_aabsℓ_a(p)-overline L_a. Then norm C B_L(p)- B_overline L(q) _1 ≤ frac2d(pmid L,overline L) max{Z(p,L),ipoverline Lq}.
Exact mathematical passage
d(p\mid L,\overline L) = \sum_{a=1}^mq_a\abs{\ell_a(p)-\overline L_a}.
\minUse in source · 16 passages
Search aliases: min
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 704
A coarse-block average of a fine-block average is the coarse-block average, and a fine-block average of a coarse-constant function leaves it unchanged. The difference identities follow from P_iP_j=P_min(i,j).
Exact mathematical passage
P_iP_j=P_{\min(i,j)}
\min_{I_\alpha}Use in source · 1 passage
Search aliases: min_I_α min alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3603
Subtract r_α,i from the local posterior formula in [source reference]. The norm identity follows because the factor multiplying η_i is nonnegative. The last estimate uses ∑_i r_α,iabsη_i ≤ osc_I_α(λ) and overlineλ_α(p)≥min_I_αλ.
Exact mathematical passage
\overline\lambda_\alpha(p)\geq\min_{I_\alpha}\lambda
\min_{i\in I_\alpha}Use in source · 2 passages
Search aliases: min_i∈ I_α min in alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3588
] For every packet with positive mass and evidence, q_α,i-r_α,i = fracr_α,iη_i(p) overlineλ_α(p). Consequently, normq_α-r_α_1 = frac ∑_i∈ I_αr_α,iabsη_i(p) overlineλ_α(p). If λ is positive on the packet, then normq_α-r_α_1 ≤ fracosc_I_α(λ) min_i∈ I_αλ_i.
Exact mathematical passage
\norm{q_\alpha-r_\alpha}_1 \leq \frac{\osc_{I_\alpha}(\lambda)} {\min_{i\in I_\alpha}\lambda_i}.
NUse in source · 43 passages
Search aliases: N
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2154
[Resolvent expansion over packet paths] With A=D+N as in the directed path decomposition, let R_D(z)=(zI-D)^-1. If normR_D(z)N<1, then (zI-A)^-1 = ∑_m=0^∞ (R_D(z)N)^mR_D(z). After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
A=D+N
n^{-1}Use in source · 1 passage
Search aliases: n^-1
Use in source · summability.tex, line 33
dotted,line width=.9pt] plot coordinates (1.33125,1.107143) (1.8625,1.221429) (2.39375,1.272222) (2.925,1.300794) (3.45625,1.319079) (3.9875,1.331778) (4.51875,1.341107) (5.05,1.34825) (5.58125,1.353894) (6.1125,1.358465) (6.64375,1.362243) (7.175,1.365418) (7.70625,1.368123) (8.2375,1.370455) (8.76875,1.372487) (9.3,1.374273); node[font=small,align=center,font=small] at (11.7,3.95) solid: ∑_n≤ Nn^-1; node[font=small,align=center,font=small] at (11.7,2.95) dashed: ∑_n≤ Nn^-2; node[font=small,ali…
Exact mathematical passage
\sum_{n\le N}n^{-1}
n^{-2}Use in source · 3 passages
Search aliases: n^-2
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4432
Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.
Exact mathematical passage
\e^{tB/n}\e^{tC/n} = I+\frac{t}{n}(B+C)+O(n^{-2})
N^2Use in source · 4 passages
Search aliases: N^2
Use in source · atlas-pages.tex, line 516
vpage23A repeated pole remembers a Jordan chainNONDIAGONAL HIDDEN DYNAMICS / POLYNOMIAL EXPONENTIALSTraditional Part I: Theorem 13.2 and Section 15. Functional calculus must retain nilpotent data. Now use a two-dimensional hidden block J=-2I+N, N=mat0 · 1 0 · 0, N^2=0. Choose the entrance column V=(0,1)^ T and exit row U=(1,0). The hidden exponential is not e^-2tI: e^tJ=e^-2t(I+tN), K_rm mem(t)=Ue^tJV=t e^-2t. The nilpotent link carries an input from the second hidden coordinate to the first. vf…
Exact mathematical passage
J=-2I+N,\qquad N=\mat{0&1\\0&0},\qquad N^2=0.
n^2Use in source · 5 passages
Search aliases: n^2
Use in source · atlas-pages.tex, line 370
On pagerefV27, the sequences 1/n^2 and 1/n turn this change in error scale into a concrete distinction between summable and nonsummable defects. takeawayA fixed gap is quantitative information. It prevents the derivative of the fractional function from becoming singular along the comparison.
Exact mathematical passage
1/n^2
n^\alphaUse in source · 1 passage
Search aliases: n^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3306
[Fractional Hermite transfer] Let - L be the nonnegative Ornstein Uhlenbeck number operator with - L h_n=nh_n. For every α>0, e^-t(- L)^αf = ∑_n=0^∞ e^-tn^αP_nf. This is a contraction semigroup on L^2(γ).
Exact mathematical passage
\e^{-t(-\Lcal)^\alpha}f = \sum_{n=0}^\infty \e^{-tn^\alpha}P_nf.
N_jUse in source · 2 passages
Search aliases: N_j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4645
[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r}…
Exact mathematical passage
N_j=(A_{QQ}-\mu_jI)Q_j.
N_j^kUse in source · 3 passages
Search aliases: N_j^k
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4649
mory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the compl…
Exact mathematical passage
B_{j,k} = A_{PQ}N_j^kQ_jA_{QP} \qquad (0\leq k<\nu_j).
N_j^{\nu_j}Use in source · 1 passage
Search aliases: N_j^ν_j nu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4648
rem[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,……
Exact mathematical passage
N_j^{\nu_j}=0
n_\alphaUse in source · 3 passages
Search aliases: n_α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3654
on decomposition] Let p and q be probability vectors on a finite partition (I_α)_α, and assume q_i=0 whenever p_i=0. Let m=Cp, n=Cq, and let p_α and q_α be the conditional distributions inside packet I_α whenever the corresponding mass is positive. Then D(qVert p) = D(nVert m) + ∑_α:n_α>0 n_α D(q_αVert p_α). The first term is the information retained by the coarse masses. The second term is the exact information carried by changes inside the packets.
Exact mathematical passage
D(q\Vert p) = D(n\Vert m) + \sum_{\alpha:n_\alpha>0} n_\alpha D(q_\alpha\Vert p_\alpha).
OUse in source · 7 passages
Search aliases: O
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1417
[Second-order low-mode convergence] For fixed k, λ_k,h=(kπ)^2+O(h^2). More precisely, for sufficiently small h, 0≤ (kπ)^2-λ_k,h ≤ frac(kπ)^412h^2.
Exact mathematical passage
\lambda_{k,h}=(k\pi)^2+O(h^2).
oUse in source · 9 passages
Search aliases: o
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2338
[Infinitesimal recovery of the defect] The generator defect is the first order semigroup defect: lim_t→0 frac D(t)t = Δ in operator norm. Consequently, C e^tA_f= e^tA_cC hboxfor every t if and only if CA_f=A_cC.
Exact mathematical passage
C\e^{tA_f}=\e^{tA_c}C \quad\hbox{for every }t
\OcalUse in source · 12 passages
Search aliases: Ocal O mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3120
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
(\Ocal_t f)(x) = \int_\RR f\left( \e^{-t}x+\sqrt{1-\e^{-2t}}\,y \right) \gamma(\dd y).
\Ocal_tUse in source · 12 passages
Search aliases: Ocal_t O_t Ocal mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3120
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
(\Ocal_t f)(x) = \int_\RR f\left( \e^{-t}x+\sqrt{1-\e^{-2t}}\,y \right) \gamma(\dd y).
\odotUse in source · 12 passages
Search aliases: odot
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 730
Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.
Exact mathematical passage
\boxed{ \cB_\lambda(p) = \frac{\lambda\odot p}{Z_\lambda(p)}. }
\ointUse in source · 1 passage
Search aliases: oint
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1666
(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ)⊂ U, and Cauchy's formula gives fracF^(k)(μ)k! =frac12π ioint_|z-μ|=r_μ fracF(z)(z-μ)^k+1 dz. Consequently the same canonical operator is obtained from any such local Cauchy circle…
Exact mathematical passage
\frac{F^{(k)}(\mu)}{k!} =\frac{1}{2\pi i}\oint_{|z-\mu|=r_\mu} \frac{F(z)}{(z-\mu)^{k+1}}\,dz.
\oint_{|z-\mu|=r_\mu}Use in source · 1 passage
Search aliases: oint_|z-μ|=r_μ oint mu mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1666
(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ)⊂ U, and Cauchy's formula gives fracF^(k)(μ)k! =frac12π ioint_|z-μ|=r_μ fracF(z)(z-μ)^k+1 dz. Consequently the same canonical operator is obtained from any such local Cauchy circle…
Exact mathematical passage
\frac{F^{(k)}(\mu)}{k!} =\frac{1}{2\pi i}\oint_{|z-\mu|=r_\mu} \frac{F(z)}{(z-\mu)^{k+1}}\,dz.
\oneUse in source · 20 passages
Search aliases: one 1 bm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2840
Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.
Exact mathematical passage
\one^TQ=0
\one^TUse in source · 15 passages
Search aliases: one^T 1^T one bm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2840
Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.
Exact mathematical passage
\one^TQ=0
\one_{i\in B_a}Use in source · 1 passage
Search aliases: one_i∈ B_a 1_i∈ B_a one in bm in
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5348
The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement. FloatBarrier Reversible coarse dynamics. Let Q be a finite Markov generator acting on column probability vectors, let (B_a) be a partition into nonempty blocks, and let C_ai= 1_i∈ B_a. Strong lumpability means CQ=overline Q C for a coarse Markov generator. For observables, the backward generator is Q^T; weighted L^2 in…
Exact mathematical passage
C_{ai}=\one_{i\in B_a}
\one_{\{i\in B_\alpha\}}Use in source · 1 passage
Search aliases: one_{i∈ B_α} 1_{i∈ B_α} one in alpha bm in alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2847
Let {B_1,…,B_m} be a partition of the fine state set. Its aggregation matrix is C_α i= 1_{i∈ B_α}.
Exact mathematical passage
C_{\alpha i}=\one_{\{i\in B_\alpha\}}.
\onesUse in source · 5 passages
Search aliases: ones
Use in source · atlas-pages.tex, line 28
midrule p · A nonnegative column with ones^ Tp=1. · It is not a vector of function values.
Exact mathematical passage
\ones^{\mathsf T}p=1
\ones^{\mathsf T}Use in source · 4 passages
Search aliases: ones^ T ones mathsf
Use in source · atlas-pages.tex, line 28
midrule p · A nonnegative column with ones^ Tp=1. · It is not a vector of function values.
Exact mathematical passage
\ones^{\mathsf T}p=1
\oscUse in source · 2 passages
Search aliases: osc operatorname
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3588
] For every packet with positive mass and evidence, q_α,i-r_α,i = fracr_α,iη_i(p) overlineλ_α(p). Consequently, normq_α-r_α_1 = frac ∑_i∈ I_αr_α,iabsη_i(p) overlineλ_α(p). If λ is positive on the packet, then normq_α-r_α_1 ≤ fracosc_I_α(λ) min_i∈ I_αλ_i.
Exact mathematical passage
\norm{q_\alpha-r_\alpha}_1 \leq \frac{\osc_{I_\alpha}(\lambda)} {\min_{i\in I_\alpha}\lambda_i}.
\osc_{I_\alpha}Use in source · 2 passages
Search aliases: osc_I_α osc alpha operatorname alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3588
] For every packet with positive mass and evidence, q_α,i-r_α,i = fracr_α,iη_i(p) overlineλ_α(p). Consequently, normq_α-r_α_1 = frac ∑_i∈ I_αr_α,iabsη_i(p) overlineλ_α(p). If λ is positive on the packet, then normq_α-r_α_1 ≤ fracosc_I_α(λ) min_i∈ I_αλ_i.
Exact mathematical passage
\norm{q_\alpha-r_\alpha}_1 \leq \frac{\osc_{I_\alpha}(\lambda)} {\min_{i\in I_\alpha}\lambda_i}.
\overline DUse in source · 1 passage
Search aliases: overline D overline
Use in source · atlas-pages.tex, line 614
Two distinct equalities justify the coarse computation. First, CK=K_cC transports prediction. Second, the likelihood is constant on each packet, so CD_r=overline D_rC transports observation weighting. Their composition preserves both the posterior packet totals and the evidence factors.
Exact mathematical passage
CD_r=\overline D_rC
\overline D_rUse in source · 1 passage
Search aliases: overline D_r overline
Use in source · atlas-pages.tex, line 614
Two distinct equalities justify the coarse computation. First, CK=K_cC transports prediction. Second, the likelihood is constant on each packet, so CD_r=overline D_rC transports observation weighting. Their composition preserves both the posterior packet totals and the evidence factors.
Exact mathematical passage
CD_r=\overline D_rC
\overline KUse in source · 27 passages
Search aliases: overline K overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2863
[Descent through an aggregation map] Let C:X_f→ X_c be a surjective linear map between finite dimensional spaces. A unique operator overline K:X_c→ X_c satisfying CK=overline KC exists if and only if Ob_C(K)=0. For any right inverse R of C, the only possible coarse operator is overline K=CKR, and CK-overline KC = CK(I-RC).
Exact mathematical passage
\overline K:X_c\to X_c
\overline K^nUse in source · 2 passages
Search aliases: overline K^n overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2960
[Finite horizon lumping defect] For arbitrary fine and coarse operators K and overline K, with the fixed aggregation map C, set Δ_K=CK-overline KC. Then CK^n-overline K^nC = ∑_j=0^n-1 overline K^n-1-jΔ_KK^j. If K and overline K are column stochastic, then normCK^n-overline K^nC_1→1 ≤ nnormΔ_K_1→1. For every probability vector p, the total variation discrepancy is at most frac n2normΔ_K_1→1.
Exact mathematical passage
CK^n-\overline K^nC = \sum_{j=0}^{n-1} \overline K^{n-1-j}\Delta_KK^j.
\overline K^{n-1-j}Use in source · 2 passages
Search aliases: overline K^n-1-j overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2960
[Finite horizon lumping defect] For arbitrary fine and coarse operators K and overline K, with the fixed aggregation map C, set Δ_K=CK-overline KC. Then CK^n-overline K^nC = ∑_j=0^n-1 overline K^n-1-jΔ_KK^j. If K and overline K are column stochastic, then normCK^n-overline K^nC_1→1 ≤ nnormΔ_K_1→1. For every probability vector p, the total variation discrepancy is at most frac n2normΔ_K_1→1.
Exact mathematical passage
CK^n-\overline K^nC = \sum_{j=0}^{n-1} \overline K^{n-1-j}\Delta_KK^j.
\overline K^{n-j}Use in source · 1 passage
Search aliases: overline K^n-j overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2980
Sum the telescoping identities overline K^n-1-jCK^j+1 - overline K^n-jCK^j = overline K^n-1-jΔ_KK^j. Stochastic matrices are contractions in the ℓ^1 norm. The last statement uses total variation equal to one half of the ℓ^1 distance.
Exact mathematical passage
\overline K^{n-1-j}CK^{j+1} - \overline K^{n-j}CK^j = \overline K^{n-1-j}\Delta_KK^j.
\overline K_{\alpha\beta}Use in source · 1 passage
Search aliases: overline K_αβ overline alpha beta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2922
Let K be column stochastic. A coarse column stochastic matrix overline K satisfying CK=overline KC exists if and only if, for every source block B_β, every pair i,j∈ B_β, and every target block B_α, ∑_r∈ B_αK_ri = ∑_r∈ B_αK_rj. When it exists, overline K is unique and overline K_αβ = ∑_r∈ B_αK_ri for any i∈ B_β.
Exact mathematical passage
\overline K_{\alpha\beta} = \sum_{r\in B_\alpha}K_{ri}
\overline LUse in source · 18 passages
Search aliases: overline L overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5404
tly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
\overline L_a = \sum_{i\in B_a}L_ir_i^{(a)}.
\overline L_aUse in source · 6 passages
Search aliases: overline L_a overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5404
tly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
\overline L_a = \sum_{i\in B_a}L_ir_i^{(a)}.
\overline L_yUse in source · 6 passages
Search aliases: overline L_y overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5534
For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
L_y,\overline L_y
\overline MUse in source · 2 passages
Search aliases: overline M overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5558
[One step filter defect] Let r=Kp, q=Cr, and let ℓ_y(r) be the exact coarse likelihood obtained by averaging L_y inside each block with respect to r. Assume 0<overline m≤(overline L_y)_a≤overline M. Then normCΦ_y(p)-overlineΦ_y(Cp)_1 ≤ · frac2d(rmid L_y,overline L_y) max{ipL_yr,ipoverline L_yq}
Exact mathematical passage
0<\overline m\leq(\overline L_y)_a\leq\overline M.
\overline mUse in source · 2 passages
Search aliases: overline m overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5558
[One step filter defect] Let r=Kp, q=Cr, and let ℓ_y(r) be the exact coarse likelihood obtained by averaging L_y inside each block with respect to r. Assume 0<overline m≤(overline L_y)_a≤overline M. Then normCΦ_y(p)-overlineΦ_y(Cp)_1 ≤ · frac2d(rmid L_y,overline L_y) max{ipL_yr,ipoverline L_yq}
Exact mathematical passage
0<\overline m\leq(\overline L_y)_a\leq\overline M.
\overline QUse in source · 17 passages
Search aliases: overline Q overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2935
[Continuous time lumping] A coarse Markov generator overline Q satisfying CQ=overline QC exists if and only if the aggregated rates ∑_r∈ B_αQ_ri depend only on the source block of i. In that case C e^tQ= e^toverline QC for every t≥0.
Exact mathematical passage
\overline Q
\overline Q^TUse in source · 4 passages
Search aliases: overline Q^T overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5387
[Coarse Dirichlet form] Under the hypotheses of [source reference], let Jg be the fine function that is constant with value g_a on B_a. Then -ipJgQ^TJg_L^2(π) = -ipgoverline Q^Tg_L^2(overlineπ).
Exact mathematical passage
-\ip{Jg}{Q^TJg}_{L^2(\pi)} = -\ip{g}{\overline Q^Tg}_{L^2(\overline\pi)}.
\overline Q_{ab}Use in source · 2 passages
Search aliases: overline Q_ab overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5359
the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
\overline Q_{ab}\overline\pi_b = \overline Q_{ba}\overline\pi_a.
\overline Q_{ba}Use in source · 2 passages
Search aliases: overline Q_ba overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5359
the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
\overline Q_{ab}\overline\pi_b = \overline Q_{ba}\overline\pi_a.
\overline VUse in source · 4 passages
Search aliases: overline V overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5644
[Coarse Feynman Kac transfer] Assume CQ=overline QC, and let J=C^T lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so VJ=Joverline V. Then e^t(Q^T-V)J = J e^t(overline Q^T-overline V) for every t≥0.
Exact mathematical passage
VJ=J\overline V.
\overline{\left(\sum_{j\in B}\pi_j g_j\right)}Use in source · 1 passage
Search aliases: overline(∑_j∈ Bπ_j g_j) overline left sum in pi right
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 674
After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block; both equal ∑_B∈ Q frac1π(B) (∑_i∈ Bπ_i f_i) overline(∑_j∈ Bπ_j g_j).
Exact mathematical passage
\sum_{B\in\mathcal Q} \frac{1}{\pi(B)} \left(\sum_{i\in B}\pi_i f_i\right) \overline{\left(\sum_{j\in B}\pi_j g_j\right)}.
\overline{\left(\sum_{j\in B}\pi_jg_j\right)}Use in source · 1 passage
Search aliases: overline(∑_j∈ Bπ_jg_j) overline left sum in pi right
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3036
One application produces a block constant function, so a second application changes nothing. Expanding both weighted inner products block by block gives ipP_ Qfg_π = ∑_B∈ Q frac (∑_i∈ Bπ_if_i) overline(∑_j∈ Bπ_jg_j) π(B) = ipfP_ Qg_π. The range is exactly the block constant subspace.
Exact mathematical passage
\ip{P_\mathcal Qf}{g}_\pi = \sum_{B\in\mathcal Q} \frac{ \left(\sum_{i\in B}\pi_if_i\right) \overline{\left(\sum_{j\in B}\pi_jg_j\right)} }{\pi(B)} = \ip{f}{P_\mathcal Qg}_\pi.
\overline{D(\mu,r_\mu)}Use in source · 1 passage
Search aliases: overlineD(μ,r_μ) overline mu mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1665
.F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ)⊂ U, and Cauchy's formula gives fracF^(k)(μ)k! =frac12π ioint_|z-μ|=r_μ fracF(z)(z-μ)^k+1 dz. Consequently the same canonical operator is obtained fr…
Exact mathematical passage
\overline{D(\mu,r_\mu)}\subset U
\overline{g(x)}Use in source · 1 passage
Search aliases: overlineg(x) overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3261
-xf'(x))+o(t). Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is o(t), so the expansion may be integrated term by term. Therefore .frac d d t O_tf|_t=0 =f''-xf'. The Hermite eigenvalue identity also follows by differentiating O_th_n= e^-nth_n at zero. Gaussian integration by parts gives ∫_ R f'(x)overlineg(x)γ( d x) = ∫_ R f(x)overlinexg(x)-g'(x)γ( d x). Set g=f' and rearrange.
Exact mathematical passage
\int_\RR f'(x)\overline{g(x)}\gamma(\dd x) = \int_\RR f(x)\overline{xg(x)-g'(x)}\gamma(\dd x).
\overline{xg(x)-g'(x)}Use in source · 1 passage
Search aliases: overlinexg(x)-g'(x) overline
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3261
-xf'(x))+o(t). Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is o(t), so the expansion may be integrated term by term. Therefore .frac d d t O_tf|_t=0 =f''-xf'. The Hermite eigenvalue identity also follows by differentiating O_th_n= e^-nth_n at zero. Gaussian integration by parts gives ∫_ R f'(x)overlineg(x)γ( d x) = ∫_ R f(x)overlinexg(x)-g'(x)γ( d x). Set g=f' and rearrange.
Exact mathematical passage
\int_\RR f'(x)\overline{g(x)}\gamma(\dd x) = \int_\RR f(x)\overline{xg(x)-g'(x)}\gamma(\dd x).
\overline\lambda_\betaUse in source · 1 passage
Search aliases: overlineλ_β overline lambda beta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3500
The posterior mass of packet I_α is frac∑_i∈ I_αλ_ip_i ∑_jλ_jp_j = frac(Cp)_αoverlineλ_α(p) ∑_β(Cp)_βoverlineλ_β(p). This is the coarse Bayesian formula. Dividing the fine posterior coordinates by the packet mass gives the local formula.
Exact mathematical passage
\frac{\sum_{i\in I_\alpha}\lambda_ip_i} {\sum_j\lambda_jp_j} = \frac{(Cp)_\alpha\overline\lambda_\alpha(p)} {\sum_\beta(Cp)_\beta\overline\lambda_\beta(p)}.
\overline\piUse in source · 6 passages
Search aliases: overlineπ overline pi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5355
se Markov generator. For observables, the backward generator is Q^T; weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
\overline\pi_a=\sum_{i\in B_a}\pi_i.
\overline\pi_aUse in source · 3 passages
Search aliases: overlineπ_a overline pi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5355
se Markov generator. For observables, the backward generator is Q^T; weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
\overline\pi_a=\sum_{i\in B_a}\pi_i.
\overline\pi_bUse in source · 2 passages
Search aliases: overlineπ_b overline pi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5359
the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
\overline Q_{ab}\overline\pi_b = \overline Q_{ba}\overline\pi_a.
\overline\PhiUse in source · 4 passages
Search aliases: overlineΦ overline Phi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5534
For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
\overline\Phi_y=\Bcal_{\overline L_y}\circ\overline K
\overline\Phi_yUse in source · 3 passages
Search aliases: overlineΦ_y overline Phi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5534
For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
\overline\Phi_y=\Bcal_{\overline L_y}\circ\overline K
\overline\Phi_{y_1}Use in source · 1 passage
Search aliases: overlineΦ_y_1 overline Phi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5545
[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
C\Phi_{y_r}\circ\cdots\circ\Phi_{y_1} = \overline\Phi_{y_r}\circ\cdots\circ\overline\Phi_{y_1}C.
\overline\Phi_{y_r}Use in source · 1 passage
Search aliases: overlineΦ_y_r overline Phi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5545
[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
C\Phi_{y_r}\circ\cdots\circ\Phi_{y_1} = \overline\Phi_{y_r}\circ\cdots\circ\overline\Phi_{y_1}C.
P^{\mathsf T}Use in source · 5 passages
Search aliases: P^ T mathsf
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 926
[Bayesian information-loss packet] Assume nonnegative fine and declared coarse likelihoods, with both Z_r>0 and Z_barλ>0. The coarsening defect is C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood; the full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together. Moreover, C_π B_λ(p)- B_barλ(P) …
Exact mathematical passage
\boxed{ C_\pi\cB_\lambda(p) - \cB_{\bar\lambda}(P) = \frac{1}{Z_r} \left(\diag(P)-\bar bP^{\mathsf T}\right) \delta(p).}
p^{(a)}Use in source · 3 passages
Search aliases: p^(a)
Use in source · atlas-pages.tex, line 216
vpage10Two priors that fool the coarse modelSUFFICIENCY / A COMPUTABLE FAILURE WITNESSTraditional Part I: Theorem 8.2 and the one-packet exception. Keep the same likelihood λ=(4,1,1,1)^ T and partition A={0,1}, B={2,3}. Compare p^(a)=(1/2,0,1/2,0)^ T, p^(b)=(0,1/2,1/2,0)^ T. Both have coarse prior (1/2,1/2)^ T. A procedure given only that two-entry vector cannot distinguish them. vfiginsufficiency platecaptionAfter the identical fine likelihood is applied, the coarse posteriors are (4/5,1/5) and…
Exact mathematical passage
p^{(a)}=(1/2,0,1/2,0)^{\mathsf T},\qquad p^{(b)}=(0,1/2,1/2,0)^{\mathsf T}.
p^{(b)}Use in source · 3 passages
Search aliases: p^(b)
Use in source · atlas-pages.tex, line 216
vpage10Two priors that fool the coarse modelSUFFICIENCY / A COMPUTABLE FAILURE WITNESSTraditional Part I: Theorem 8.2 and the one-packet exception. Keep the same likelihood λ=(4,1,1,1)^ T and partition A={0,1}, B={2,3}. Compare p^(a)=(1/2,0,1/2,0)^ T, p^(b)=(0,1/2,1/2,0)^ T. Both have coarse prior (1/2,1/2)^ T. A procedure given only that two-entry vector cannot distinguish them. vfiginsufficiency platecaptionAfter the identical fine likelihood is applied, the coarse posteriors are (4/5,1/5) and…
Exact mathematical passage
p^{(a)}=(1/2,0,1/2,0)^{\mathsf T},\qquad p^{(b)}=(0,1/2,1/2,0)^{\mathsf T}.
P^2Use in source · 1 passage
Search aliases: P^2
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2754
[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.
Exact mathematical passage
\norm{QT(t)P} \leq M M_Q\norm P^2\norm{QAP} \int_0^t \e^{\omega_Q(t-s)+\omega s} \dd s.
P_{\mathcal Q}Use in source · 5 passages
Search aliases: P_ Q mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 649
Let Ω={1,…,n} carry strictly positive probabilities π_i, and let Q={B_1,…,B_m} be a partition. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j∑_j∈ B(i)π_j, where B(i) is the block containing i.
Exact mathematical passage
(P_{\mathcal Q}f)_i = \frac{\sum_{j\in B(i)}\pi_jf_j}{\sum_{j\in B(i)}\pi_j},
P_{\mathcal Q}^2Use in source · 1 passage
Search aliases: P_ Q^2 mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 662
[Finite conditional-expectation projector] With the weighted inner product ⟨ f,g⟩_π=∑_iπ_if_ioverlineg_i, the operator P_ Q satisfies P_ Q^2=P_ Q, ⟨ P_ Qf,g⟩_π = ⟨ f,P_ Qg⟩_π. Its range is the space of functions constant on each partition block.
Exact mathematical passage
P_{\mathcal Q}^2=P_{\mathcal Q}, \qquad \langle P_{\mathcal Q}f,g\rangle_\pi = \langle f,P_{\mathcal Q}g\rangle_\pi.
P_0Use in source · 8 passages
Search aliases: P_0
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 689
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
P_0\le P_1\le\cdots\le P_r
p_0Use in source · 36 passages
Search aliases: p_0
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1585
Let A be a finite subgenerator: its off-diagonal entries are nonnegative and its column sums are nonpositive. These sign conditions alone allow nonabsorbing states; eventual absorption is a separate hypothesis. Let p_0 be an initial transient-state distribution. Define p(t)=e^tAp_0, S(t)=1^Tp(t). The exit-rate row vector is a^T=-1^TA≥0.
Exact mathematical passage
p_0
p_0^TUse in source · 1 passage
Search aliases: p_0^T
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4942
Average [source reference] with terminal function f= 1 against the initial distribution. Transposition gives p_0^T e^t(Q^T-D_V) 1 = 1^T e^t(Q-D_V)p_0. Since 1^TQ=0, differentiation gives S_V'(t) = - 1^TD_V e^tAp_0.
Exact mathematical passage
p_0^T\e^{t(Q^T-D_V)}\one = \one^T\e^{t(Q-D_V)}p_0.
P_1Use in source · 6 passages
Search aliases: P_1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 689
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
P_0\le P_1\le\cdots\le P_r
p_1Use in source · 6 passages
Search aliases: p_1
Use in source · atlas-pages.tex, line 10
one probability vector and its two block masses. The right panel is a separate retained/hidden model, developed on pagerefV22. Every curve is drawn from an explicit formula. headCoordinates retain; aggregation forgets; elimination remembers Keeping every component P_a x is an exact change of representation when ∑_aP_a=I. Reconstruct by summing the components. In contrast, replacing (p_0,p_1,p_2,p_3) by (p_0+p_1,p_2+p_3) forgets how mass was distributed inside either pair. That information can ma…
Exact mathematical passage
(p_0,p_1,p_2,p_3)
P_2Use in source · 3 passages
Search aliases: P_2
Use in source · atlas-pages.tex, line 84
basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next. headThe same evolution has a closed formula Writing q=e^-2t gives e^tQ=P_0+qP_1+q^2P_2+q^3P_3. At q=1/2, the point seed e_0 becomes (15,5,9,3)^ T/32. The polynomial identity T(q)T(r)=T(qr) checks comp…
Exact mathematical passage
e^{tQ}=P_0+qP_1+q^2P_2+q^3P_3.
p_2Use in source · 4 passages
Search aliases: p_2
Use in source · atlas-pages.tex, line 10
one probability vector and its two block masses. The right panel is a separate retained/hidden model, developed on pagerefV22. Every curve is drawn from an explicit formula. headCoordinates retain; aggregation forgets; elimination remembers Keeping every component P_a x is an exact change of representation when ∑_aP_a=I. Reconstruct by summing the components. In contrast, replacing (p_0,p_1,p_2,p_3) by (p_0+p_1,p_2+p_3) forgets how mass was distributed inside either pair. That information can ma…
Exact mathematical passage
(p_0,p_1,p_2,p_3)
P_3Use in source · 3 passages
Search aliases: P_3
Use in source · atlas-pages.tex, line 84
basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next. headThe same evolution has a closed formula Writing q=e^-2t gives e^tQ=P_0+qP_1+q^2P_2+q^3P_3. At q=1/2, the point seed e_0 becomes (15,5,9,3)^ T/32. The polynomial identity T(q)T(r)=T(qr) checks comp…
Exact mathematical passage
e^{tQ}=P_0+qP_1+q^2P_2+q^3P_3.
p_3Use in source · 2 passages
Search aliases: p_3
Use in source · atlas-pages.tex, line 10
one probability vector and its two block masses. The right panel is a separate retained/hidden model, developed on pagerefV22. Every curve is drawn from an explicit formula. headCoordinates retain; aggregation forgets; elimination remembers Keeping every component P_a x is an exact change of representation when ∑_aP_a=I. Reconstruct by summing the components. In contrast, replacing (p_0,p_1,p_2,p_3) by (p_0+p_1,p_2+p_3) forgets how mass was distributed inside either pair. That information can ma…
Exact mathematical passage
(p_0,p_1,p_2,p_3)
P_aUse in source · 15 passages
Search aliases: P_a
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 840
Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero.
Exact mathematical passage
(C_\pi p)_a=P_a=\sum_{i\in I_a}p_i.
P_bUse in source · 2 passages
Search aliases: P_b
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 865
For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.
Exact mathematical passage
\bigl(C_\pi\cB_\lambda(p)\bigr)_a = \frac{\sum_{i\in I_a}\lambda_ip_i}{Z} = \frac{P_ar_a(p)}{\sum_bP_br_b(p)} = \bigl(\cB_{r(p)}(P)\bigr)_a.
P_cUse in source · 2 passages
Search aliases: P_c
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 684
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
\boxed{ P_cP_f=P_fP_c=P_c. }
P_fUse in source · 1 passage
Search aliases: P_f
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 684
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
\boxed{ P_cP_f=P_fP_c=P_c. }
P_iUse in source · 7 passages
Search aliases: P_i
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 704
A coarse-block average of a fine-block average is the coarse-block average, and a fine-block average of a coarse-constant function leaves it unchanged. The difference identities follow from P_iP_j=P_min(i,j).
Exact mathematical passage
P_iP_j=P_{\min(i,j)}
p_i'Use in source · 1 passage
Search aliases: p_i'
Use in source · atlas-pages.tex, line 843
tatistical meanings The first derivative of log evidence is the current mean score, and the second is its variance: fracddtlog Z(t)= E_p(t)u, fracd^2dt^2log Z(t)=Var_p(t)(u)≥0. At q=2 the exact values are E u=frac107, E u^2=frac187, Var(u)=frac2649. Thus the derivative of the mean is 26/49 at this time. The flow equation p_i'=(u_i- E u)p_i gives derivatives (-10,-6,16)/49, which sum to zero.
Exact mathematical passage
p_i'=(u_i-\EE u)p_i
P_jUse in source · 19 passages
Search aliases: P_j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 690
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
D_0=P_0, \qquad D_j=P_j-P_{j-1}.
p_jUse in source · 7 passages
Search aliases: p_j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 752
The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.
Exact mathematical passage
\frac{\mu_i(\lambda_ip_i/Z_\lambda)}{\sum_j\mu_j\lambda_jp_j/Z_\lambda} = \frac{\mu_i\lambda_ip_i}{\sum_j\mu_j\lambda_jp_j}.
P_{j-1}Use in source · 2 passages
Search aliases: P_j-1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 690
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
D_0=P_0, \qquad D_j=P_j-P_{j-1}.
P_kUse in source · 1 passage
Search aliases: P_k
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1148
All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.
Exact mathematical passage
L^\alpha L^\beta =\sum_{j,k}\lambda_j^\alpha\lambda_k^\beta P_jP_k =\sum_j\lambda_j^{\alpha+\beta}P_j.
p_MUse in source · 4 passages
Search aliases: p_M
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3838
[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).
Exact mathematical passage
1-\sum_{i\in M}p_i(t) \leq \frac{1-p_M(0)}{p_M(0)}\e^{-\delta t},
P_{m,n}Use in source · 11 passages
Search aliases: P_m,n
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2649
[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally uniform and every T_n is norm continuous, then U_m is a norm continuous semigroup.
Exact mathematical passage
P_{m,n}=R_{n,m}C_{m,n}
P_{\min(i,j)}Use in source · 3 passages
Search aliases: P_min(i,j) min
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 704
A coarse-block average of a fine-block average is the coarse-block average, and a fine-block average of a coarse-constant function leaves it unchanged. The difference identities follow from P_iP_j=P_min(i,j).
Exact mathematical passage
P_iP_j=P_{\min(i,j)}
P_nUse in source · 5 passages
Search aliases: P_n
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3193
[Hermite packet reconstruction] Let P_n be the orthogonal projection onto C h_n. Then I=∑_n=0^∞ P_n in the strong operator topology and O_t = ∑_n=0^∞ e^-ntP_n. For f∈ L^2(γ), norm O_t f_2^2 = ∑_n=0^∞ e^-2ntnormP_nf_2^2.
Exact mathematical passage
P_n
P_rUse in source · 8 passages
Search aliases: P_r
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 689
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
P_0\le P_1\le\cdots\le P_r
p_rUse in source · 8 passages
Search aliases: p_r
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1092
Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).
Exact mathematical passage
\widehat p_r=Kp_{r-1}, \qquad p_r=\cB_{\lambda^{(r)}}(\widehat p_r).
p_{r-1}Use in source · 4 passages
Search aliases: p_r-1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1092
Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).
Exact mathematical passage
\widehat p_r=Kp_{r-1}, \qquad p_r=\cB_{\lambda^{(r)}}(\widehat p_r).
p_tUse in source · 2 passages
Search aliases: p_t
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5019
[Residual life as a restarted semigroup] For S(t)>0, define the conditional transient distribution p_t = frac e^tAp_0S(t). Then fracS(t+s)S(t) = 1^T e^sAp_t.
Exact mathematical passage
p_t = \frac{\e^{tA}p_0}{S(t)}.
p_\alphaUse in source · 3 passages
Search aliases: p_α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3653
[Exact information decomposition] Let p and q be probability vectors on a finite partition (I_α)_α, and assume q_i=0 whenever p_i=0. Let m=Cp, n=Cq, and let p_α and q_α be the conditional distributions inside packet I_α whenever the corresponding mass is positive. Then D(qVert p) = D(nVert m) + ∑_α:n_α>0 n_α D(q_αVert p_α). The first term is the information retained by the coarse masses. The second term is the exact information carried by changes inside the packets.
Exact mathematical passage
p_\alpha
p_{\alpha,i}Use in source · 2 passages
Search aliases: p_α,i alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3667
For i∈ I_α, write p_i=m_α p_α,i, q_i=n_α q_α,i. Then logfracq_ip_i = logfracn_αm_α + logfracq_α,ip_α,i. Multiply by q_i, sum first inside each packet, and then sum over the packets. This is the finite relative entropy chain rule. Standard information theoretic treatments include [source reference].
Exact mathematical passage
p_i=m_\alpha p_{\alpha,i}, \qquad q_i=n_\alpha q_{\alpha,i}.
P_{\alpha_0}Use in source · 1 passage
Search aliases: P_α_0 alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 431
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
P_{\alpha_1}Use in source · 1 passage
Search aliases: P_α_1 alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 431
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
P_{\alpha_m}Use in source · 1 passage
Search aliases: P_α_m alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 431
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
P_{\alpha_{m-1}}Use in source · 1 passage
Search aliases: P_α_m-1 alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 431
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
P_\gammaUse in source · 3 passages
Search aliases: P_γ gamma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 399
Insert the resolution of the identity between the two semigroup factors: P_β e^(s+t)AP_α · =P_β e^sA(∑_γ P_γ)e^tAP_α · =∑_γ (P_β e^sAP_γ)(P_γ e^tAP_α).
Exact mathematical passage
\begin{aligned} P_\beta e^{(s+t)A}P_\alpha &=P_\beta e^{sA}\left(\sum_\gamma P_\gamma\right)e^{tA}P_\alpha\\ &=\sum_\gamma (P_\beta e^{sA}P_\gamma)(P_\gamma e^{tA}P_\alpha). \end{aligned}
\pmodUse in source · 1 passage
Search aliases: pmod
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 578
[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.
Exact mathematical passage
k\equiv n\pmod2
q'Use in source · 1 passage
Search aliases: q'
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2779
For x∈ PX, write p(t)=PT(t)x, q(t)=QT(t)x. The hidden component satisfies q'(t)=QAQq(t)+QAPp(t), q(0)=0. Variation of constants gives q(t)= ∫_0^t e^(t-s)(QAQ)|_QX QAPp(s) d s. Since normp(s)≤norm PnormT(s)norm x and x=Px, the first estimate follows. The contraction case is immediate.
Exact mathematical passage
q'(t)=QAQq(t)+QAPp(t), \qquad q(0)=0.
Q^*Use in source · 1 passage
Search aliases: Q^*
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3027
[Finite conditional expectation] The operator P_ Q is the orthogonal projector onto the functions that are constant on every block of Q. In particular, P_ Q^2=P_ Q, P_ Q^*=P_ Q.
Exact mathematical passage
P_\mathcal Q^2=P_\mathcal Q, \qquad P_\mathcal Q^*=P_\mathcal Q.
Q^2Use in source · 1 passage
Search aliases: Q^2
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3027
[Finite conditional expectation] The operator P_ Q is the orthogonal projector onto the functions that are constant on every block of Q. In particular, P_ Q^2=P_ Q, P_ Q^*=P_ Q.
Exact mathematical passage
P_\mathcal Q^2=P_\mathcal Q, \qquad P_\mathcal Q^*=P_\mathcal Q.
q^2Use in source · 2 passages
Search aliases: q^2
Use in source · atlas-pages.tex, line 84
basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next. headThe same evolution has a closed formula Writing q=e^-2t gives e^tQ=P_0+qP_1+q^2P_2+q^3P_3. At q=1/2, the point seed e_0 becomes (15,5,9,3)^ T/32. The polynomial identity T(q)T(r)=T(qr) checks comp…
Exact mathematical passage
e^{tQ}=P_0+qP_1+q^2P_2+q^3P_3.
q^3Use in source · 1 passage
Search aliases: q^3
Use in source · atlas-pages.tex, line 84
basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next. headThe same evolution has a closed formula Writing q=e^-2t gives e^tQ=P_0+qP_1+q^2P_2+q^3P_3. At q=1/2, the point seed e_0 becomes (15,5,9,3)^ T/32. The polynomial identity T(q)T(r)=T(qr) checks comp…
Exact mathematical passage
e^{tQ}=P_0+qP_1+q^2P_2+q^3P_3.
Q^TUse in source · 13 passages
Search aliases: Q^T
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4381
[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.
Exact mathematical passage
\left( \e^{t(Q^T-D_V)}f \right)_i = \Exp_i\left[ \exp\left( -\int_0^tV(X_s)\dd s \right) f(X_t) \right].
q_{\mathrm A}Use in source · 2 passages
Search aliases: q_ A mathrm
Use in source · atlas-pages.tex, line 205
vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditional shapes are m_ A=5/7, q_ A=(4/5,1/5,0,0)^ T, m_ B=2/7, q_ B=(0,0,1/2,1/2)^ T. Thus m_ Aq_ A+m_ Bq_ B is exactly the displayed posterior. Keeping o…
Exact mathematical passage
m_{\mathrm A}=5/7,\quad q_{\mathrm A}=(4/5,1/5,0,0)^{\mathsf T}, \qquad m_{\mathrm B}=2/7,\quad q_{\mathrm B}=(0,0,1/2,1/2)^{\mathsf T}.
q_{\mathrm B}Use in source · 2 passages
Search aliases: q_ B mathrm
Use in source · atlas-pages.tex, line 205
vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditional shapes are m_ A=5/7, q_ A=(4/5,1/5,0,0)^ T, m_ B=2/7, q_ B=(0,0,1/2,1/2)^ T. Thus m_ Aq_ A+m_ Bq_ B is exactly the displayed posterior. Keeping o…
Exact mathematical passage
m_{\mathrm A}=5/7,\quad q_{\mathrm A}=(4/5,1/5,0,0)^{\mathsf T}, \qquad m_{\mathrm B}=2/7,\quad q_{\mathrm B}=(0,0,1/2,1/2)^{\mathsf T}.
\mathcal Q_{\mathcal L}Use in source · 3 passages
Search aliases: Q_ L mathcal mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3550
[Coarsest fixed sufficient partition] Let L be a nonempty family of strictly positive likelihood vectors. Define i∼_ Lj Longleftrightarrow λ_i=λ_j hboxfor every λ∈ L. Let Q_ L be the partition into equivalence classes. A partition Q gives a fixed coarse sufficient description in the sense of [source reference] for every λ∈ L if and only if every block of Q is contained in a block of Q_ L. Hence Q_ L is the unique coarsest partition with this property.
Exact mathematical passage
\mathcal Q_{\mathcal L}
Q_aUse in source · 1 passage
Search aliases: Q_a
Use in source · atlas-pages.tex, line 818
The weighting is by the packet mass P_a from the first argument of D(pVert q). Replacing it by Q_a generally breaks the identity. Likewise, forgetting zero-mass conventions or allowing a positive p_i over a zero q_i changes which expressions are finite. takeawayCoarse relative entropy measures distinguishability that survives aggregation. The conditional term records exactly the distinguishability removed inside the packets.
Exact mathematical passage
Q_a
q_aUse in source · 10 passages
Search aliases: q_a
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5401
FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
p_i=q_a r_i^{(a)}
Q_cUse in source · 1 passage
Search aliases: Q_c
Use in source · atlas-pages.tex, line 344
-1 · 1C. The coarse matrix is the positive square root of -Q_c. This is a concrete naturality identity, checked on the full matrices. It is not a claim that an arbitrary partition commutes with every fractional construction.
Exact mathematical passage
-Q_c
\mathcal Q_cUse in source · 1 passage
Search aliases: Q_c mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 683
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
\mathcal Q_c
\mathcal Q_fUse in source · 1 passage
Search aliases: Q_f mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 683
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
\mathcal Q_f
Q_{ii}Use in source · 2 passages
Search aliases: Q_ii
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 550
Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.
Exact mathematical passage
\lambda\ge\max_i(-Q_{ii})
Q_{ij}Use in source · 2 passages
Search aliases: Q_ij
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5351
ty means CQ=overline Q C for a coarse Markov generator. For observables, the backward generator is Q^T; weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
Q_{ij}\pi_j=Q_{ji}\pi_i.
Q_jUse in source · 9 passages
Search aliases: Q_j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4644
[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r}…
Exact mathematical passage
Q_j
Q_{ji}Use in source · 2 passages
Search aliases: Q_ji
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5351
ty means CQ=overline Q C for a coarse Markov generator. For observables, the backward generator is Q^T; weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
Q_{ij}\pi_j=Q_{ji}\pi_i.
Q_{m,n}Use in source · 3 passages
Search aliases: Q_m,n
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2809
A bounded generator of a contraction semigroup on a Hilbert space is dissipative. If Q_m,n=I-P_m,n and y∈ Q_m,nX_n, then ReipQ_m,nA_nQ_m,nyy = ReipA_nyy ≤0. Hence the compressed hidden semigroup is contractive. Apply [source reference] to P_m,n. Since P_m,nR_n,m=R_n,m, η_m,n(T) ≤ Tnorm(I-P_m,n)A_nP_m,n normR_n,m. The right side tends to zero. Use [source reference].
Exact mathematical passage
Q_{m,n}=I-P_{m,n}
Q_{ri}Use in source · 1 passage
Search aliases: Q_ri
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2940
[Continuous time lumping] A coarse Markov generator overline Q satisfying CQ=overline QC exists if and only if the aggregated rates ∑_r∈ B_αQ_ri depend only on the source block of i. In that case C e^tQ= e^toverline QC for every t≥0.
Exact mathematical passage
\sum_{r\in B_\alpha}Q_{ri}
q_\alphaUse in source · 12 passages
Search aliases: q_α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 774
[Bayesian packet reconstruction] B_λ(p) = ∑_α m_α q_α. Only packets with positive evidence carry a conditional probability vector. A zero-evidence packet has zero posterior mass and contributes the zero vector. The nonlinear coupling between packets occurs only through the total evidence used to normalize the packet evidences.
Exact mathematical passage
\boxed{ \cB_\lambda(p) = \sum_\alpha m_\alpha q_\alpha. }
q_{\alpha,i}Use in source · 10 passages
Search aliases: q_α,i alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 762
Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.
Exact mathematical passage
Z_\alpha=\sum_{i\in I_\alpha}\lambda_ip_i, \qquad m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta}, \qquad q_{\alpha,i}=\begin{cases} \lambda_ip_i/Z_\alpha,&Z_\alpha>0,\ i\in I_\alpha,\\ 0,&\text{otherwise}. \end{cases}
\mathbb RUse in source · 1 passage
Search aliases: R mathbb
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5179
e renewcommandtheHfigureR.arabicfigure renewcommandthetableRarabictable renewcommandtheHtableR.arabictable FloatBarrier Transfer paths and product-topology limits. Let T(t) be an arbitrary strongly continuous semigroup on a complex Banach space X, with generator A, and suppose normT(t)≤ M e^ω t for M≥1 and ω∈ R. Let (P_α)_α∈ I_0 be a finite family of bounded pairwise orthogonal idempotents summing to I, and put T_βα(t)=P_β T(t)P_α. The generator need not be bounded. In the following resolvent st…
Exact mathematical passage
\omega\in\mathbb R
r^{(a)}Use in source · 1 passage
Search aliases: r^(a)
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5401
FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
r^{(a)}
R_0Use in source · 2 passages
Search aliases: R_0
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2689
[Summable defects do not force a semigroup limit] Let X_0= R and let X_n= R^2 for every n≥1. Set C_0(x,y)=x, R_0x=(x,0), and use identity connecting maps and reconstructions above level one. Let T_0(t)=I and let every higher level carry the rotation group T_n(t) = cos t · -sin t
Exact mathematical passage
C_0(x,y)=x, \qquad R_0x=(x,0),
r_1Use in source · 1 passage
Search aliases: r_1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5241
Every nonzero term in [source reference] contains at least m_0 time integrations. The simplex 0<r_1<⋯<r_m<t has volume t^m/m!, so the first possible order is t^m_0. If there is no path, every packet term vanishes.
Exact mathematical passage
0<r_1<\cdots<r_m<t
r_aUse in source · 5 passages
Search aliases: r_a
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 844
Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero.
Exact mathematical passage
r_a(p) = \frac{\sum_{i\in I_a}\lambda_i p_i}{P_a} \qquad(P_a>0).
r_bUse in source · 1 passage
Search aliases: r_b
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 865
For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.
Exact mathematical passage
\bigl(C_\pi\cB_\lambda(p)\bigr)_a = \frac{\sum_{i\in I_a}\lambda_ip_i}{Z} = \frac{P_ar_a(p)}{\sum_bP_br_b(p)} = \bigl(\cB_{r(p)}(P)\bigr)_a.
R_cUse in source · 10 passages
Search aliases: R_c
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 514
For z in the common resolvent set, let R_f(z)=(zI-A_f)^-1, R_c(z)=(zI-A_c)^-1.
Exact mathematical passage
R_f(z)=(zI-A_f)^{-1}, \qquad R_c(z)=(zI-A_c)^{-1}.
R_DUse in source · 4 passages
Search aliases: R_D
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2155
[Resolvent expansion over packet paths] With A=D+N as in the directed path decomposition, let R_D(z)=(zI-D)^-1. If normR_D(z)N<1, then (zI-A)^-1 = ∑_m=0^∞ (R_D(z)N)^mR_D(z). After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
R_D(z)=(zI-D)^{-1}.
R_fUse in source · 10 passages
Search aliases: R_f
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 514
For z in the common resolvent set, let R_f(z)=(zI-A_f)^-1, R_c(z)=(zI-A_c)^-1.
Exact mathematical passage
R_f(z)=(zI-A_f)^{-1}, \qquad R_c(z)=(zI-A_c)^{-1}.
r_i^{(a)}Use in source · 3 passages
Search aliases: r_i^(a)
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5401
FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
p_i=q_a r_i^{(a)}
R_mUse in source · 1 passage
Search aliases: R_m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2522
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,n}=C_mC_{m+1}\cdots C_{n-1}, \qquad R_{n,m}=R_{n-1}\cdots R_m.
r_mUse in source · 1 passage
Search aliases: r_m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5241
Every nonzero term in [source reference] contains at least m_0 time integrations. The simplex 0<r_1<⋯<r_m<t has volume t^m/m!, so the first possible order is t^m_0. If there is no path, every packet term vanishes.
Exact mathematical passage
0<r_1<\cdots<r_m<t
R_{m,m}Use in source · 1 passage
Search aliases: R_m,m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2528
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,m}=I_{X_m}, \qquad R_{m,m}=I_{X_m}.
R_nUse in source · 4 passages
Search aliases: R_n
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2547
[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
w_{m,n} = \norm{C_{m,n}}\, \norm{R_nR_{n,m}}.
R_{n-1}Use in source · 1 passage
Search aliases: R_n-1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2522
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,n}=C_mC_{m+1}\cdots C_{n-1}, \qquad R_{n,m}=R_{n-1}\cdots R_m.
R_xUse in source · 3 passages
Search aliases: R_x
Use in source · atlas-pages.tex, line 472
1 · -1. Retain the first coordinate and eliminate the second. For z away from the relevant poles, the retained entry of (zI-A)^-1 is R_x(z)=[z+1-frac1z+1]^-1 =fracz+1z(z+2). The term Σ(z)=1/(z+1) is the hidden coordinate's self-energy. vfigresolvent platecaptionThe full inverse gives a rational response with two poles, at 0 and -2. Its partial fractions are R_x(z)=frac12z^-1+frac12(z+2)^-1. The hidden intermediate inverse has its own excluded point z=-1. headAn eliminated variable has not disapp…
Exact mathematical passage
R_x(z)=\left[z+1-\frac1{z+1}\right]^{-1} =\frac{z+1}{z(z+2)}.
r_\alphaUse in source · 6 passages
Search aliases: r_α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3526
If λ is constant on each packet, [source reference] has packet averages independent of the prior and its local posterior formula gives q_α=r_α. This proves the first condition.
Exact mathematical passage
q_\alpha=r_\alpha
r_\muUse in source · 2 passages
Search aliases: r_μ mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1664
ator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ)⊂ U, and Cauchy's formula gives fracF^(k)(μ)k! =frac12π ioint_|z-μ|=r_μ fracF(z)(z-μ)^k+1 dz. Consequently the same canonical operator i…
Exact mathematical passage
r_\mu>0
R_\ellUse in source · 3 passages
Search aliases: R_ℓ ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2516
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
R_\ell:X_\ell\to X_{\ell+1}, \qquad C_\ell R_\ell=I_{X_\ell}.
\RanUse in source · 3 passages
Search aliases: Ran operatorname
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1503
[Schur--Feshbach projected resolvent] Assume both zI-A and zI-A_QQ are invertible. Define F_P(z) =zI-A_PP-A_PQ(zI-A_QQ)^-1A_QP. Then .P(zI-A)^-1|_Ran P=F_P(z)^-1.
Exact mathematical passage
\boxed{ \left.P(zI-A)^{-1}\right|_{\Ran P}=F_P(z)^{-1}. }
\rmUse in source · 4 passages
Search aliases: rm
Use in source · atlas-pages.tex, line 424
vpage19Convergence with a nonzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMTraditional Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil differentiates the quadratic x(1-x) exactly, the solution u_h to the sampled forcing 12x_i^2 satisfies u_h(x_i)-u(x_i)=-h^2x_i(1-x_i). This gives both …
Exact mathematical passage
L_hu|_{\rm nodes}=12x_i^2+2h^2.
\RR^2Use in source · 2 passages
Search aliases: RR^2 R^2 RR mathbb
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2688
[Summable defects do not force a semigroup limit] Let X_0= R and let X_n= R^2 for every n≥1. Set C_0(x,y)=x, R_0x=(x,0), and use identity connecting maps and reconstructions above level one. Let T_0(t)=I and let every higher level carry the rotation group T_n(t) = cos t · -sin t
Exact mathematical passage
X_n=\RR^2
\RR^nUse in source · 3 passages
Search aliases: RR^n R^n RR mathbb
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3330
Let Δ_n-1 = {p∈ R^n: p_i≥0, ∑_ip_i=1 } be the probability simplex. Let λ∈[0,∞)^n have positive evidence Z_λ(p)=∑_iλ_ip_i. The Bayesian transfer is B_λ(p) = fracλodot pZ_λ(p).
Exact mathematical passage
\Delta_{n-1} = \left\{p\in\RR^n: p_i\geq0, \quad \sum_ip_i=1 \right\}
SUse in source · 43 passages
Search aliases: S
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 570
[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.
Exact mathematical passage
T=\frac12(S+S^{-1})
S'Use in source · 5 passages
Search aliases: S'
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1598
[Semigroup survival identities] For t≥0, the finite-time exit density and, wherever S(t)>0, the hazard are f(t)=a^Te^tAp_0=-S'(t), h(t)=fracf(t)S(t). Consequently, S is nonincreasing and S(t)=exp(-∫_0^th(r) dr) whenever S(t)>0. The density may be defective: its total mass is 1-lim_t→∞S(t). It is a probability density when absorption is eventual, so that this limit is zero.
Exact mathematical passage
\boxed{ f(t)=a^Te^{tA}p_0=-S'(t), \qquad h(t)=\frac{f(t)}{S(t)}. }
S^{-1}Use in source · 1 passage
Search aliases: S^-1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 570
[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.
Exact mathematical passage
T=\frac12(S+S^{-1})
s^{-1/2}Use in source · 1 passage
Search aliases: s^-1/2
Use in source · atlas-pages.tex, line 394
A normalized density need not have a finite mean. For large s, the exponential factor tends to one and η_t(s) is proportional to s^-3/2. Consequently sη_t(s) has a nonintegrable s^-1/2 tail. The clock has infinite mean even though its total mass is one. takeawaySubordination is a weighted mixture of diffusion times. Replacing the clock distribution by a single average time is not justified, and here that average is infinite.
Exact mathematical passage
s^{-1/2}
s^{-3/2}Use in source · 6 passages
Search aliases: s^-3/2
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1166
[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.
Exact mathematical passage
\boxed{ e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)}e^{-sL}\,ds. }
s^mUse in source · 1 passage
Search aliases: s^m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 342
Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.
Exact mathematical passage
e^{sA}e^{tA} =\sum_{m,n\ge0}\frac{s^mt^n}{m!n!}A^{m+n} =\sum_{r\ge0}\frac{(s+t)^r}{r!}A^r.
s^\alphaUse in source · 1 passage
Search aliases: s^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4056
Set R_f(s)=(sI+L_f)^-1, R_c(s)=(sI+L_c)^-1. The resolvent identity gives CR_f(s)-R_c(s)C = -R_c(s)Δ_LR_f(s). Since LR=I-sR, CL_fR_f(s)-L_cR_c(s)C = sR_c(s)Δ_LR_f(s). Insert this into [source reference]. The spectral lower bound gives normR_f(s),normR_c(s) ≤ frac1s+m. Therefore normCL_f^α-L_c^α C · ≤ fracsin(πα)π normΔ_L ∫_0^∞ fracs^α(s+m)^2 d s · = α m^α-1normΔ_L. The semigroup estimate follows from [source reference], applied to the contraction generators -L_f^α and -L_c^α.
Exact mathematical passage
\begin{aligned} \norm{CL_f^\alpha-L_c^\alpha C} &\leq \frac{\sin(\pi\alpha)}{\pi} \norm{\Delta_L} \int_0^\infty \frac{s^\alpha}{(s+m)^2}\dd s\\ &= \alpha m^{\alpha-1}\norm{\Delta_L}. \end{aligned}
s^{\alpha-1}Use in source · 2 passages
Search aliases: s^α-1 alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3955
[Balakrishnan formula in finite dimension] For 0<α<1, L^α = fracsin(πα)π ∫_0^∞ s^α-1L(sI+L)^-1 d s. The integral converges in operator norm.
Exact mathematical passage
L^\alpha = \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty s^{\alpha-1}L(sI+L)^{-1} \dd s.
s_1Use in source · 1 passage
Search aliases: s_1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2235
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
s_mUse in source · 1 passage
Search aliases: s_m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2235
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
s_{m-1}Use in source · 1 passage
Search aliases: s_m-1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2235
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
S_VUse in source · 2 passages
Search aliases: S_V
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4917
[State dependent killing as survival] Let Q be a finite Markov generator, let V≥0, and set A=Q-D_V. For an initial distribution p_0, define S_V(t)= 1^T e^tAp_0. Then S_V(t) = E_p_0 [ exp( -∫_0^tV(X_s) d s ) ]. Also, -S_V'(t) = 1^TD_V e^tAp_0. The matrix A=Q-D_V has nonnegative off diagonal entries and nonpositive column sums, so it is the subgenerator of the killed, possibly nontransient chain. Thus the potential in the finite Feynman Kac formula is a state dependent killing rate in the forward …
Exact mathematical passage
S_V(t)=\one^T\e^{tA}p_0.
S_V'Use in source · 2 passages
Search aliases: S_V'
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4932
[State dependent killing as survival] Let Q be a finite Markov generator, let V≥0, and set A=Q-D_V. For an initial distribution p_0, define S_V(t)= 1^T e^tAp_0. Then S_V(t) = E_p_0 [ exp( -∫_0^tV(X_s) d s ) ]. Also, -S_V'(t) = 1^TD_V e^tAp_0. The matrix A=Q-D_V has nonnegative off diagonal entries and nonpositive column sums, so it is the subgenerator of the killed, possibly nontransient chain. Thus the potential in the finite Feynman Kac formula is a state dependent killing rate in the forward …
Exact mathematical passage
-S_V'(t) = \one^TD_V\e^{tA}p_0.
S_zUse in source · 2 passages
Search aliases: S_z
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4505
0 · I and S_z= F_P(z) · 0 -A_QP · zI-A_QQ . Direct multiplication gives zI-A=L_zS_z. The first factor is invertible. The claims follow from triangular inversion and the determinant of the second factor. This is the finite block form of the Feshbach reduction [source reference].
Exact mathematical passage
S_z= \begin{pmatrix} F_P(z)&0\\ -A_{QP}&zI-A_{QQ} \end{pmatrix}.
\SfunUse in source · 3 passages
Search aliases: Sfun S mathsf
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5913
ceives the operator-substitution result deliberately moved out of Paper IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=log D_N once and define S_N^(r)(z)=exp(-r g(z)) for complex r. The bounded Banach-space result below uses classical holomorphic functional calculus; it is separate fro…
Exact mathematical passage
\Sfun_N^{(r)}(z)=\exp(-r g(z))
\Sfun_N^{(r)}Use in source · 3 passages
Search aliases: Sfun_N^(r) S_N^(r) Sfun mathsf
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5913
ceives the operator-substitution result deliberately moved out of Paper IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=log D_N once and define S_N^(r)(z)=exp(-r g(z)) for complex r. The bounded Banach-space result below uses classical holomorphic functional calculus; it is separate fro…
Exact mathematical passage
\Sfun_N^{(r)}(z)=\exp(-r g(z))
\Sfun_N^{(r+s)}Use in source · 1 passage
Search aliases: Sfun_N^(r+s) S_N^(r+s) Sfun mathsf
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5935
[Operator reciprocal family] Let A be bounded and suppose Spec(A) lies in a simply connected region on which D_N has no zeros. Fix a holomorphic branch g=log D_N there. Then S_N^(r)(A) = exp(-r g(A)) = ( D_N^-r)(A) is well defined and S_N^(r)(A) S_N^(s)(A) = S_N^(r+s)(A). If (P_i) is a finite complete family of bounded idempotents, then P_i f(A)P_j = frac12π i ∫_Γ f(z)P_i(zI-A)^-1P_j dz for every holomorphic f on a neighborhood of the spectrum.
Exact mathematical passage
\Sfun_N^{(r)}(A)\Sfun_N^{(s)}(A) = \Sfun_N^{(r+s)}(A).
\Sfun_N^{(s)}Use in source · 1 passage
Search aliases: Sfun_N^(s) S_N^(s) Sfun mathsf
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5935
[Operator reciprocal family] Let A be bounded and suppose Spec(A) lies in a simply connected region on which D_N has no zeros. Fix a holomorphic branch g=log D_N there. Then S_N^(r)(A) = exp(-r g(A)) = ( D_N^-r)(A) is well defined and S_N^(r)(A) S_N^(s)(A) = S_N^(r+s)(A). If (P_i) is a finite complete family of bounded idempotents, then P_i f(A)P_j = frac12π i ∫_Γ f(z)P_i(zI-A)^-1P_j dz for every holomorphic f on a neighborhood of the spectrum.
Exact mathematical passage
\Sfun_N^{(r)}(A)\Sfun_N^{(s)}(A) = \Sfun_N^{(r+s)}(A).
\sinUse in source · 19 passages
Search aliases: sin
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1202
[Fractional intertwining theorem] For bounded nonnegative self-adjoint A and B, D_α = fracsin(πα)π ∫_0^∞ t^α(tI+B)^-1D(tI+A)^-1 dt. The improper integral converges in operator norm. Moreover, normD_α ≤ K_αnormC^1-αnormD^α, K_α= frac2^1-αsin(πα) πα(1-α). If A,B≥ mI for some m>0, then the stronger linear estimate holds: normD_α ≤ α m^α-1normD. In particular, exact generator intertwining CA=BC implies exact fractional intertwining CA^α=B^α C.
Exact mathematical passage
\boxed{ D_\alpha = \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty t^\alpha(tI+B)^{-1}D(tI+A)^{-1}\,dt. }
\sin^2Use in source · 6 passages
Search aliases: sin^2 sin
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1410
The kth discrete eigenvalue is λ_k,h=frac4h^2sin^2(frackπ h2), whereas the continuum eigenvalue is (kπ)^2.
Exact mathematical passage
\lambda_{k,h}=\frac4{h^2}\sin^2\left(\frac{k\pi h}{2}\right),
\sqcupUse in source · 2 passages
Search aliases: sqcup
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1017
[Bayesian refinement] For the finite test partition {0,1}sqcup{2,3}, a blockwise-constant likelihood gave commutation residual 1.11×10^-16. A nonconstant likelihood produced an exact defect-factorization residual 7.29×10^-17. Its measured L^1 defect was 4.8787×10^-2, below the certified bound 1.1016×10^-1. Two priors with identical coarse masses but different hidden within-packet shapes produced coarse posteriors differing by 1.2400×10^-1, which is a direct necessity witness.
Exact mathematical passage
\{0,1\}\sqcup\{2,3\}
\supUse in source · 23 passages
Search aliases: sup
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1328
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces; no operator sum across unrelated spaces…
Exact mathematical passage
\sup_\ell\norm{C_\ell}\le M
\sup_{0\leq r\leq t}Use in source · 1 passage
Search aliases: sup_0≤ r≤ t sup leq leq
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4857
Let E(t)=sup_0≤ r≤ tnormx(r)-widetilde x(r). Subtract the two mild equations. Fubini's theorem and the triangle inequality give E(t) ≤ M_TL∫_0^tE(r) d r + M_TBtε. For t≤ T, replace the last factor t by T and apply Gronwall's inequality.
Exact mathematical passage
E(t)=\sup_{0\leq r\leq t}\norm{x(r)-\widetilde x(r)}.
\sup_{0\leq s,t\leq T}Use in source · 1 passage
Search aliases: sup_0≤ s,t≤ T sup leq leq
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2653
[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally uniform and every T_n is norm continuous, then U_m is a norm continuous semigroup.
Exact mathematical passage
\rho_{m,n}(T) = \sup_{0\leq s,t\leq T} \norm{ C_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m} } \longrightarrow0.
\sup_{0\le t\le T}Use in source · 2 passages
Search aliases: sup_0≤ t≤ T sup le le
Use in source · atlas-pages.tex, line 724
2^-(n+1), ∑_j=n^∞|A_j+1-A_j|=2^-n. vfigext_tails platecaptionThe bars show the certified tail 2^-n at horizon T=1. They are bounds on a whole time interval, not errors sampled at the right endpoint. headThe estimate follows before any graph is drawn For 0≤ t≤ T, the derivative of a↦ e^-at has absolute value at most T when a≥0. Thus sup_0≤ t≤ T|T_n(t)-e^-t|≤ T2^-n. The limit is e^-t, which does have the semigroup law. Here every reconstruction is the identity, so the hidden-range excursion is ide…
Exact mathematical passage
\sup_{0\le t\le T}|T_n(t)-e^{-t}|\le T2^{-n}.
\sup_{0\leq t\leq T}Use in source · 11 passages
Search aliases: sup_0≤ t≤ T sup leq leq
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2554
[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
\sum_{n=m}^{\infty} w_{m,n} \sup_{0\leq t\leq T}\norm{E_n(t)}<\infty,
\sup_{z\in\Gamma}Use in source · 1 passage
Search aliases: sup_z∈Γ sup in Gamma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2388
a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that Γ lies in the common resolvent set. Thus its winding number is one about every spectral point. If g is holomorphic on a neighborhood of the closure of the union, then Cg(A_f)-g(A_c)C = frac12π i ∫_Γ g(z)R_c(z)Δ R_f(z) d z. In particular, normCg(A_f)-g(A_c)C ≤ · fraclength(Γ)2π sup_z∈Γabsg(z) sup_z∈ΓnormR_c(z) sup_z∈ΓnormR_f(z) normΔ.
Exact mathematical passage
\begin{aligned} \norm{Cg(A_f)-g(A_c)C} \leq{}& \frac{\operatorname{length}(\Gamma)}{2\pi} \sup_{z\in\Gamma}\abs{g(z)} \sup_{z\in\Gamma}\norm{R_c(z)} \sup_{z\in\Gamma}\norm{R_f(z)} \norm\Delta. \end{aligned}
\sup_\ellUse in source · 6 passages
Search aliases: sup_ℓ sup ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1328
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces; no operator sum across unrelated spaces…
Exact mathematical passage
\sup_\ell\norm{C_\ell}\le M
\mathsf TUse in source · 44 passages
Search aliases: T mathsf
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 926
[Bayesian information-loss packet] Assume nonnegative fine and declared coarse likelihoods, with both Z_r>0 and Z_barλ>0. The coarsening defect is C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood; the full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together. Moreover, C_π B_λ(p)- B_barλ(P) …
Exact mathematical passage
\boxed{ C_\pi\cB_\lambda(p) - \cB_{\bar\lambda}(P) = \frac{1}{Z_r} \left(\diag(P)-\bar bP^{\mathsf T}\right) \delta(p).}
T'Use in source · 3 passages
Search aliases: T'
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 337
[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.
Exact mathematical passage
T'(t)=AT(t)=T(t)A
T^2Use in source · 3 passages
Search aliases: T^2
Use in source · atlas-pages.tex, line 884
trajectory platecaptionThe first-coordinate error is exactly t^2/2. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory. headTrack every constant before applying the estimate Here M_T=1, B=sup widetilde x =1, and L= K _L^1(0,T)=T, ε= K-widetilde K _L^1(0,T)=T. The theorem yields sup_0≤ t≤ T x(t)-widetilde x(t) =fracT^22≤ T^2e^T^2. At T=1/2, the exact error is 1/8 and the certificate is e^1/4/4. Nilpotence makes this example po…
Exact mathematical passage
\sup_{0\le t\le T}\|x(t)-\widetilde x(t)\| =\frac{T^2}{2}\le T^2e^{T^2}.
t^2Use in source · 17 passages
Search aliases: t^2
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1166
[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.
Exact mathematical passage
\boxed{ e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)}e^{-sL}\,ds. }
t^kUse in source · 6 passages
Search aliases: t^k
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1558
Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace; equivalently, apply the decomposition to left multiplication by B on the matrix algebra and evaluate the exponential at I. The strict spectral inequality makes every such term absolutely integrable on (0,∞) and forces it to tend to zero. H…
Exact mathematical passage
t^k e^{(\lambda-z)t}
t^mUse in source · 2 passages
Search aliases: t^m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5009
The Laplace transform follows by integrating the density in [source reference]. Stability of A gives ∫_0^∞ e^-st e^tA d t = (sI-A)^-1. At s=0, a^T(-A)^-1p_0 = 1^Tp_0=1. For the moments, ∫_0^∞ t^m e^tA d t = m!(-A)^-m-1. Multiplying by a^T= 1^T(-A) gives the result. These are the standard matrix formulas for phase type laws [source reference].
Exact mathematical passage
\int_0^\infty t^m\e^{tA}\dd t = m!(-A)^{-m-1}.
t^{m_0}Use in source · 2 passages
Search aliases: t^m_0
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5232
in Part II. A directed edge from α to β means that P_β A P_α is nonzero, with α≠β; a packet always has the length-zero path to itself. [Minimum transfer order] Suppose every directed packet path from α to β uses at least m_0 off diagonal transfers. Then P_β e^tAP_α =O(t^m_0) as tdownarrow0. If there is no directed path from α to β, the block vanishes for all t.
Exact mathematical passage
P_\beta\e^{tA}P_\alpha =O(t^{m_0}) \quad\text{as }t\downarrow0.
t^nUse in source · 1 passage
Search aliases: t^n
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 342
Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.
Exact mathematical passage
e^{sA}e^{tA} =\sum_{m,n\ge0}\frac{s^mt^n}{m!n!}A^{m+n} =\sum_{r\ge0}\frac{(s+t)^r}{r!}A^r.
t^\alphaUse in source · 2 passages
Search aliases: t^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1202
[Fractional intertwining theorem] For bounded nonnegative self-adjoint A and B, D_α = fracsin(πα)π ∫_0^∞ t^α(tI+B)^-1D(tI+A)^-1 dt. The improper integral converges in operator norm. Moreover, normD_α ≤ K_αnormC^1-αnormD^α, K_α= frac2^1-αsin(πα) πα(1-α). If A,B≥ mI for some m>0, then the stronger linear estimate holds: normD_α ≤ α m^α-1normD. In particular, exact generator intertwining CA=BC implies exact fractional intertwining CA^α=B^α C.
Exact mathematical passage
\boxed{ D_\alpha = \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty t^\alpha(tI+B)^{-1}D(tI+A)^{-1}\,dt. }
t^{\alpha-1}Use in source · 2 passages
Search aliases: t^α-1 alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1236
For 0<α<1, the Balakrishnan formula gives A^α = fracsin(πα)π ∫_0^∞ t^α-1A(tI+A)^-1 dt, and likewise for B. Using A(tI+A)^-1=I-t(tI+A)^-1 and the rectangular resolvent identity (tI+B)^-1C-C(tI+A)^-1 = (tI+B)^-1(CA-BC)(tI+A)^-1, the difference of the two Balakrishnan integrals becomes the displayed resolvent-defect formula.
Exact mathematical passage
A^\alpha = \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty t^{\alpha-1}A(tI+A)^{-1}\,dt,
t^{\alpha-2}Use in source · 1 passage
Search aliases: t^α-2 alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1264
For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D^α. The zero cases are immediate.
Exact mathematical passage
\begin{aligned} \norm{D_\alpha} &\le \frac{\sin(\pi\alpha)}{\pi} \left( 2\norm C\int_0^r t^{\alpha-1}\,dt + \norm D\int_r^\infty t^{\alpha-2}\,dt \right)\\ &= K_\alpha\norm C^{1-\alpha}\norm D^\alpha. \end{aligned}
T_0Use in source · 1 passage
Search aliases: T_0
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2694
[Summable defects do not force a semigroup limit] Let X_0= R and let X_n= R^2 for every n≥1. Set C_0(x,y)=x, R_0x=(x,0), and use identity connecting maps and reconstructions above level one. Let T_0(t)=I and let every higher level carry the rotation group T_n(t) = cos t · -sin t
Exact mathematical passage
T_0(t)=I
T_jUse in source · 9 passages
Search aliases: T_j
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5247
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
T_j
T_{j+1}Use in source · 4 passages
Search aliases: T_j+1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5248
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
E_j=C_jT_{j+1}-T_jC_j.
T_mUse in source · 3 passages
Search aliases: T_m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5252
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
\boxed{ C_{\ell,m}T_m-T_\ell C_{\ell,m} = \sum_{j=\ell}^{m-1} C_{\ell,j}E_jC_{j+1,m}. }
T_{n+1}Use in source · 2 passages
Search aliases: T_n+1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2543
[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
E_n(t)=C_nT_{n+1}(t)-T_n(t)C_n.
T_tUse in source · 1 passage
Search aliases: T_t
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 716
[Gaussian chaos realization] For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally: T_tH_n=e^-ntH_n. Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.
Exact mathematical passage
T_tH_n=e^{-nt}H_n.
T_zUse in source · 7 passages
Search aliases: T_z
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1556
Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace; equivalently, apply the decomposition to left multiplication by B on the matrix algebra and evaluate the exponential at I. The strict spectral inequality makes every such term absolutely integrable on (0,∞) and forces it to tend to zero. H…
Exact mathematical passage
T_z(t)=e^{-zt}e^{tB}=e^{t(B-zI)}
T_z'Use in source · 1 passage
Search aliases: T_z'
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1567
Let J=∫_0^∞ T_z(t) dt. Integrating both derivative identities T_z'=(B-zI)T_z=T_z(B-zI) and using T_z(0)=I gives (zI-B)J=J(zI-B)=I. Thus J=(zI-B)^-1. Continuous left and right multiplication carry the Bochner integral through the two rectangular transfer blocks, yielding the claim.
Exact mathematical passage
T_z'=(B-zI)T_z=T_z(B-zI)
T_\alphaUse in source · 6 passages
Search aliases: T_α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1129
Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.
Exact mathematical passage
T_\alpha(t)=e^{-tL^\alpha} =\sum_je^{-t\lambda_j^\alpha}P_j.
T_{\beta\gamma}Use in source · 4 passages
Search aliases: T_βγ beta gamma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 388
[Packet semigroup convolution] For every s,t and every pair of packets α,β, T_βα(s+t) =∑_γ∈ IT_βγ(s)T_γα(t). Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.
Exact mathematical passage
\boxed{ T_{\beta\alpha}(s+t) =\sum_{\gamma\in I}T_{\beta\gamma}(s)T_{\gamma\alpha}(t). }
T_{\gamma\alpha}Use in source · 4 passages
Search aliases: T_γα gamma alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 388
[Packet semigroup convolution] For every s,t and every pair of packets α,β, T_βα(s+t) =∑_γ∈ IT_βγ(s)T_γα(t). Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.
Exact mathematical passage
\boxed{ T_{\beta\alpha}(s+t) =\sum_{\gamma\in I}T_{\beta\gamma}(s)T_{\gamma\alpha}(t). }
T_\ellUse in source · 12 passages
Search aliases: T_ℓ ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2461
[Exact projective limit semigroup] For each ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x_ℓ-x_ℓ=0 for every (x_ℓ)_ℓ∈ X_∞.
Exact mathematical passage
T_\ell(t)
T_{\ell+1}Use in source · 1 passage
Search aliases: T_ℓ+1 ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2462
[Exact projective limit semigroup] For each ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x_ℓ-x_ℓ=0 for every (x_ℓ)_ℓ∈ X_∞.
Exact mathematical passage
C_\ell T_{\ell+1}(t)=T_\ell(t)C_\ell
T_\inftyUse in source · 5 passages
Search aliases: T_∞ infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2470
[Exact projective limit semigroup] For each ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x_ℓ-x_ℓ=0 for every (x_ℓ)_ℓ∈ X_∞.
Exact mathematical passage
T_\infty(t)(x_\ell)_\ell = (T_\ell(t)x_\ell)_\ell.
uUse in source · 41 passages
Search aliases: u
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1187
By the spectral theorem it suffices to prove the scalar identity e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)-λ s ds for λ≥0. This is the classical Gaussian integral after the substitution u=t/(2sqrt s). Applying the identity to every eigenvalue and recombining the spectral projectors proves the operator formula.
Exact mathematical passage
u=t/(2\sqrt s)
u''Use in source · 3 passages
Search aliases: u''
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1357
On the interval (0,1) with Dirichlet boundary conditions, the Green kernel for -u''=f is G(x,y)=min(x,y)-xy. Thus the point source δ_y generates the solution kernel, and a general source is reconstructed by u(x)=∫_0^1G(x,y)f(y) dy.
Exact mathematical passage
-u''=f
u^{(4)}Use in source · 2 passages
Search aliases: u^(4)
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4250
[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.
Exact mathematical passage
\max_i\abs{u^h_i-u(x_i)} \leq \frac{h^2}{96} \norm{u^{(4)}}_\infty.
u^2Use in source · 5 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3130
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
\exp\left(ux-\frac{u^2}{2}\right) = \sum_{n=0}^\infty H_n(x)\frac{u^n}{n!},
u^hUse in source · 4 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4199
[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.
Exact mathematical passage
u^h
u^h_iUse in source · 1 passage
Search aliases: u^h_i
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4250
[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.
Exact mathematical passage
\max_i\abs{u^h_i-u(x_i)} \leq \frac{h^2}{96} \norm{u^{(4)}}_\infty.
u^nUse in source · 2 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3130
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
\exp\left(ux-\frac{u^2}{2}\right) = \sum_{n=0}^\infty H_n(x)\frac{u^n}{n!},
u^{\alpha-1}Use in source · 1 passage
Search aliases: u^α-1 alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3968
It is enough to prove the formula on an eigenvector with eigenvalue x≥0. For x>0, the substitution s=xu gives fracsin(πα)π ∫_0^∞ s^α-1fracxs+x d s =x^α fracsin(πα)π ∫_0^∞fracu^α-11+u d u =x^α. The value at x=0 is zero. There are finitely many eigenvalues, so the operator integral converges. This is the finite spectral form of the fractional power construction in [source reference].
Exact mathematical passage
\frac{\sin(\pi\alpha)}{\pi} \int_0^\infty s^{\alpha-1}\frac{x}{s+x}\dd s =x^\alpha \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty\frac{u^{\alpha-1}}{1+u}\dd u =x^\alpha.
U_{0,n}Use in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2704
sin t · cos t . Only the first adjacent defect is nonzero, so every defect sum is finite. Yet for n≥1, U_0,n(t)=C_0T_n(t)R_0=cos t. The limiting family is not a semigroup because in general cos(s+t)≠cos scos t. The failure is exactly the excursion from the reconstructed line into its orthogonal complement between two time steps.
Exact mathematical passage
U_{0,n}(t)=C_0T_n(t)R_0=\cos t.
u_hUse in source · 3 passages
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Use in source · atlas-pages.tex, line 427
nzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMTraditional Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil differentiates the quadratic x(1-x) exactly, the solution u_h to the sampled forcing 12x_i^2 satisfies u_h(x_i)-u(x_i)=-h^2x_i(1-x_i). This gives both the sign and size of the err…
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u_h
u_iUse in source · 2 passages
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Use in source · atlas-pages.tex, line 411
1 · 2 · 3. vfiggreen platecaptionThe tent is the central column of the continuous Green kernel. It has nodal values (1,2,1)/8; its discrete inverse column is (1,2,1)/32, smaller by the factor h. headA forcing vector is not a quadrature-weighted measure For sampled forcing f_j, the finite solution is u_i=∑_j(L_h^-1)_ijf_j =h∑_jG(x_i,x_j)f_j. The factor h converts samples into the quadrature weights appropriate to this formula. Dropping it multiplies the computed solution by four on the displayed …
Exact mathematical passage
u_i=\sum_j(L_h^{-1})_{ij}f_j =h\sum_jG(x_i,x_j)f_j.
u_i^hUse in source · 1 passage
Search aliases: u_i^h
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4204
[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.
Exact mathematical passage
u_i^h = h\sum_{j=1}^nG(x_i,x_j)f_j^h.
u_jUse in source · 2 passages
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Use in source · atlas-pages.tex, line 79
0 · 2 · 1 · -3. The graph shows rates between actual states. The four normalized sign vectors on the right form an orthonormal basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next. headThe same evolution has a closed formula Writing q=e^-2t gives e^tQ=P_…
Exact mathematical passage
u_j
u_j^{\mathsf T}Use in source · 1 passage
Search aliases: u_j^ T mathsf
Use in source · atlas-pages.tex, line 79
0 · 2 · 1 · -3. The graph shows rates between actual states. The four normalized sign vectors on the right form an orthonormal basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next. headThe same evolution has a closed formula Writing q=e^-2t gives e^tQ=P_…
Exact mathematical passage
P_j=u_ju_j^{\mathsf T}
U_mUse in source · 8 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2559
t_completion Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
U_m(t)
U_{m,n+1}Use in source · 2 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2571
Since C_nR_n=I, U_m,n+1(t)-U_m,n(t) = · C_m,n (C_nT_n+1(t)-T_n(t)C_n) R_nR_n,m. Therefore normU_m,n+1(t)-U_m,n(t) ≤ w_m,nnormE_n(t). The weighted summability hypothesis makes the sequence uniformly Cauchy. Summing the tail estimates gives the stated bound.
Exact mathematical passage
\begin{aligned} U_{m,n+1}(t)-U_{m,n}(t) ={}& C_{m,n} \bigl(C_nT_{n+1}(t)-T_n(t)C_n\bigr) R_nR_{n,m}. \end{aligned}
U_nUse in source · 1 passage
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Use in source · atlas-pages.tex, line 689
not close the dynamicsREFINEMENT / A PERSISTENT RANGE EXCURSIONTraditional Part II: Theorem E4.8 and Example E4.10. Summability and closure are different tests. Let the coarse space be R and every finer space be R^2. The first connecting maps are C(x,y)=x and Rx=(x,0); all later maps are identities. Let the coarse evolution be the identity and all fine evolutions be rotations: T(t)=matcos t · -sin t sin t · cos t, U_n(t)=CT(t)R=cos t (n≥1). The reconstructed sequence is already constant above it…
Exact mathematical passage
T(t)=\mat{\cos t&-\sin t\\\sin t&\cos t},\qquad U_n(t)=CT(t)R=\cos t\quad(n\ge1).
u_nUse in source · 2 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1446
Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.
Exact mathematical passage
u_n=M_V^nf.
VUse in source · 40 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1441
Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.
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V
vUse in source · 14 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2338
[Infinitesimal recovery of the defect] The generator defect is the first order semigroup defect: lim_t→0 frac D(t)t = Δ in operator norm. Consequently, C e^tA_f= e^tA_cC hboxfor every t if and only if CA_f=A_cC.
Exact mathematical passage
C\e^{tA_f}=\e^{tA_c}C \quad\hbox{for every }t
v_0Use in source · 3 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4393
[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.
Exact mathematical passage
V\geq v_0
v_iUse in source · 2 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4233
Nonnegativity follows from the entry formula in [source reference]. Let v_i=x_i(1-x_i)/2. Since the second difference is exact on quadratics, L_hv= 1. Hence the ith row sum of L_h^-1 is v_i. For a nonnegative matrix, the infinity operator norm is the largest row sum.
Exact mathematical passage
v_i=x_i(1-x_i)/2
v_{k,h}Use in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4276
The discrete eigenpairs are λ_k,h = frac4h^2 sin^2(frackπ h2), v_k,h(i)=sin(kπ x_i).
Exact mathematical passage
\lambda_{k,h} = \frac{4}{h^2} \sin^2\left(\frac{k\pi h}{2}\right), \qquad v_{k,h}(i)=\sin(k\pi x_i).
v_rUse in source · 1 passage
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Use in source · atlas-pages.tex, line 294
_order platecaptionBoth routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem. headKeep an unnormalized path alongside the filter For a sequence of observations, let D_r=diag(λ^(r)). The forward recursion is widehat p_r=Kp_r-1, p_r=D_rwidehat p_r/Z_r. Its unnormalized companion is v_r=D_rK⋯ D_1Kp_0, p_r=fracv_rones^ Tv_r, ones^ Tv_r=∏_j=1^r Z_j. The product of evidences records the normalization lost at each indivi…
Exact mathematical passage
v_r=D_rK\cdots D_1Kp_0, \qquad p_r=\frac{v_r}{\ones^{\mathsf T}v_r}, \qquad \ones^{\mathsf T}v_r=\prod_{j=1}^r Z_j.
\VarUse in source · 10 passages
Search aliases: Var operatorname
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 579
[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.
Exact mathematical passage
\Exp[X_n]=0, \qquad \Var(X_n)=n.
\operatorname{Var}Use in source · 1 passage
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Use in source · atlas-pages.tex, line 184
o. The vectors below specify the corresponding directions, not only their sizes. headThe energy identity follows from actual orthogonality The tower identities give E_0E_1=E_1E_0=E_0. Together with weighted self-adjointness, these imply D_iD_j=0 (i≠ j), ∑_j=0^2 D_j=I. For the chosen f, the total weighted energy is f _π^2=13 and the mean is 16/5. Consequently Var_π(f) =13-(frac165)^2 =frac6925 =frac108175+frac157. The first term is the energy of the between-block contrast, and the second is the w…
Exact mathematical passage
\operatorname{Var}_\pi(f) =13-\left(\frac{16}{5}\right)^2 =\frac{69}{25} =\frac{108}{175}+\frac{15}{7}.
\Var_{p(t)}Use in source · 4 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3808
[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).
Exact mathematical passage
\psi'(t)=\ip{g}{p(t)}, \qquad \psi''(t)=\Var_{p(t)}(g)\geq0.
\Var_\gammaUse in source · 2 passages
Search aliases: Var_γ Var gamma operatorname gamma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3277
[Gaussian spectral gap] If f∈ L^2(γ) has mean zero, then norm O_tf_2 ≤ e^-tnorm f_2. For every polynomial f, Var_γ(f) ≤ ∫_ Rabsf'(x)^2γ( d x).
Exact mathematical passage
\Var_\gamma(f) \leq \int_\RR\abs{f'(x)}^2\gamma(\dd x).
\Var_\piUse in source · 1 passage
Search aliases: Var_π Var pi operatorname pi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3092
[Law of total variance] For a real valued f, let P_0f= E_π[f] 1. Then Var_π(f) = Var_π(P_ Qf) + normf-P_ Qf_π^2.
Exact mathematical passage
\Var_\pi(f) = \Var_\pi(P_\mathcal Qf) + \norm{f-P_\mathcal Qf}_\pi^2.
\operatorname{Var}_\piUse in source · 1 passage
Search aliases: Var_π operatorname pi
Use in source · atlas-pages.tex, line 184
o. The vectors below specify the corresponding directions, not only their sizes. headThe energy identity follows from actual orthogonality The tower identities give E_0E_1=E_1E_0=E_0. Together with weighted self-adjointness, these imply D_iD_j=0 (i≠ j), ∑_j=0^2 D_j=I. For the chosen f, the total weighted energy is f _π^2=13 and the mean is 16/5. Consequently Var_π(f) =13-(frac165)^2 =frac6925 =frac108175+frac157. The first term is the energy of the between-block contrast, and the second is the w…
Exact mathematical passage
\operatorname{Var}_\pi(f) =13-\left(\frac{16}{5}\right)^2 =\frac{69}{25} =\frac{108}{175}+\frac{15}{7}.
WUse in source · 5 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5624
Let Q and overline Q be finite Markov generators in the column convention. All times in this section satisfy t≥0. [Potential perturbation bound] Let V and W be nonnegative diagonal matrices. In the supremum norm, norm e^t(Q^T-V) - e^t(Q^T-W) _∞→∞ ≤ tnormV-W_∞→∞.
Exact mathematical passage
W
w_{(1)}Use in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4739
Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.
Exact mathematical passage
w_{(1)}\geq w_{(2)}\geq\cdots\geq w_{(r)},
w_{(2)}Use in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4739
Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.
Exact mathematical passage
w_{(1)}\geq w_{(2)}\geq\cdots\geq w_{(r)},
w_{(j)}Use in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4743
normB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.
Exact mathematical passage
\sum_{j>k}w_{(j)}\leq\varepsilon.
w_{(r)}Use in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4739
Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.
Exact mathematical passage
w_{(1)}\geq w_{(2)}\geq\cdots\geq w_{(r)},
w_1Use in source · 1 passage
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Use in source · atlas-pages.tex, line 854
A DEFINED COSTTraditional Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks; block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified integrated weights are w_1=frac14, w_2=frac32+frac22^2=2, w_3=frac1/33=frac19. vfigext_selection platecaptionWith one primary block retained, choose th…
Exact mathematical passage
w_1=\frac14,\qquad w_2=\frac32+\frac{2}{2^2}=2, \qquad w_3=\frac{1/3}{3}=\frac19.
w_2Use in source · 1 passage
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Use in source · atlas-pages.tex, line 854
A DEFINED COSTTraditional Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks; block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified integrated weights are w_1=frac14, w_2=frac32+frac22^2=2, w_3=frac1/33=frac19. vfigext_selection platecaptionWith one primary block retained, choose th…
Exact mathematical passage
w_1=\frac14,\qquad w_2=\frac32+\frac{2}{2^2}=2, \qquad w_3=\frac{1/3}{3}=\frac19.
w_3Use in source · 1 passage
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Use in source · atlas-pages.tex, line 854
A DEFINED COSTTraditional Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks; block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified integrated weights are w_1=frac14, w_2=frac32+frac22^2=2, w_3=frac1/33=frac19. vfigext_selection platecaptionWith one primary block retained, choose th…
Exact mathematical passage
w_1=\frac14,\qquad w_2=\frac32+\frac{2}{2^2}=2, \qquad w_3=\frac{1/3}{3}=\frac19.
w_iUse in source · 2 passages
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Use in source · atlas-pages.tex, line 50
1/4 · 3/4, p=e_0, f=(2,1)^ T, we obtain K_2p=(3/4,1/4)^ T and f^ TK_2p=7/4=(K_2^ Tf)_0. With departure weights w_i=e^-V(i), the backward Feynman--Kac operator is diag(w)K_2^ T. Its forward dual is K_2diag(w). Interchanging these factors changes when the weight is applied. takeawayAlways name the object being transported. The transpose and the position of the likelihood or potential are mathematical choices, not typographic details.
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w_i=e^{-V(i)}
w_jUse in source · 2 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4727
[Optimal selection for the certified primary block bound] Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.
Exact mathematical passage
w_j = \sum_{k=0}^{\nu_j-1} \frac{\norm{B_{j,k}}} {(-\operatorname{Re}\mu_j)^{k+1}}.
w_{m,j}Use in source · 1 passage
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2560
fine the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
\sup_{0\leq t\leq T} \norm{U_m(t)-U_{m,n}(t)} \leq \sum_{j=n}^{\infty}w_{m,j} \sup_{0\leq t\leq T}\norm{E_j(t)}.
w_{m,n}Use in source · 4 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2547
[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
w_{m,n} = \norm{C_{m,n}}\, \norm{R_nR_{n,m}}.
\widehat KUse in source · 1 passage
Search aliases: widehat K widehat
Use in source · jordan.tex, line 38
24183) (8.62,0.815842) (8.705,0.807882) (8.79,0.800286) (8.875,0.79304) (8.96,0.786128) (9.045,0.779536) (9.13,0.77325) (9.215,0.767257) (9.3,0.761544); path[fill=black] (2.216667,3.409096) circle (0.065); node[font=small,align=center] at (11.7,3.9) J=-2I+N, N^2=0; node[font=small,align=center] at (11.7,2.8) e^tJ=e^-2t(I+tN); node[font=small,align=center] at (11.7,1.35) widehat K(z)=dfrac1(z+2)^2; node[font=small,align=center,font=scriptsize] at (11.7,0.4) A repeated pole carries a jet.;
Exact mathematical passage
\widehat K(z)=\dfrac1{(z+2)^2}
\widehat pUse in source · 6 passages
Search aliases: widehat p widehat
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1092
Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).
Exact mathematical passage
\widehat p_r=Kp_{r-1}, \qquad p_r=\cB_{\lambda^{(r)}}(\widehat p_r).
\widehat p_rUse in source · 6 passages
Search aliases: widehat p_r widehat
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1092
Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).
Exact mathematical passage
\widehat p_r=Kp_{r-1}, \qquad p_r=\cB_{\lambda^{(r)}}(\widehat p_r).
\widehat\KcalUse in source · 1 passage
Search aliases: widehatKcal widehat K widehat Kcal widehat mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5734
tional characterization concerns continuous kernels of exponential order, so uniqueness of the ordinary Laplace transform applies. [Rational memory realization] A matrix valued memory transform has a finite auxiliary realization if and only if it is a strictly proper rational matrix function. More precisely, widehat K(z)=B(zI-D)^-1C if and only if K(t)=B e^tDC.
Exact mathematical passage
\boxed{ \widehat\Kcal(z)=B(zI-D)^{-1}C }
\widetilde KUse in source · 9 passages
Search aliases: widetilde K widetilde
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3876
[One step filter stability] Let K and widetilde K be column stochastic. Assume 0<a≤λ_i≤ b. Then norm B_λ(Kp)- B_λ(widetilde Kq) _1 ≤ frac2ba ( normp-q_1 + normK-widetilde K_1→1 ).
Exact mathematical passage
\widetilde K
\widetilde xUse in source · 13 passages
Search aliases: widetilde x widetilde
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4824
[Volterra stability under kernel compression] Let S(t) be a strongly continuous semigroup on a Banach space X, fix T>0, and assume normS(t)≤ M_T (0≤ t≤ T). Let K,widetilde K∈ L^1((0,T), B(X)), let g∈ L^1((0,T),X), and let x,widetilde x∈ C([0,T],X). Suppose these are mild solutions with the same initial state and forcing, except that K is replaced by widetilde K: x(t) = · S(t)x_0+ ∫_0^tS(t-s)g(s) d s
Exact mathematical passage
x,\widetilde x\in C([0,T],X)
\widetilde x'Use in source · 1 passage
Search aliases: widetilde x' widetilde
Use in source · atlas-pages.tex, line 870
0 · 0, so N^2=0 and N _2=1. On [0,T], compare the memory kernels K(t)=N and widetilde K(t)=0, with identity free semigroup, no forcing, and initial state x_0=e_1=(0,1)^ T. The equations are x'(t)=∫_0^t N x(r) dr, widetilde x'(t)=0. Their exact solutions are x(t)=(t^2/2,1)^ T and widetilde x(t)=(0,1)^ T. vfigext_trajectory platecaptionThe first-coordinate error is exactly t^2/2. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajecto…
Exact mathematical passage
x'(t)=\int_0^t N x(r)\,dr,\qquad \widetilde x'(t)=0.
\widetilde\KcalUse in source · 3 passages
Search aliases: widetildeKcal widetilde K widetilde Kcal widetilde mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5673
FloatBarrier Memory compression, rational realization, and phase-type consequences. Fix T>0. For the next estimate let A be a bounded operator on a Banach space X, let K,widetilde K∈ L^1((0,T), B(X)), and let f be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II. [Stability of memory compression] Let x and widetilde x solve dot x(t) =…
Exact mathematical passage
\Kcal,\widetilde\Kcal\in L^1((0,T),\mathcal B(X))
\widetilde\lambdaUse in source · 3 passages
Search aliases: widetildeλ widetilde lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3608
[Approximate coarse Bayesian rule] Choose one positive number ℓ_α for every packet and define the block constant fine likelihood widetildeλ_i=ℓ_α (i∈ I_α). Then norm C B_λ(p)- B_ℓ(Cp) _1 ≤ frac 2∑_i p_iabsλ_i-widetildeλ_i max{Z_λ(p),Z_widetildeλ(p)} .
Exact mathematical passage
\widetilde\lambda_i=\ell_\alpha \quad(i\in I_\alpha).
\widetilde\lambda_iUse in source · 2 passages
Search aliases: widetildeλ_i widetilde lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3608
[Approximate coarse Bayesian rule] Choose one positive number ℓ_α for every packet and define the block constant fine likelihood widetildeλ_i=ℓ_α (i∈ I_α). Then norm C B_λ(p)- B_ℓ(Cp) _1 ≤ frac 2∑_i p_iabsλ_i-widetildeλ_i max{Z_λ(p),Z_widetildeλ(p)} .
Exact mathematical passage
\widetilde\lambda_i=\ell_\alpha \quad(i\in I_\alpha).
x'Use in source · 4 passages
Search aliases: x'
Use in source · atlas-pages.tex, line 489
vpage22Hidden state returns as memoryEXACT RETAINED DYNAMICS / NOT A SCALAR SEMIGROUPTraditional Part I: Theorem 13.2. The retained-state memory identity, including hidden initial data. Write the same system as x'=-x+y, y'=x-y. Solving the second equation first gives y(t)=e^-ty(0)+∫_0^t e^-(t-r)x(r) dr. Substitution is exact: x'(t)=-x(t)+e^-ty(0)+∫_0^t e^-(t-r)x(r) dr. The hidden initial state produces a separate forcing term. It must be retained even when the memory kernel is known. vfigmemory …
Exact mathematical passage
x'=-x+y,\qquad y'=x-y.
x^2Use in source · 4 passages
Search aliases: x^2
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1428
For x≥0, 0≤ x^2-sin^2x≤ x^4/3. Set x=kπ h/2 and multiply by 4/h^2.
Exact mathematical passage
0\le x^2-\sin^2x\le x^4/3
x^3Use in source · 1 passage
Search aliases: x^3
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4300
Set x=kπ h/2. The condition kh≤1 gives 0≤ x≤π/2. On this interval, sin x≥ x-fracx^36, so 0≤ x^2-sin^2x≤fracx^43. Multiply by 4/h^2.
Exact mathematical passage
\sin x\geq x-\frac{x^3}{6},
x^4Use in source · 4 passages
Search aliases: x^4
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1428
For x≥0, 0≤ x^2-sin^2x≤ x^4/3. Set x=kπ h/2 and multiply by 4/h^2.
Exact mathematical passage
0\le x^2-\sin^2x\le x^4/3
x^\alphaUse in source · 6 passages
Search aliases: x^α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3968
It is enough to prove the formula on an eigenvector with eigenvalue x≥0. For x>0, the substitution s=xu gives fracsin(πα)π ∫_0^∞ s^α-1fracxs+x d s =x^α fracsin(πα)π ∫_0^∞fracu^α-11+u d u =x^α. The value at x=0 is zero. There are finitely many eigenvalues, so the operator integral converges. This is the finite spectral form of the fractional power construction in [source reference].
Exact mathematical passage
\frac{\sin(\pi\alpha)}{\pi} \int_0^\infty s^{\alpha-1}\frac{x}{s+x}\dd s =x^\alpha \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty\frac{u^{\alpha-1}}{1+u}\dd u =x^\alpha.
X_0Use in source · 2 passages
Search aliases: X_0
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2436
Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.
Exact mathematical passage
X_0\xleftarrow{C_0}X_1\xleftarrow{C_1}X_2\xleftarrow{}\cdots
x_0Use in source · 2 passages
Search aliases: x_0
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4825
ssion Let S(t) be a strongly continuous semigroup on a Banach space X, fix T>0, and assume normS(t)≤ M_T (0≤ t≤ T). Let K,widetilde K∈ L^1((0,T), B(X)), let g∈ L^1((0,T),X), and let x,widetilde x∈ C([0,T],X). Suppose these are mild solutions with the same initial state and forcing, except that K is replaced by widetilde K: x(t) = · S(t)x_0+ ∫_0^tS(t-s)g(s) d s · + ∫_0^tS(t-s) ∫_0^sK(s-r)x(r) d r d s. Assume sup_0≤ t≤ Tnormwidetilde x(t)≤ B. Set L=norm K_L^1(0,T), ε=normK-widetilde K_L^1(0,T). Th…
Exact mathematical passage
\begin{aligned} x(t) ={}&S(t)x_0+ \int_0^tS(t-s)g(s)\dd s\\ &+ \int_0^tS(t-s) \int_0^sK(s-r)x(r)\dd r\dd s. \end{aligned}
X_1Use in source · 1 passage
Search aliases: X_1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2436
Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.
Exact mathematical passage
X_0\xleftarrow{C_0}X_1\xleftarrow{C_1}X_2\xleftarrow{}\cdots
x_1Use in source · 1 passage
Search aliases: x_1
Use in source · atlas-pages.tex, line 680
path, one-edge paths, and the two-edge path exist. The coefficient 3t^2 includes both the edge product 2·3 and the ordered-time simplex area t^2/2. headPath length counts transfers, not elapsed time Starting at e_0, the complete state is (1,2t,3t^2)^ T. No fourth packet or repeated cycle can contribute. Direct differentiation gives x'(t)=Ax(t), including the relation x_2'(t)=3x_1(t).
Exact mathematical passage
x_2'(t)=3x_1(t)
X_2Use in source · 1 passage
Search aliases: X_2
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2436
Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.
Exact mathematical passage
X_0\xleftarrow{C_0}X_1\xleftarrow{C_1}X_2\xleftarrow{}\cdots
x_2'Use in source · 1 passage
Search aliases: x_2'
Use in source · atlas-pages.tex, line 680
path, one-edge paths, and the two-edge path exist. The coefficient 3t^2 includes both the edge product 2·3 and the ordered-time simplex area t^2/2. headPath length counts transfers, not elapsed time Starting at e_0, the complete state is (1,2t,3t^2)^ T. No fourth packet or repeated cycle can contribute. Direct differentiation gives x'(t)=Ax(t), including the relation x_2'(t)=3x_1(t).
Exact mathematical passage
x_2'(t)=3x_1(t)
X_3Use in source · 1 passage
Search aliases: X_3
Use in source · atlas-pages.tex, line 448
1 · 3, start at state 0, use weights w=(1,1/2), and terminal observable f=(2,1). For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate E_0[∏_r=0^2w(X_r) f(X_3)]. Each path contributes its transition probability times its three departure weights times its terminal value. vfigpaths platecaptionEvery possible path is included. The probability column sums to one. The contribution column sums to 311/256, agreeing exactly with the transfer calculati…
Exact mathematical passage
\EE_0\left[\prod_{r=0}^{2}w(X_r)\,f(X_3)\right].
X_cUse in source · 8 passages
Search aliases: X_c
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 447
Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.
Exact mathematical passage
X_c
X_fUse in source · 6 passages
Search aliases: X_f
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 447
Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.
Exact mathematical passage
X_f
x_iUse in source · 18 passages
Search aliases: x_i
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1378
[Exact discrete Green reconstruction at grid nodes] Let x_i=ih. Then frac1h(L_h^-1)_ij =G(x_i,x_j). Thus the scaled discrete inverse reproduces the continuum Dirichlet Green kernel exactly at the grid nodes.
Exact mathematical passage
x_i=ih
x_i^2Use in source · 2 passages
Search aliases: x_i^2
Use in source · atlas-pages.tex, line 424
vpage19Convergence with a nonzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMTraditional Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil differentiates the quadratic x(1-x) exactly, the solution u_h to the sampled forcing 12x_i^2 satisfies u_h(x_i)-u(x_i)=-h^2x_i(1-x_i). This gives both …
Exact mathematical passage
L_hu|_{\rm nodes}=12x_i^2+2h^2.
X_mUse in source · 1 passage
Search aliases: X_m
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2528
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,m}=I_{X_m}, \qquad R_{m,m}=I_{X_m}.
X_PUse in source · 1 passage
Search aliases: X_P
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4459
Let X=X_P⊕ X_Q and write A= A_PP · A_PQ
Exact mathematical passage
X=X_P\oplus X_Q
x_PUse in source · 6 passages
Search aliases: x_P
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1512
Solve the hidden block equation (zI-A_QQ)x_Q=A_QPx_P+y_Q for x_Q and substitute it into the retained equation. The resulting retained coefficient is F_P(z). Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting y_Q=0 gives the stated identity as an operator on Ran P. Equivalently, this is the block inverse formula.
Exact mathematical passage
(zI-A_{QQ})x_Q=A_{QP}x_P+y_Q
x_QUse in source · 7 passages
Search aliases: x_Q
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1512
Solve the hidden block equation (zI-A_QQ)x_Q=A_QPx_P+y_Q for x_Q and substitute it into the retained equation. The resulting retained coefficient is F_P(z). Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting y_Q=0 gives the stated identity as an operator on Ran P. Equivalently, this is the block inverse formula.
Exact mathematical passage
(zI-A_{QQ})x_Q=A_{QP}x_P+y_Q
X_rUse in source · 3 passages
Search aliases: X_r
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1452
[Discrete Feynman--Kac path formula] For an initial state i, (u_n)_i = E_i[ exp(-∑_r=0^n-1V(X_r))f(X_n) ].
Exact mathematical passage
\boxed{ (u_n)_i = \Exp_i\left[ \exp\left(-\sum_{r=0}^{n-1}V(X_r)\right)f(X_n) \right]. }
X_sUse in source · 3 passages
Search aliases: X_s
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4381
[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.
Exact mathematical passage
\left( \e^{t(Q^T-D_V)}f \right)_i = \Exp_i\left[ \exp\left( -\int_0^tV(X_s)\dd s \right) f(X_t) \right].
X_tUse in source · 2 passages
Search aliases: X_t
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4380
[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.
Exact mathematical passage
(X_t)_{t\geq0}
X_\ellUse in source · 4 passages
Search aliases: X_ℓ ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2457
The product of the spaces X_ℓ, equipped with the bounded supremum norm, is Banach. The equations C_ℓ x_ℓ+1=x_ℓ define a closed linear subspace.
Exact mathematical passage
X_\ell
x_\ellUse in source · 9 passages
Search aliases: x_ℓ ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2440
Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.
Exact mathematical passage
X_\infty = \left\{ (x_\ell)_{\ell\geq0}: C_\ell x_{\ell+1}=x_\ell, \quad \sup_\ell\norm{x_\ell}<\infty \right\}
X_{\ell+1}Use in source · 2 passages
Search aliases: X_ℓ+1 ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2515
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_\ell:X_{\ell+1}\to X_\ell
x_{\ell+1}Use in source · 2 passages
Search aliases: x_ℓ+1 ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2440
Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.
Exact mathematical passage
X_\infty = \left\{ (x_\ell)_{\ell\geq0}: C_\ell x_{\ell+1}=x_\ell, \quad \sup_\ell\norm{x_\ell}<\infty \right\}
\xleftarrowUse in source · 1 passage
Search aliases: xleftarrow
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2436
Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.
Exact mathematical passage
X_0\xleftarrow{C_0}X_1\xleftarrow{C_1}X_2\xleftarrow{}\cdots
YUse in source · 5 passages
Search aliases: Y
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 988
[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its evidence is positive. If all likelihoods are strictly positive and there are at least two profile classes, this is also the unique coarsest nontrivia…
Exact mathematical passage
\{\lambda^{(y)}\}_{y\in Y}
yUse in source · 50 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 988
[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its evidence is positive. If all likelihoods are strictly positive and there are at least two profile classes, this is also the unique coarsest nontrivia…
Exact mathematical passage
\{\lambda^{(y)}\}_{y\in Y}
y'Use in source · 1 passage
Search aliases: y'
Use in source · atlas-pages.tex, line 489
vpage22Hidden state returns as memoryEXACT RETAINED DYNAMICS / NOT A SCALAR SEMIGROUPTraditional Part I: Theorem 13.2. The retained-state memory identity, including hidden initial data. Write the same system as x'=-x+y, y'=x-y. Solving the second equation first gives y(t)=e^-ty(0)+∫_0^t e^-(t-r)x(r) dr. Substitution is exact: x'(t)=-x(t)+e^-ty(0)+∫_0^t e^-(t-r)x(r) dr. The hidden initial state produces a separate forcing term. It must be retained even when the memory kernel is known. vfigmemory …
Exact mathematical passage
x'=-x+y,\qquad y'=x-y.
y_1Use in source · 6 passages
Search aliases: y_1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5544
c K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
y_1,\ldots,y_r
y_2Use in source · 4 passages
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Use in source · atlas-pages.tex, line 895
vpage39Two auxiliary states realize a repeated poleFINITE REALIZATION / AN EXACT DIFFERENTIAL MODELTraditional Part II: Corollaries E15.4 and E15.5. This is a supplied rational realization. The kernel K(t)=te^-2t from V--23 has transform 1/(z+2)^2. It is realized by two auxiliary states: dot y_1 · =-2y_1+y_2, · y_1(0) · =0, dot y_2 · =-2y_2+x, · y_2(0) · =0, dot x · =a x+y_1. · With D=mat-2 · 1
Exact mathematical passage
\begin{aligned} \dot y_1&=-2y_1+y_2,& y_1(0)&=0,\\ \dot y_2&=-2y_2+x,& y_2(0)&=0,\\ \dot x&=a x+y_1.& \end{aligned}
y_QUse in source · 2 passages
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Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1512
Solve the hidden block equation (zI-A_QQ)x_Q=A_QPx_P+y_Q for x_Q and substitute it into the retained equation. The resulting retained coefficient is F_P(z). Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting y_Q=0 gives the stated identity as an operator on Ran P. Equivalently, this is the block inverse formula.
Exact mathematical passage
(zI-A_{QQ})x_Q=A_{QP}x_P+y_Q
y_rUse in source · 2 passages
Search aliases: y_r
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5544
c K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
y_1,\ldots,y_r
ZUse in source · 62 passages
Search aliases: Z
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 726
Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.
Exact mathematical passage
Z_\lambda(p)=\sum_i\lambda_ip_i.
z^{-1}Use in source · 1 passage
Search aliases: z^-1
Use in source · atlas-pages.tex, line 478
tain the first coordinate and eliminate the second. For z away from the relevant poles, the retained entry of (zI-A)^-1 is R_x(z)=[z+1-frac1z+1]^-1 =fracz+1z(z+2). The term Σ(z)=1/(z+1) is the hidden coordinate's self-energy. vfigresolvent platecaptionThe full inverse gives a rational response with two poles, at 0 and -2. Its partial fractions are R_x(z)=frac12z^-1+frac12(z+2)^-1. The hidden intermediate inverse has its own excluded point z=-1. headAn eliminated variable has not disappeared If t…
Exact mathematical passage
R_x(z)=\frac12z^{-1}+\frac12(z+2)^{-1}
z^{k+1}Use in source · 1 passage
Search aliases: z^k+1
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4615
For large z, Σ_P(z) = ∑_k=0^∞ fracA_PQA_QQ^kA_QPz^k+1. Thus the second condition implies the moment identities. Conversely, the first d identities imply all later ones by the Cayley Hamilton theorem, so the series vanishes.
Exact mathematical passage
\Sigma_P(z) = \sum_{k=0}^{\infty} \frac{A_{PQ}A_{QQ}^kA_{QP}}{z^{k+1}}.
z^{Nm}Use in source · 1 passage
Search aliases: z^Nm
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5910
This section receives the operator-substitution result deliberately moved out of Paper IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=log D_N once and define S_N^(r)(z)=exp(-r g(z)) for complex r. The bounded Banach-space result below uses classical holomorphic functional calculus; it …
Exact mathematical passage
\Dfun_N(z)=\sum_{m=0}^{\infty}\frac{(-1)^m z^{Nm}}{(Nm)!}.
Z_{\mathrm A}Use in source · 1 passage
Search aliases: Z_ A mathrm
Use in source · atlas-pages.tex, line 204
im. Multiplying all likelihoods by one common positive factor leaves this posterior unchanged. vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditional shapes are m_ A=5/7, q_ A=(4/5,1/5,0,0)^ T, m_ B=2…
Exact mathematical passage
Z_{\mathrm A}=5/4
Z_{\mathrm B}Use in source · 1 passage
Search aliases: Z_ B mathrm
Use in source · atlas-pages.tex, line 204
lihoods by one common positive factor leaves this posterior unchanged. vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditional shapes are m_ A=5/7, q_ A=(4/5,1/5,0,0)^ T, m_ B=2/7, q_ B=(0,0,1/2,1/2)^ …
Exact mathematical passage
Z_{\mathrm B}=1/2
Z_1Use in source · 4 passages
Search aliases: Z_1
Use in source · atlas-pages.tex, line 608
,3,2,0)^ T, p_2=tfrac1104(41,15,40,8)^ T. vfigintegrated_filter platecaptionWithin-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is (7/13,6/13). headCheck the normalizers, not only the final vector The first and second predictive evidences are Z_1=frac58, Z_2=frac1320, Z_1Z_2=frac1332. The unnormalized product D_2KD_1Kp_0 has mass 13/32 and normalizes to p_2. Because the likelihood en…
Exact mathematical passage
Z_1=\frac58,\qquad Z_2=\frac{13}{20}, \qquad Z_1Z_2=\frac{13}{32}.
Z_2Use in source · 4 passages
Search aliases: Z_2
Use in source · atlas-pages.tex, line 608
,3,2,0)^ T, p_2=tfrac1104(41,15,40,8)^ T. vfigintegrated_filter platecaptionWithin-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is (7/13,6/13). headCheck the normalizers, not only the final vector The first and second predictive evidences are Z_1=frac58, Z_2=frac1320, Z_1Z_2=frac1332. The unnormalized product D_2KD_1Kp_0 has mass 13/32 and normalizes to p_2. Because the likelihood en…
Exact mathematical passage
Z_1=\frac58,\qquad Z_2=\frac{13}{20}, \qquad Z_1Z_2=\frac{13}{32}.
Z_aUse in source · 1 passage
Search aliases: Z_a
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5419
The evidence in block B_a is Z_a =q_a∑_i∈ B_aL_ir_i^(a) =q_aoverline L_a. Apply [source reference].
Exact mathematical passage
Z_a =q_a\sum_{i\in B_a}L_ir_i^{(a)} =q_a\overline L_a.
Z_{\bar\lambda}Use in source · 4 passages
Search aliases: Z_barλ bar lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 925
[Bayesian information-loss packet] Assume nonnegative fine and declared coarse likelihoods, with both Z_r>0 and Z_barλ>0. The coarsening defect is C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood; the full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together. Moreover, C_π B_λ(p)- B_barλ(P) …
Exact mathematical passage
Z_{\bar\lambda}>0
Z_jUse in source · 1 passage
Search aliases: Z_j
Use in source · atlas-pages.tex, line 294
_order platecaptionBoth routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem. headKeep an unnormalized path alongside the filter For a sequence of observations, let D_r=diag(λ^(r)). The forward recursion is widehat p_r=Kp_r-1, p_r=D_rwidehat p_r/Z_r. Its unnormalized companion is v_r=D_rK⋯ D_1Kp_0, p_r=fracv_rones^ Tv_r, ones^ Tv_r=∏_j=1^r Z_j. The product of evidences records the normalization lost at each indivi…
Exact mathematical passage
v_r=D_rK\cdots D_1Kp_0, \qquad p_r=\frac{v_r}{\ones^{\mathsf T}v_r}, \qquad \ones^{\mathsf T}v_r=\prod_{j=1}^r Z_j.
Z_rUse in source · 9 passages
Search aliases: Z_r
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 914
When sufficiency fails, the defect still has an exact finite factorization. Fix a declared coarse likelihood barλ, put P=C_π p, δ(p)=r(p)-barλ, bar b= B_barλ(P), Z_r=⟨ r(p),P⟩.
Exact mathematical passage
P=C_\pi p, \qquad \delta(p)=r(p)-\bar\lambda, \qquad \bar b=\cB_{\bar\lambda}(P), \qquad Z_r=\langle r(p),P\rangle.
Z_{\widetilde\lambda}Use in source · 1 passage
Search aliases: Z_widetildeλ widetilde lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3613
[Approximate coarse Bayesian rule] Choose one positive number ℓ_α for every packet and define the block constant fine likelihood widetildeλ_i=ℓ_α (i∈ I_α). Then norm C B_λ(p)- B_ℓ(Cp) _1 ≤ frac 2∑_i p_iabsλ_i-widetildeλ_i max{Z_λ(p),Z_widetildeλ(p)} .
Exact mathematical passage
\norm{ C\Bcal_\lambda(p)-\Bcal_\ell(Cp) }_1 \leq \frac{ 2\sum_i p_i\abs{\lambda_i-\widetilde\lambda_i} }{ \max\{Z_\lambda(p),Z_{\widetilde\lambda}(p)\} }.
Z_\alphaUse in source · 9 passages
Search aliases: Z_α alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 762
Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.
Exact mathematical passage
Z_\alpha=\sum_{i\in I_\alpha}\lambda_ip_i, \qquad m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta}, \qquad q_{\alpha,i}=\begin{cases} \lambda_ip_i/Z_\alpha,&Z_\alpha>0,\ i\in I_\alpha,\\ 0,&\text{otherwise}. \end{cases}
Z_\betaUse in source · 3 passages
Search aliases: Z_β beta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 762
Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.
Exact mathematical passage
Z_\alpha=\sum_{i\in I_\alpha}\lambda_ip_i, \qquad m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta}, \qquad q_{\alpha,i}=\begin{cases} \lambda_ip_i/Z_\alpha,&Z_\alpha>0,\ i\in I_\alpha,\\ 0,&\text{otherwise}. \end{cases}
Z_\lambdaUse in source · 20 passages
Search aliases: Z_λ lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 726
Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.
Exact mathematical passage
Z_\lambda(p)=\sum_i\lambda_ip_i.
Z_{\lambda\odot\mu}Use in source · 1 passage
Search aliases: Z_λodotμ lambda odot mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 740
[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).
Exact mathematical passage
Z_{\lambda\odot\mu}(p)>0
Z_\muUse in source · 6 passages
Search aliases: Z_μ mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 804
[Likelihood perturbation bound] Let p be fixed and let λ,μ≥0 have positive evidences Z_λ,Z_μ. Then norm B_λ(p)- B_μ(p)_1 ≤ frac2∑_i|λ_i-μ_i|p_imin(Z_λ,Z_μ).
Exact mathematical passage
Z_\lambda,Z_\mu
|_{E_\mu}Use in source · 1 passage
Search aliases: |_E_μ mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1657
For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. I…
Exact mathematical passage
\left.F(A)\right|_{E_\mu} =\left.\sum_{k=0}^{d-1}\frac{F^{(k)}(\mu)}{k!}(A-\mu I)^k\right|_{E_\mu}.
|_{P_\alpha X}Use in source · 1 passage
Search aliases: |_P_α X alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2189
thm:reducing_packets The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.
Exact mathematical passage
T(t)|_{P_\alpha X} = \e^{tA_{\alpha\alpha}}.
|_{QX}Use in source · 2 passages
Search aliases: |_QX
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2749
[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.
Exact mathematical passage
\norm{\e^{t(QAQ)|_{QX}}} \leq M_Q\e^{\omega_Qt}.
|_{\Ran P}Use in source · 1 passage
Search aliases: |_Ran P Ran operatorname
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1503
[Schur--Feshbach projected resolvent] Assume both zI-A and zI-A_QQ are invertible. Define F_P(z) =zI-A_PP-A_PQ(zI-A_QQ)^-1A_QP. Then .P(zI-A)^-1|_Ran P=F_P(z)^-1.
Exact mathematical passage
\boxed{ \left.P(zI-A)^{-1}\right|_{\Ran P}=F_P(z)^{-1}. }
|_{\rm nodes}Use in source · 1 passage
Search aliases: |_rm nodes rm
Use in source · atlas-pages.tex, line 424
vpage19Convergence with a nonzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMTraditional Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil differentiates the quadratic x(1-x) exactly, the solution u_h to the sampled forcing 12x_i^2 satisfies u_h(x_i)-u(x_i)=-h^2x_i(1-x_i). This gives both …
Exact mathematical passage
L_hu|_{\rm nodes}=12x_i^2+2h^2.
|_{t=0}Use in source · 1 passage
Search aliases: |_t=0
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3252
= · f(x)+t(f''(x)-xf'(x))+o(t). Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is o(t), so the expansion may be integrated term by term. Therefore .frac d d t O_tf|_t=0 =f''-xf'. The Hermite eigenvalue identity also follows by differentiating O_th_n= e^-nth_n at zero. Gaussian integration by parts gives ∫_ R f'(x)overlineg(x)γ( d x) = ∫_ R f(x)overlinexg(x)-g'(x)γ( d x). Set g=f' and rearrange.
Exact mathematical passage
\left.\frac{\dd}{\dd t}\Ocal_tf\right|_{t=0} =f''-xf'.
\cdotUse in source · 3 passages
Search aliases: · cdot
Use in source · atlas-pages.tex, line 678
6 · 0 · 0, A^3=0. Therefore the exponential is the exact polynomial e^tA=I+tA+fract^22A^2. vfigext_paths platecaptionOnly the empty path, one-edge paths, and the two-edge path exist. The coefficient 3t^2 includes both the edge product 2·3 and the ordered-time simplex area t^2/2. headPath length counts transfers, not elapsed time Starting at e_0, the complete state is (1,2t,3t^2)^ T. No fourth packet or repeated cycle can contribute. Direct differentiation gives x'(t)=Ax(t), including the relatio…
Exact mathematical passage
2\cdot3
\timesUse in source · 32 passages
Search aliases: × times
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1017
[Bayesian refinement] For the finite test partition {0,1}sqcup{2,3}, a blockwise-constant likelihood gave commutation residual 1.11×10^-16. A nonconstant likelihood produced an exact defect-factorization residual 7.29×10^-17. Its measured L^1 defect was 4.8787×10^-2, below the certified bound 1.1016×10^-1. Two priors with identical coarse masses but different hidden within-packet shapes produced coarse posteriors differing by 1.2400×10^-1, which is a direct necessity witness.
Exact mathematical passage
1.11\times10^{-16}
\alpha_0Use in source · 1 passage
Search aliases: α_0 alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 431
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
\alpha_1Use in source · 1 passage
Search aliases: α_1 alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 431
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
\alpha_mUse in source · 1 passage
Search aliases: α_m alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 431
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
\alpha_{m-1}Use in source · 1 passage
Search aliases: α_m-1 alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 431
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
\GammaUse in source · 7 passages
Search aliases: Γ Gamma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2378
[Holomorphic defect formula] Let Γ be the positively oriented rectifiable boundary of a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that Γ lies in the common resolvent set. Thus its winding number is one about every spectral point. If g is holomorphic on a neighborhood of the closure of the union, then Cg(A_f)-g(A_c)C = frac12π i ∫_Γ g(z)R_c(z)Δ R_f(z) d z. In particular, normCg(A_f)-g(A_c)C ≤ · fraclength(Γ)2π sup_z∈Γabsg(z) sup_z…
Exact mathematical passage
\Gamma
\gammaUse in source · 29 passages
Search aliases: γ gamma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 388
[Packet semigroup convolution] For every s,t and every pair of packets α,β, T_βα(s+t) =∑_γ∈ IT_βγ(s)T_γα(t). Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.
Exact mathematical passage
\boxed{ T_{\beta\alpha}(s+t) =\sum_{\gamma\in I}T_{\beta\gamma}(s)T_{\gamma\alpha}(t). }
\DeltaUse in source · 48 passages
Search aliases: Δ Delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 452
Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.
Exact mathematical passage
\boxed{\Delta=CA_f-A_cC.}
\deltaUse in source · 32 passages
Search aliases: δ delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 368
Let P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For any operator B, define B_βα=P_β BP_α. Then B=∑_α,βB_βα. For the semigroup, write T_βα(t)=P_β e^tAP_α.
Exact mathematical passage
P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
\delta_0Use in source · 2 passages
Search aliases: δ_0 delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 564
[htbp] centering includegraphics[width=0.78textwidth]03_random_walk_transfer.pdf captionThe point mass δ_0 propagated by repeated symmetric random-walk transfer.
Exact mathematical passage
\delta_0
\Delta_{13}Use in source · 2 passages
Search aliases: Δ_13 Delta
Use in source · atlas-pages.tex, line 948
0 · 1 · 0. The generator defect is Δ=C_η A-BC_η, with only one nonzero entry, Δ_13=7η (matrix indices begin at one). vfigext_substitution platecaptionThe perturbation lets the discarded eigenvalue enter the comparison. Exact naturality holds at η=0; a nonzero defect is transformed, not erased, by the rational calculus. headA resolvent identity supplies the sign and both inverse factors For F(z)=1/(1+z), all required inverses exist, and C_η F(A)-F(B)C_η =-(I+B)^-1Δ(I+A)^-1 =mat0 · 0 · -7η/30
Exact mathematical passage
\Delta_{13}=7\eta
\delta_aUse in source · 3 passages
Search aliases: δ_a delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 937
C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood; the full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together. Moreover, C_π B_λ(p)- B_barλ(P) _1 ≤ frac2∑_aP_a|δ_a(p)| min{Z_r,Z_barλ} and ∑_aP_a|δ_a(p)| ≤ ∑_i p_i|λ_i-barλ_π(i)|.
Exact mathematical passage
\boxed{ \left\|C_\pi\cB_\lambda(p)-\cB_{\bar\lambda}(P)\right\|_1 \le \frac{2\sum_aP_a|\delta_a(p)|} {\min\{Z_r,Z_{\bar\lambda}\}}}
\delta_iUse in source · 1 passage
Search aliases: δ_i delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1639
Markov · point mass δ_i · K^n or e^tQ · state lumping and CK^n=bar K^nC
Exact mathematical passage
\delta_i
\delta_{ij}Use in source · 2 passages
Search aliases: δ_ij delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 696
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
D_iD_j=\delta_{ij}D_j, \qquad P_r=\sum_{j=0}^rD_j.
\delta_jUse in source · 4 passages
Search aliases: δ_j delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5274
[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.
Exact mathematical passage
\delta_j=\norm{C_jA_{j+1}-A_jC_j}.
\Delta_KUse in source · 5 passages
Search aliases: Δ_K Delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2956
[Finite horizon lumping defect] For arbitrary fine and coarse operators K and overline K, with the fixed aggregation map C, set Δ_K=CK-overline KC. Then CK^n-overline K^nC = ∑_j=0^n-1 overline K^n-1-jΔ_KK^j. If K and overline K are column stochastic, then normCK^n-overline K^nC_1→1 ≤ nnormΔ_K_1→1. For every probability vector p, the total variation discrepancy is at most frac n2normΔ_K_1→1.
Exact mathematical passage
\Delta_K=CK-\overline KC.
\Delta_LUse in source · 6 passages
Search aliases: Δ_L Delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4009
[Fractional defect bound] Let 0<α<1. Suppose L_f≥ mI, L_c≥ mI for some m>0. Define Δ_L=CL_f-L_cC. Then normCL_f^α-L_c^α C ≤ α m^α-1normΔ_L. Consequently, norm C e^-tL_f^α - e^-tL_c^αC ≤ α t m^α-1normΔ_L.
Exact mathematical passage
\Delta_L=CL_f-L_cC.
\delta_{mn}Use in source · 1 passage
Search aliases: δ_mn delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3186
· = ∑_n=0^∞ e^-ntH_n(x)fracu^nn!. Comparison of coefficients proves the eigenvalue formula. The same generating function shows ∫ H_mH_n dγ=n!δ_mn. The Hermite polynomials span all polynomials, and polynomials are dense in L^2(γ). Hence the normalized family is an orthonormal basis. See [source reference] for the general Gaussian Hilbert space setting.
Exact mathematical passage
\int H_mH_n\dd\gamma=n!\delta_{mn}.
\Delta_{n-1}Use in source · 2 passages
Search aliases: Δ_n-1 Delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3330
Let Δ_n-1 = {p∈ R^n: p_i≥0, ∑_ip_i=1 } be the probability simplex. Let λ∈[0,∞)^n have positive evidence Z_λ(p)=∑_iλ_ip_i. The Bayesian transfer is B_λ(p) = fracλodot pZ_λ(p).
Exact mathematical passage
\Delta_{n-1} = \left\{p\in\RR^n: p_i\geq0, \quad \sum_ip_i=1 \right\}
\delta_yUse in source · 2 passages
Search aliases: δ_y delta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1361
On the interval (0,1) with Dirichlet boundary conditions, the Green kernel for -u''=f is G(x,y)=min(x,y)-xy. Thus the point source δ_y generates the solution kernel, and a general source is reconstructed by u(x)=∫_0^1G(x,y)f(y) dy.
Exact mathematical passage
\delta_y
\delta_{\alpha\beta}Use in source · 2 passages
Search aliases: δ_αβ delta alpha beta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 368
Let P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For any operator B, define B_βα=P_β BP_α. Then B=∑_α,βB_βα. For the semigroup, write T_βα(t)=P_β e^tAP_α.
Exact mathematical passage
P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
\delta_{\beta\alpha}Use in source · 1 passage
Search aliases: δ_βα delta beta alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2102
[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.
Exact mathematical passage
T_{\beta\alpha}(0)=\delta_{\beta\alpha}P_\alpha.
\Delta_\ellUse in source · 3 passages
Search aliases: Δ_ℓ Delta ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2625
[Generator defect criterion] Assume normC_ℓ≤1, normR_ℓ≤1, and suppose T_ℓ(t)= e^tA_ℓ are contraction semigroups with bounded generators. Set Δ_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ. If ∑_ℓ=0^∞normΔ_ℓ<∞, then the conclusion of [source reference] holds on every bounded time interval, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ T∑_ℓ=n^∞normΔ_ℓ.
Exact mathematical passage
\Delta_\ell=C_\ell A_{\ell+1}-A_\ell C_\ell.
\varepsilonUse in source · 24 passages
Search aliases: ε varepsilon
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4742
weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.
Exact mathematical passage
\varepsilon
\eta_1Use in source · 2 passages
Search aliases: η_1 eta
Use in source · atlas-pages.tex, line 382
Square-root evolution can be written as an average of ordinary diffusion over a positive clock variable s: e^-tsqrt L=∫_0^∞ η_t(s)e^-sL ds, η_t(s)=fract2sqrtπs^-3/2e^-t^2/(4s), t>0. Here t is the external time; s is the ordinary diffusion time being mixed. They are not interchangeable variables. vfigrandom_clock platecaptionThe density η_1 peaks at s=1/6 but has a long right tail. The drawing ends at s=2; it does not claim that the missing tail has zero mass. headCheck the normalizer before inte…
Exact mathematical passage
\eta_1
\eta_{m,n}Use in source · 4 passages
Search aliases: η_m,n eta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2716
[Range leakage criterion] Under the contractive assumptions of [source reference], define η_m,n(T) = sup_0≤ t≤ T norm(I-P_m,n)T_n(t)R_n,m. Then ρ_m,n(T)≤η_m,n(T). Consequently, η_m,n(T)→0 for every T>0 is sufficient for the limiting family U_m to satisfy the semigroup law.
Exact mathematical passage
\eta_{m,n}(T) = \sup_{0\leq t\leq T} \norm{(I-P_{m,n})T_n(t)R_{n,m}}.
\eta_tUse in source · 8 passages
Search aliases: η_t eta
Use in source · atlas-pages.tex, line 375
vpage17A random diffusion clockSUBORDINATION / A DISTRIBUTION OF ORDINARY TIMESTraditional Part I: Theorem 10.2. The Poisson subordination formula for the finite positive model. Square-root evolution can be written as an average of ordinary diffusion over a positive clock variable s: e^-tsqrt L=∫_0^∞ η_t(s)e^-sL ds, η_t(s)=fract2sqrtπs^-3/2e^-t^2/(4s), t>0. Here t is the external time; s is the ordinary diffusion time being mixed. They are not interchangeable variables. vfigrandom_clock platecap…
Exact mathematical passage
e^{-t\sqrt L}=\int_0^\infty \eta_t(s)e^{-sL}\,ds, \qquad \eta_t(s)=\frac{t}{2\sqrt\pi}s^{-3/2}e^{-t^2/(4s)},\quad t>0.
\eta_{\alpha,s}Use in source · 1 passage
Search aliases: η_α,s eta alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4090
d t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\eta_{\alpha,s}*\eta_{\alpha,t} = \eta_{\alpha,s+t}.
\eta_{\alpha,s+t}Use in source · 1 passage
Search aliases: η_α,s+t eta alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4090
d t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\eta_{\alpha,s}*\eta_{\alpha,t} = \eta_{\alpha,s+t}.
\eta_{\alpha,t}Use in source · 5 passages
Search aliases: η_α,t eta alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4076
[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\eta_{\alpha,t}
\thetaUse in source · 8 passages
Search aliases: θ theta
Use in source · atlas-pages.tex, line 306
vpage14Conjugacy tracks exponentsBETA MODEL / POSTERIOR AND EVIDENCETraditional Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-θ). Multiplication adds exponents, so the posterior belongs to Beta(4,4) with density 140θ^3(1-θ)^3. vfigbeta platecaptionThe dashed prior is asymmetr…
Exact mathematical passage
\rho(\theta)=12\theta(1-\theta)^2,\qquad0\le\theta\le1.
\theta^2Use in source · 2 passages
Search aliases: θ^2 theta
Use in source · atlas-pages.tex, line 309
vpage14Conjugacy tracks exponentsBETA MODEL / POSTERIOR AND EVIDENCETraditional Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-θ). Multiplication adds exponents, so the posterior belongs to Beta(4,4) with density 140θ^3(1-θ)^3. vfigbeta platecaptionThe dashed prior is asymmetr…
Exact mathematical passage
\theta^2(1-\theta)
\theta^3Use in source · 2 passages
Search aliases: θ^3 theta
Use in source · atlas-pages.tex, line 309
ite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-θ). Multiplication adds exponents, so the posterior belongs to Beta(4,4) with density 140θ^3(1-θ)^3. vfigbeta platecaptionThe dashed prior is asymmetric. The solid posterior is symmetric with mean 1/2. Polynomial integration verifies the normalizers exactl…
Exact mathematical passage
140\theta^3(1-\theta)^3
\lambda^{\mathsf T}Use in source · 1 passage
Search aliases: λ^ T lambda mathsf
Use in source · atlas-pages.tex, line 35
B_λ · diag(λ)p/(λ^ Tp). · Nonnegative likelihood and positive total evidence.
Exact mathematical passage
\operatorname{diag}(\lambda)p/(\lambda^{\mathsf T}p)
\lambda^{(1)}Use in source · 2 passages
Search aliases: λ^(1) lambda
Use in source · atlas-pages.tex, line 595
vpage26One complete coarse filterEND-TO-END MODEL / TWO OBSERVATIONSTraditional Part I: Proposition 9.3 with Theorems 5.1 and 8.2. Markov closure and likelihood sufficiency are separate. Use the four-state kernel K and aggregation C from pagerefV5, starting at p_0=e_0. Observe two likelihood vectors constant inside both packets: λ^(1)=(3/4,3/4,1/4,1/4), λ^(2)=(1/2,1/2,1,1). Predict, then update at each observation. The fine calculation gives p_1=tfrac120(15,3,2,0)^ T, p_2=tfrac1104(41,15,40,8)^ …
Exact mathematical passage
\lambda^{(1)}=(3/4,3/4,1/4,1/4),\qquad \lambda^{(2)}=(1/2,1/2,1,1).
\lambda^{(1)}_iUse in source · 1 passage
Search aliases: λ^(1)_i lambda
Use in source · ext_profiles.tex, line 9
(1,1); draw[decorate,decoration=brace,mirror,amplitude=5pt](-.7,.55)--(2.7,.55)node[midway,below=8pt]one packet; node[note]at(4.8,.1)separate packet;node[note]at(7.2,.1)separate packet; node[note]at(3.6,-.75)Joint likelihood profile =(λ^(1)_i,λ^(2)_i);
Exact mathematical passage
=(\lambda^{(1)}_i,\lambda^{(2)}_i)
\lambda^{(2)}Use in source · 2 passages
Search aliases: λ^(2) lambda
Use in source · atlas-pages.tex, line 595
vpage26One complete coarse filterEND-TO-END MODEL / TWO OBSERVATIONSTraditional Part I: Proposition 9.3 with Theorems 5.1 and 8.2. Markov closure and likelihood sufficiency are separate. Use the four-state kernel K and aggregation C from pagerefV5, starting at p_0=e_0. Observe two likelihood vectors constant inside both packets: λ^(1)=(3/4,3/4,1/4,1/4), λ^(2)=(1/2,1/2,1,1). Predict, then update at each observation. The fine calculation gives p_1=tfrac120(15,3,2,0)^ T, p_2=tfrac1104(41,15,40,8)^ …
Exact mathematical passage
\lambda^{(1)}=(3/4,3/4,1/4,1/4),\qquad \lambda^{(2)}=(1/2,1/2,1,1).
\lambda^{(2)}_iUse in source · 1 passage
Search aliases: λ^(2)_i lambda
Use in source · ext_profiles.tex, line 9
(1,1); draw[decorate,decoration=brace,mirror,amplitude=5pt](-.7,.55)--(2.7,.55)node[midway,below=8pt]one packet; node[note]at(4.8,.1)separate packet;node[note]at(7.2,.1)separate packet; node[note]at(3.6,-.75)Joint likelihood profile =(λ^(1)_i,λ^(2)_i);
Exact mathematical passage
=(\lambda^{(1)}_i,\lambda^{(2)}_i)
\lambda^{(r)}Use in source · 5 passages
Search aliases: λ^(r) lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1091
Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).
Exact mathematical passage
\lambda^{(r)}
\lambda^{(y)}Use in source · 1 passage
Search aliases: λ^(y) lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 988
[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its evidence is positive. If all likelihoods are strictly positive and there are at least two profile classes, this is also the unique coarsest nontrivia…
Exact mathematical passage
\{\lambda^{(y)}\}_{y\in Y}
\lambda^\alphaUse in source · 2 passages
Search aliases: λ^α lambda alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4077
[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\int_0^\infty \e^{-s\lambda}\eta_{\alpha,t}(\dd s) = \e^{-t\lambda^\alpha}
\lambda_aUse in source · 3 passages
Search aliases: λ_a lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 898
i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the conditional average of λ_i and λ_i'. It can be independent of that variation only if λ_i=λ_i'. Hence λ is constant on every packet; write the value on I_a as λ_a.
Exact mathematical passage
\lambda_a
\lambda_bUse in source · 1 passage
Search aliases: λ_b lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 901
Now use priors supported on one state of packet a and one state of packet b. Equality of the two coarse posterior odds gives fracλ_aλ_b = fracbarλ_abarλ_b. This holds for every pair of coarse packets, so λ_a=cbarλ_a for one common c>0.
Exact mathematical passage
\frac{\lambda_a}{\lambda_b} = \frac{\bar\lambda_a}{\bar\lambda_b}.
\lambda_{i'}Use in source · 2 passages
Search aliases: λ_i' lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 898
sely, fix two fine states i,i'∈ I_a and one state j∈ I_b in another coarse packet. Vary the distribution of a fixed coarse mass inside I_a between i and i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the conditional average of λ_i and λ_i'. It can be independent of that variation only if λ_i=λ_i'. Hence λ is constant on every packet; write the value on I_a as λ…
Exact mathematical passage
\lambda_{i'}
\lambda_i^{(y)}Use in source · 1 passage
Search aliases: λ_i^(y) lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 989
[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its evidence is positive. If all likelihoods are strictly positive and there are at least two profile classes, this is also the unique coarsest nontrivia…
Exact mathematical passage
i\sim j \quad\Longleftrightarrow\quad \lambda_i^{(y)}=\lambda_j^{(y)} \text{ for every }y\in Y.
\lambda_jUse in source · 7 passages
Search aliases: λ_j lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 752
The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.
Exact mathematical passage
\frac{\mu_i(\lambda_ip_i/Z_\lambda)}{\sum_j\mu_j\lambda_jp_j/Z_\lambda} = \frac{\mu_i\lambda_ip_i}{\sum_j\mu_j\lambda_jp_j}.
\lambda_j^{(y)}Use in source · 1 passage
Search aliases: λ_j^(y) lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 989
[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its evidence is positive. If all likelihoods are strictly positive and there are at least two profile classes, this is also the unique coarsest nontrivia…
Exact mathematical passage
i\sim j \quad\Longleftrightarrow\quad \lambda_i^{(y)}=\lambda_j^{(y)} \text{ for every }y\in Y.
\lambda_j^\alphaUse in source · 5 passages
Search aliases: λ_j^α lambda alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1125
Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.
Exact mathematical passage
\boxed{L^\alpha=\sum_j\lambda_j^\alpha P_j.}
\lambda_j^{\alpha+\beta}Use in source · 1 passage
Search aliases: λ_j^α+β lambda alpha beta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1148
All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.
Exact mathematical passage
L^\alpha L^\beta =\sum_{j,k}\lambda_j^\alpha\lambda_k^\beta P_jP_k =\sum_j\lambda_j^{\alpha+\beta}P_j.
\lambda_kUse in source · 2 passages
Search aliases: λ_k lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4311
[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.
Exact mathematical passage
\lambda_k=(k\pi)^2
\lambda_{k,h}Use in source · 12 passages
Search aliases: λ_k,h lambda
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1410
The kth discrete eigenvalue is λ_k,h=frac4h^2sin^2(frackπ h2), whereas the continuum eigenvalue is (kπ)^2.
Exact mathematical passage
\lambda_{k,h}=\frac4{h^2}\sin^2\left(\frac{k\pi h}{2}\right),
\lambda_{k,h}^\alphaUse in source · 5 passages
Search aliases: λ_k,h^α lambda alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4319
[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.
Exact mathematical passage
\lambda_{k,h}^\alpha = \lambda_k^\alpha+O(h^2)
\lambda_{k,h}^{\alpha-1}Use in source · 1 passage
Search aliases: λ_k,h^α-1 lambda alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5606
The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.
Exact mathematical passage
(k\pi)^{2\alpha}-\lambda_{k,h}^\alpha \leq \alpha\lambda_{k,h}^{\alpha-1} \bigl((k\pi)^2-\lambda_{k,h}\bigr).
\lambda_k^\alphaUse in source · 2 passages
Search aliases: λ_k^α lambda alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4319
[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.
Exact mathematical passage
\lambda_{k,h}^\alpha = \lambda_k^\alpha+O(h^2)
\lambda_k^\betaUse in source · 1 passage
Search aliases: λ_k^β lambda beta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1148
All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.
Exact mathematical passage
L^\alpha L^\beta =\sum_{j,k}\lambda_j^\alpha\lambda_k^\beta P_jP_k =\sum_j\lambda_j^{\alpha+\beta}P_j.
\muUse in source · 51 passages
Search aliases: μ mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 740
[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).
Exact mathematical passage
\mu
\mu_1Use in source · 1 passage
Search aliases: μ_1 mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4644
[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r}…
Exact mathematical passage
\mu_1,\ldots,\mu_r
\mu_iUse in source · 3 passages
Search aliases: μ_i mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 752
The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.
Exact mathematical passage
\frac{\mu_i(\lambda_ip_i/Z_\lambda)}{\sum_j\mu_j\lambda_jp_j/Z_\lambda} = \frac{\mu_i\lambda_ip_i}{\sum_j\mu_j\lambda_jp_j}.
\mu_jUse in source · 19 passages
Search aliases: μ_j mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 752
The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.
Exact mathematical passage
\frac{\mu_i(\lambda_ip_i/Z_\lambda)}{\sum_j\mu_j\lambda_jp_j/Z_\lambda} = \frac{\mu_i\lambda_ip_i}{\sum_j\mu_j\lambda_jp_j}.
\mu_rUse in source · 1 passage
Search aliases: μ_r mu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4644
[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r}…
Exact mathematical passage
\mu_1,\ldots,\mu_r
\nuUse in source · 12 passages
Search aliases: ν nu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4648
[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r}…
Exact mathematical passage
\nu_j
\nu_jUse in source · 12 passages
Search aliases: ν_j nu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4648
[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r}…
Exact mathematical passage
\nu_j
\pi^2Use in source · 4 passages
Search aliases: π^2 pi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3071
nded:record:037 Let P_0,P_1,…,P_r come from increasingly fine partitions, so that Ran P_0⊆Ran P_1⊆⋯⊆Ran P_r. Define D_0=P_0, D_j=P_j-P_j-1 (1≤ j≤ r). Then P_iP_j=P_min(i,j), D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j. Every D_j is an orthogonal projector, and normP_rf_π^2 = ∑_j=0^rnormD_jf_π^2 and norm f_π^2 = ∑_j=0^rnormD_jf_π^2 + normf-P_rf_π^2.
Exact mathematical passage
\norm{P_rf}_\pi^2 = \sum_{j=0}^r\norm{D_jf}_\pi^2
\rhoUse in source · 7 passages
Search aliases: ρ rho
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2653
[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally uniform and every T_n is norm continuous, then U_m is a norm continuous semigroup.
Exact mathematical passage
\rho_{m,n}(T) = \sup_{0\leq s,t\leq T} \norm{ C_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m} } \longrightarrow0.
\rho_{m,n}Use in source · 4 passages
Search aliases: ρ_m,n rho
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2653
[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally uniform and every T_n is norm continuous, then U_m is a norm continuous semigroup.
Exact mathematical passage
\rho_{m,n}(T) = \sup_{0\leq s,t\leq T} \norm{ C_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m} } \longrightarrow0.
\rho_tUse in source · 2 passages
Search aliases: ρ_t rho
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3235
Write ρ_t= e^-t=1-t+O(t^2), σ_t^2=1- e^-2t=2t+O(t^2). For a polynomial f, Taylor expansion at x and Gaussian integration give O_tf(x) = · f(x) +f'(x)(ρ_t-1)x +frac12f''(x)σ_t^2 +o(t)
Exact mathematical passage
\rho_t=\e^{-t}=1-t+O(t^2), \qquad \sigma_t^2=1-\e^{-2t}=2t+O(t^2).
\SigmaUse in source · 12 passages
Search aliases: Σ Sigma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1519
The self-energy Σ_P(z)=A_PQ(zI-A_QQ)^-1A_QP contains the exact effect of the eliminated packets. A bare truncation A_PP discards this term and is generally not exact.
Exact mathematical passage
\Sigma_P(z)=A_{PQ}(zI-A_{QQ})^{-1}A_{QP}
\sigmaUse in source · 3 passages
Search aliases: σ sigma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1545
[Memory/self-energy correspondence] For finite complex block matrices, assume Rez>Reλ for every λ∈σ(A_QQ). Then the following improper integral is absolutely convergent and ∫_0^∞ e^-zt K(t) dt =A_PQ(zI-A_QQ)^-1A_QP =Σ_P(z).
Exact mathematical passage
\lambda\in\sigma(A_{QQ})
\Sigma_PUse in source · 10 passages
Search aliases: Σ_P Sigma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1519
The self-energy Σ_P(z)=A_PQ(zI-A_QQ)^-1A_QP contains the exact effect of the eliminated packets. A bare truncation A_PP discards this term and is generally not exact.
Exact mathematical passage
\Sigma_P(z)=A_{PQ}(zI-A_{QQ})^{-1}A_{QP}
\sigma_t^2Use in source · 2 passages
Search aliases: σ_t^2 sigma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3235
Write ρ_t= e^-t=1-t+O(t^2), σ_t^2=1- e^-2t=2t+O(t^2). For a polynomial f, Taylor expansion at x and Gaussian integration give O_tf(x) = · f(x) +f'(x)(ρ_t-1)x +frac12f''(x)σ_t^2 +o(t)
Exact mathematical passage
\rho_t=\e^{-t}=1-t+O(t^2), \qquad \sigma_t^2=1-\e^{-2t}=2t+O(t^2).
\tau^2Use in source · 1 passage
Search aliases: τ^2 tau
Use in source · atlas-pages.tex, line 558
Exact integration gives ∫_0^∞ f(t) dt=1, E[τ]=2, E[τ^2]=6, Var(τ)=2. Equivalently, the mean is ones^ T(-T)^-1e_0. A single exponential waiting time with mean two would have constant hazard 1/2 and variance four, so matching the mean alone would not reproduce this distribution.
Exact mathematical passage
\int_0^\infty f(t)\,dt=1,\qquad \EE[\tau]=2,\qquad \EE[\tau^2]=6, \qquad \Var(\tau)=2.
\tau^mUse in source · 1 passage
Search aliases: τ^m tau
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4987
[Laplace transform and moments] Assume the spectrum of A lies in the open left half plane. Then absorption occurs with probability one. For s≥0, E[ e^-sτ] = a^T(sI-A)^-1p_0. For every integer m≥1, E[τ^m] = m! 1^T(-A)^-mp_0.
Exact mathematical passage
\Exp[\tau^m] = m!\one^T(-A)^{-m}p_0.
\PhiUse in source · 9 passages
Search aliases: Φ Phi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1033
Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.
Exact mathematical passage
\Phi_t(p)_i = \frac{e^{tg_i}p_i}{\sum_je^{tg_j}p_j}.
\phiUse in source · 5 passages
Search aliases: ϕ phi
Use in source · atlas-pages.tex, line 268
nentially, while the probabilities remain inside the simplex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Direct algebra gives ϕ_a(ϕ_b(x))=ϕ_ab(x). Thus a=e^t gives a nonlinear semigroup. Nevertheless,…
Exact mathematical passage
\phi_a(x)=\frac{ax}{1+(a-1)x}.
\phi_3Use in source · 3 passages
Search aliases: ϕ_3 phi
Use in source · atlas-pages.tex, line 271
id and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Direct algebra gives ϕ_a(ϕ_b(x))=ϕ_ab(x). Thus a=e^t gives a nonlinear semigroup. Nevertheless, ϕ_3(1/2)=3/4, while the mean of ϕ_3(0) and ϕ_3(1) is 1/2. Superposition fails.
Exact mathematical passage
\phi_3(1/2)=3/4
\phi_aUse in source · 2 passages
Search aliases: ϕ_a phi
Use in source · atlas-pages.tex, line 268
nentially, while the probabilities remain inside the simplex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Direct algebra gives ϕ_a(ϕ_b(x))=ϕ_ab(x). Thus a=e^t gives a nonlinear semigroup. Nevertheless,…
Exact mathematical passage
\phi_a(x)=\frac{ax}{1+(a-1)x}.
\phi_{ab}Use in source · 1 passage
Search aliases: ϕ_ab phi
Use in source · atlas-pages.tex, line 271
lex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Direct algebra gives ϕ_a(ϕ_b(x))=ϕ_ab(x). Thus a=e^t gives a nonlinear semigroup. Nevertheless, ϕ_3(1/2)=3/4, while the mean of ϕ_3(0) and ϕ_3(1) is 1/2…
Exact mathematical passage
\phi_a(\phi_b(x))=\phi_{ab}(x)
\phi_bUse in source · 1 passage
Search aliases: ϕ_b phi
Use in source · atlas-pages.tex, line 271
lex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Direct algebra gives ϕ_a(ϕ_b(x))=ϕ_ab(x). Thus a=e^t gives a nonlinear semigroup. Nevertheless, ϕ_3(1/2)=3/4, while the mean of ϕ_3(0) and ϕ_3(1) is 1/2…
Exact mathematical passage
\phi_a(\phi_b(x))=\phi_{ab}(x)
\Phi_sUse in source · 2 passages
Search aliases: Φ_s Phi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1040
[Bayesian nonlinear semigroup] Φ_s+t=Φ_s∘Φ_t. Its generator is the centered multiplicative field fracddtp_i(t) = (g_i-⟨ g,p(t)⟩)p_i(t).
Exact mathematical passage
\boxed{ \Phi_{s+t}=\Phi_s\circ\Phi_t. }
\Phi_{s+t}Use in source · 2 passages
Search aliases: Φ_s+t Phi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1040
[Bayesian nonlinear semigroup] Φ_s+t=Φ_s∘Φ_t. Its generator is the centered multiplicative field fracddtp_i(t) = (g_i-⟨ g,p(t)⟩)p_i(t).
Exact mathematical passage
\boxed{ \Phi_{s+t}=\Phi_s\circ\Phi_t. }
\Phi_tUse in source · 5 passages
Search aliases: Φ_t Phi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1033
Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.
Exact mathematical passage
\Phi_t(p)_i = \frac{e^{tg_i}p_i}{\sum_je^{tg_j}p_j}.
\Phi_yUse in source · 3 passages
Search aliases: Φ_y Phi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5534
For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
\Phi_y=\Bcal_{L_y}\circ K
\Phi_{y_1}Use in source · 1 passage
Search aliases: Φ_y_1 Phi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5545
[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
C\Phi_{y_r}\circ\cdots\circ\Phi_{y_1} = \overline\Phi_{y_r}\circ\cdots\circ\overline\Phi_{y_1}C.
\Phi_{y_r}Use in source · 1 passage
Search aliases: Φ_y_r Phi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5545
[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
C\Phi_{y_r}\circ\cdots\circ\Phi_{y_1} = \overline\Phi_{y_r}\circ\cdots\circ\overline\Phi_{y_1}C.
\psiUse in source · 2 passages
Search aliases: ψ psi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3800
[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).
Exact mathematical passage
\psi(t) = \log\left( \sum_i\e^{tg_i}p_i(0) \right).
\psi'Use in source · 1 passage
Search aliases: ψ' psi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3808
[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).
Exact mathematical passage
\psi'(t)=\ip{g}{p(t)}, \qquad \psi''(t)=\Var_{p(t)}(g)\geq0.
\psi''Use in source · 1 passage
Search aliases: ψ'' psi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3808
[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).
Exact mathematical passage
\psi'(t)=\ip{g}{p(t)}, \qquad \psi''(t)=\Var_{p(t)}(g)\geq0.
\omegaUse in source · 13 passages
Search aliases: ω omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 486
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{e^{tA_f}}\le M_f e^{\omega_f t}, \qquad \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
\omega_aUse in source · 3 passages
Search aliases: ω_a omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5490
[Oscillation bound] Suppose overline L_a is any representative likelihood for block B_a, and set ω_a = max_i∈ B_aabsL_i-overline L_a. Then d(pmid L,overline L) ≤ ∑_aq_aω_a. Thus small within block likelihood oscillation gives a small coarse posterior defect whenever the evidence is bounded away from zero.
Exact mathematical passage
\omega_a = \max_{i\in B_a}\abs{L_i-\overline L_a}.
\omega_cUse in source · 4 passages
Search aliases: ω_c omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 486
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{e^{tA_f}}\le M_f e^{\omega_f t}, \qquad \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
\omega_fUse in source · 4 passages
Search aliases: ω_f omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 486
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{e^{tA_f}}\le M_f e^{\omega_f t}, \qquad \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
\omega_QUse in source · 2 passages
Search aliases: ω_Q omega
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2749
[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.
Exact mathematical passage
\norm{\e^{t(QAQ)|_{QX}}} \leq M_Q\e^{\omega_Qt}.
\ellUse in source · 69 passages
Search aliases: ℓ ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1328
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces; no operator sum across unrelated spaces…
Exact mathematical passage
D_\ell=C_\ell A_{\ell+1}-A_\ell C_\ell
\ell^1Use in source · 8 passages
Search aliases: ℓ^1 ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2987
Sum the telescoping identities overline K^n-1-jCK^j+1 - overline K^n-jCK^j = overline K^n-1-jΔ_KK^j. Stochastic matrices are contractions in the ℓ^1 norm. The last statement uses total variation equal to one half of the ℓ^1 distance.
Exact mathematical passage
\ell^1
\ell_1Use in source · 1 passage
Search aliases: ℓ_1 ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5849
[A visible likelihood loss] Let B_1={1,2}, let B_2={3}, and take p=(3/10,2/10,5/10), L=(1,3,2). The coarse prior is (1/2,1/2). The exact block likelihoods are ℓ_1(p)=frac(3/10)+3(2/10)1/2=frac95, ℓ_2(p)=2. Thus the exact coarse posterior is (9/19,10/19). If the first block is replaced by the fixed representative value 2, the approximate coarse posterior is (1/2,1/2). The ℓ^1 defect is 1/19, while [source reference] gives the valid upper bound 1/10.
Exact mathematical passage
\ell_1(p)=\frac{(3/10)+3(2/10)}{1/2}=\frac95, \qquad \ell_2(p)=2.
\ell_2Use in source · 1 passage
Search aliases: ℓ_2 ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5849
[A visible likelihood loss] Let B_1={1,2}, let B_2={3}, and take p=(3/10,2/10,5/10), L=(1,3,2). The coarse prior is (1/2,1/2). The exact block likelihoods are ℓ_1(p)=frac(3/10)+3(2/10)1/2=frac95, ℓ_2(p)=2. Thus the exact coarse posterior is (9/19,10/19). If the first block is replaced by the fixed representative value 2, the approximate coarse posterior is (1/2,1/2). The ℓ^1 defect is 1/19, while [source reference] gives the valid upper bound 1/10.
Exact mathematical passage
\ell_1(p)=\frac{(3/10)+3(2/10)}{1/2}=\frac95, \qquad \ell_2(p)=2.
\ell_aUse in source · 6 passages
Search aliases: ℓ_a ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5451
For an arbitrary prior p, put q=Cp and Z(p,L)=ipLp. Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0; choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_aabsℓ_a(p)-overline L_a. Then norm C B_L(p)- B_overline L(q) _1 ≤ frac2d(pmid L,overline L) max{Z(p,L),ipoverline Lq}.
Exact mathematical passage
\ell_a(p)=\sum_{i\in B_a}L_ip_i/q_a
\ell_yUse in source · 1 passage
Search aliases: ℓ_y ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5557
[One step filter defect] Let r=Kp, q=Cr, and let ℓ_y(r) be the exact coarse likelihood obtained by averaging L_y inside each block with respect to r. Assume 0<overline m≤(overline L_y)_a≤overline M. Then normCΦ_y(p)-overlineΦ_y(Cp)_1 ≤ · frac2d(rmid L_y,overline L_y) max{ipL_yr,ipoverline L_yq}
Exact mathematical passage
\ell_y(r)
\ell_\alphaUse in source · 3 passages
Search aliases: ℓ_α ell alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3522
ons are equivalent. item There is a positive coarse likelihood ℓ∈(0,∞)^m such that C B_λ(p) = B_ℓ(Cp) for every strictly positive prior p, and the posterior conditional distribution inside every packet equals its prior conditional distribution. item The likelihood is constant on every packet. When these conditions hold, ℓ_α can be chosen as the common value of λ on I_α, and it is unique up to a common positive scale.
Exact mathematical passage
\ell_\alpha
\mapstoUse in source · 3 passages
Search aliases: ↦ mapsto
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5605
The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.
Exact mathematical passage
x\mapsto x^\alpha
\notinUse in source · 5 passages
Search aliases: ∉ notin
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3831
[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).
Exact mathematical passage
\delta = \min\{G-g_i:p_i(0)>0, \quad i\notin M\}>0.
\prod_{j=1}^rUse in source · 1 passage
Search aliases: ∏_j=1^r prod
Use in source · atlas-pages.tex, line 294
_order platecaptionBoth routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem. headKeep an unnormalized path alongside the filter For a sequence of observations, let D_r=diag(λ^(r)). The forward recursion is widehat p_r=Kp_r-1, p_r=D_rwidehat p_r/Z_r. Its unnormalized companion is v_r=D_rK⋯ D_1Kp_0, p_r=fracv_rones^ Tv_r, ones^ Tv_r=∏_j=1^r Z_j. The product of evidences records the normalization lost at each indivi…
Exact mathematical passage
v_r=D_rK\cdots D_1Kp_0, \qquad p_r=\frac{v_r}{\ones^{\mathsf T}v_r}, \qquad \ones^{\mathsf T}v_r=\prod_{j=1}^r Z_j.
\prod_rUse in source · 2 passages
Search aliases: ∏_r prod
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1468
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
\prod_rK_{i_{r+1},i_r}
\prod_{r=0}^{2}Use in source · 1 passage
Search aliases: ∏_r=0^2 prod
Use in source · atlas-pages.tex, line 448
1 · 3, start at state 0, use weights w=(1,1/2), and terminal observable f=(2,1). For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate E_0[∏_r=0^2w(X_r) f(X_3)]. Each path contributes its transition probability times its three departure weights times its terminal value. vfigpaths platecaptionEvery possible path is included. The probability column sums to one. The contribution column sums to 311/256, agreeing exactly with the transfer calculati…
Exact mathematical passage
\EE_0\left[\prod_{r=0}^{2}w(X_r)\,f(X_3)\right].
\sum_aUse in source · 6 passages
Search aliases: ∑_a sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 864
For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.
Exact mathematical passage
Z=\sum_i\lambda_ip_i=\sum_aP_ar_a(p)
\sum_{a=1}^mUse in source · 1 passage
Search aliases: ∑_a=1^m sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5462
Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0; choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_aabsℓ_a(p)-overline L_a. Then norm C B_L(p)- B_overline L(q) _1 ≤ frac2d(pmid L,overline L) max{Z(p,L),ipoverline Lq}.
Exact mathematical passage
d(p\mid L,\overline L) = \sum_{a=1}^mq_a\abs{\ell_a(p)-\overline L_a}.
\sum_bUse in source · 1 passage
Search aliases: ∑_b sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 865
For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.
Exact mathematical passage
\bigl(C_\pi\cB_\lambda(p)\bigr)_a = \frac{\sum_{i\in I_a}\lambda_ip_i}{Z} = \frac{P_ar_a(p)}{\sum_bP_br_b(p)} = \bigl(\cB_{r(p)}(P)\bigr)_a.
\sum_{B\in\mathcal Q}Use in source · 2 passages
Search aliases: ∑_B∈ Q sum in mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 674
After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block; both equal ∑_B∈ Q frac1π(B) (∑_i∈ Bπ_i f_i) overline(∑_j∈ Bπ_j g_j).
Exact mathematical passage
\sum_{B\in\mathcal Q} \frac{1}{\pi(B)} \left(\sum_{i\in B}\pi_i f_i\right) \overline{\left(\sum_{j\in B}\pi_j g_j\right)}.
\sum_{c\in\{\mathrm A,\mathrm B\}}Use in source · 1 passage
Search aliases: ∑_c∈{ A, B} sum in mathrm mathrm
Use in source · atlas-pages.tex, line 90
Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.
Exact mathematical passage
P_b e^{(s+t)Q}P_a=\sum_{c\in\{\mathrm A,\mathrm B\}}(P_b e^{sQ}P_c)(P_c e^{tQ}P_a).
\sum_{i\in B}Use in source · 2 passages
Search aliases: ∑_i∈ B sum in
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 674
After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block; both equal ∑_B∈ Q frac1π(B) (∑_i∈ Bπ_i f_i) overline(∑_j∈ Bπ_j g_j).
Exact mathematical passage
\sum_{B\in\mathcal Q} \frac{1}{\pi(B)} \left(\sum_{i\in B}\pi_i f_i\right) \overline{\left(\sum_{j\in B}\pi_j g_j\right)}.
\sum_{i\in B_a}Use in source · 5 passages
Search aliases: ∑_i∈ B_a sum in
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5355
se Markov generator. For observables, the backward generator is Q^T; weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
\overline\pi_a=\sum_{i\in B_a}\pi_i.
\sum_{i\in I_a}Use in source · 4 passages
Search aliases: ∑_i∈ I_a sum in
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 840
Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero.
Exact mathematical passage
(C_\pi p)_a=P_a=\sum_{i\in I_a}p_i.
\sum_{i\in M}Use in source · 2 passages
Search aliases: ∑_i∈ M sum in
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3838
[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).
Exact mathematical passage
1-\sum_{i\in M}p_i(t) \leq \frac{1-p_M(0)}{p_M(0)}\e^{-\delta t},
\sum_jUse in source · 16 passages
Search aliases: ∑_j sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 752
The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.
Exact mathematical passage
\frac{\mu_i(\lambda_ip_i/Z_\lambda)}{\sum_j\mu_j\lambda_jp_j/Z_\lambda} = \frac{\mu_i\lambda_ip_i}{\sum_j\mu_j\lambda_jp_j}.
\sum_{j,k}Use in source · 1 passage
Search aliases: ∑_j,k sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1148
All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.
Exact mathematical passage
L^\alpha L^\beta =\sum_{j,k}\lambda_j^\alpha\lambda_k^\beta P_jP_k =\sum_j\lambda_j^{\alpha+\beta}P_j.
\sum_{j=0}^2Use in source · 1 passage
Search aliases: ∑_j=0^2 sum
Use in source · atlas-pages.tex, line 180
ck. vfigvariance platecaptionThe strip lengths are proportional to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes. headThe energy identity follows from actual orthogonality The tower identities give E_0E_1=E_1E_0=E_0. Together with weighted self-adjointness, these imply D_iD_j=0 (i≠ j), ∑_j=0^2 D_j=I. For the chosen f, the total weighted energy is f _π^2=13 and the mean is 16/5. Co…
Exact mathematical passage
D_iD_j=0\ (i\ne j),\qquad \sum_{j=0}^2 D_j=I.
\sum_{j=0}^{n-1}Use in source · 3 passages
Search aliases: ∑_j=0^n-1 sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 612
[Finite-horizon defect formula] For arbitrary Δ=CK-bar KC, CK^n-bar K^nC = ∑_j=0^n-1bar K^n-1-jΔ K^j. If K and bar K are contractions in the chosen norm, then normCK^n-bar K^nC≤ nnormΔ.
Exact mathematical passage
\boxed{ CK^n-\bar K^nC = \sum_{j=0}^{n-1}\bar K^{n-1-j}\Delta K^j. }
\sum_{j=0}^rUse in source · 4 passages
Search aliases: ∑_j=0^r sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 696
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
D_iD_j=\delta_{ij}D_j, \qquad P_r=\sum_{j=0}^rD_j.
\sum_{j=1}^{m}Use in source · 1 passage
Search aliases: ∑_j=1^m sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5759
[Exponential sum memory] If K(t)=∑_j=1^mB_j e^λ_jtC_j, then K has a finite auxiliary realization with one auxiliary block for each exponential. Repeated poles produce polynomial factors in t and are realized by Jordan blocks.
Exact mathematical passage
\Kcal(t)=\sum_{j=1}^{m}B_j\e^{\lambda_jt}C_j,
\sum_{j=1}^nUse in source · 1 passage
Search aliases: ∑_j=1^n sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4204
[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.
Exact mathematical passage
u_i^h = h\sum_{j=1}^nG(x_i,x_j)f_j^h.
\sum_{j=1}^rUse in source · 4 passages
Search aliases: ∑_j=1^r sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4657
xtended:thm:modal_memory Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the complete pri…
Exact mathematical passage
\Kcal(t) = \sum_{j=1}^r \e^{\mu_jt} \sum_{k=0}^{\nu_j-1} \frac{t^k}{k!}B_{j,k}
\sum_{j=n}^{\infty}Use in source · 2 passages
Search aliases: ∑_j=n^∞ sum infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2560
fine the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
\sup_{0\leq t\leq T} \norm{U_m(t)-U_{m,n}(t)} \leq \sum_{j=n}^{\infty}w_{m,j} \sup_{0\leq t\leq T}\norm{E_j(t)}.
\sum_{j=\ell}^{m-1}Use in source · 2 passages
Search aliases: ∑_j=ℓ^m-1 sum ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5252
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
\boxed{ C_{\ell,m}T_m-T_\ell C_{\ell,m} = \sum_{j=\ell}^{m-1} C_{\ell,j}E_jC_{j+1,m}. }
\sum_{j>k}Use in source · 1 passage
Search aliases: ∑_j>k sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4743
normB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.
Exact mathematical passage
\sum_{j>k}w_{(j)}\leq\varepsilon.
\sum_{j\in B_b}Use in source · 1 passage
Search aliases: ∑_j∈ B_b sum in
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5375
Stationarity follows from overline Qoverlineπ =overline QCπ =CQπ=0. For reversibility, use the aggregate rate formula: overline Q_aboverlineπ_b · =∑_j∈ B_bπ_j∑_i∈ B_aQ_ij · =∑_i∈ B_a∑_j∈ B_bπ_iQ_ji · =overline Q_baoverlineπ_a.
Exact mathematical passage
\begin{aligned} \overline Q_{ab}\overline\pi_b &=\sum_{j\in B_b}\pi_j\sum_{i\in B_a}Q_{ij}\\ &=\sum_{i\in B_a}\sum_{j\in B_b}\pi_iQ_{ji}\\ &=\overline Q_{ba}\overline\pi_a. \end{aligned}
\sum_{j\notin J}Use in source · 4 passages
Search aliases: ∑_j∉ J sum notin
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4674
j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the complete primary blocks indexed by J, then norm K- K_J_L^1(0,∞) ≤ ∑_j∉ J ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1.
Exact mathematical passage
\norm{\Kcal-\Kcal_J}_{L^1(0,\infty)} \leq \sum_{j\notin J} \sum_{k=0}^{\nu_j-1} \frac{\norm{B_{j,k}}} {(-\operatorname{Re}\mu_j)^{k+1}}.
\sum_{j\geq0}Use in source · 1 passage
Search aliases: ∑_j≥0 sum geq
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5283
[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.
Exact mathematical passage
\sum_{j\geq0}\delta_j<\infty
\sum_{k=0}^{d-1}Use in source · 1 passage
Search aliases: ∑_k=0^d-1 sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1657
For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. I…
Exact mathematical passage
\left.F(A)\right|_{E_\mu} =\left.\sum_{k=0}^{d-1}\frac{F^{(k)}(\mu)}{k!}(A-\mu I)^k\right|_{E_\mu}.
\sum_{k=0}^{\nu_j-1}Use in source · 7 passages
Search aliases: ∑_k=0^ν_j-1 sum nu
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4657
xtended:thm:modal_memory Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the complete pri…
Exact mathematical passage
\Kcal(t) = \sum_{j=1}^r \e^{\mu_jt} \sum_{k=0}^{\nu_j-1} \frac{t^k}{k!}B_{j,k}
\sum_{k=0}^{\infty}Use in source · 1 passage
Search aliases: ∑_k=0^∞ sum infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 4615
For large z, Σ_P(z) = ∑_k=0^∞ fracA_PQA_QQ^kA_QPz^k+1. Thus the second condition implies the moment identities. Conversely, the first d identities imply all later ones by the Cayley Hamilton theorem, so the series vanishes.
Exact mathematical passage
\Sigma_P(z) = \sum_{k=0}^{\infty} \frac{A_{PQ}A_{QQ}^kA_{QP}}{z^{k+1}}.
\sum_{m,n\ge0}Use in source · 1 passage
Search aliases: ∑_m,n≥0 sum ge
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 342
Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.
Exact mathematical passage
e^{sA}e^{tA} =\sum_{m,n\ge0}\frac{s^mt^n}{m!n!}A^{m+n} =\sum_{r\ge0}\frac{(s+t)^r}{r!}A^r.
\sum_{m=0}^\inftyUse in source · 1 passage
Search aliases: ∑_m=0^∞ sum infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2163
[Resolvent expansion over packet paths] With A=D+N as in the directed path decomposition, let R_D(z)=(zI-D)^-1. If normR_D(z)N<1, then (zI-A)^-1 = ∑_m=0^∞ (R_D(z)N)^mR_D(z). After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
(zI-A)^{-1} = \sum_{m=0}^\infty \bigl(R_D(z)N\bigr)^mR_D(z).
\sum_{m=0}^{\infty}Use in source · 1 passage
Search aliases: ∑_m=0^∞ sum infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5910
This section receives the operator-substitution result deliberately moved out of Paper IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=log D_N once and define S_N^(r)(z)=exp(-r g(z)) for complex r. The bounded Banach-space result below uses classical holomorphic functional calculus; it …
Exact mathematical passage
\Dfun_N(z)=\sum_{m=0}^{\infty}\frac{(-1)^m z^{Nm}}{(Nm)!}.
\sum_{m=1}^{\infty}Use in source · 1 passage
Search aliases: ∑_m=1^∞ sum infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2235
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
\sum_nUse in source · 1 passage
Search aliases: ∑_n sum
Use in source · atlas-pages.tex, line 621
at each level is not summable across levelsREFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVETraditional Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data. For each n≥1, compare the two scalar operators A_n=0 and B_n=1/n^2 through the identity. Their ordinary defects have sizes 1/n^2, while their square-root defects have sizes 1/n: ∑_n|A_n-B_n|<∞, ∑_n|A_n^1/2-B_n^1/2|=∞. These are independent scalar comparisons, not a constructed inverse system. They…
Exact mathematical passage
\sum_n|A_n-B_n|<\infty, \qquad \sum_n|A_n^{1/2}-B_n^{1/2}|=\infty.
\sum_{n=0}^\inftyUse in source · 6 passages
Search aliases: ∑_n=0^∞ sum infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3130
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
\exp\left(ux-\frac{u^2}{2}\right) = \sum_{n=0}^\infty H_n(x)\frac{u^n}{n!},
\sum_{n=m}^{\infty}Use in source · 1 passage
Search aliases: ∑_n=m^∞ sum infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2554
[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
\sum_{n=m}^{\infty} w_{m,n} \sup_{0\leq t\leq T}\norm{E_n(t)}<\infty,
\sum_{n\le N}Use in source · 3 passages
Search aliases: ∑_n≤ N sum le
Use in source · summability.tex, line 33
dotted,line width=.9pt] plot coordinates (1.33125,1.107143) (1.8625,1.221429) (2.39375,1.272222) (2.925,1.300794) (3.45625,1.319079) (3.9875,1.331778) (4.51875,1.341107) (5.05,1.34825) (5.58125,1.353894) (6.1125,1.358465) (6.64375,1.362243) (7.175,1.365418) (7.70625,1.368123) (8.2375,1.370455) (8.76875,1.372487) (9.3,1.374273); node[font=small,align=center,font=small] at (11.7,3.95) solid: ∑_n≤ Nn^-1; node[font=small,align=center,font=small] at (11.7,2.95) dashed: ∑_n≤ Nn^-2; node[font=small,ali…
Exact mathematical passage
\sum_{n\le N}n^{-1}
\sum_{n\ge0}Use in source · 1 passage
Search aliases: ∑_n≥0 sum ge
Use in source · atlas-pages.tex, line 95
For later discrete calculations define K=I+Q/8. This matrix is stochastic, but it is not e^Q/8. It is the step kernel in the uniformization identity e^tQ=e^-8t∑_n≥0(8t)^nK^n/n!. takeawayChanging packet coordinates does not change the evolution. Replacing continuous time by one discrete step does.
Exact mathematical passage
e^{tQ}=e^{-8t}\sum_{n\ge0}(8t)^nK^n/n!
\sum_{n\geq1}Use in source · 2 passages
Search aliases: ∑_n≥1 sum geq
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3286
A mean zero function has no h_0 component, so every surviving Hermite multiplier is at most e^-t. For a polynomial f- E_γ f=∑_n≥1c_nh_n. Then Var_γ(f)=∑_n≥1absc_n^2 ≤ ∑_n≥1nabsc_n^2 = ip- L ff. Apply [source reference].
Exact mathematical passage
f-\Exp_\gamma f=\sum_{n\geq1}c_nh_n.
\sum_{r=0}^{n-1}Use in source · 2 passages
Search aliases: ∑_r=0^n-1 sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1452
[Discrete Feynman--Kac path formula] For an initial state i, (u_n)_i = E_i[ exp(-∑_r=0^n-1V(X_r))f(X_n) ].
Exact mathematical passage
\boxed{ (u_n)_i = \Exp_i\left[ \exp\left(-\sum_{r=0}^{n-1}V(X_r)\right)f(X_n) \right]. }
\sum_{r=1}^qUse in source · 3 passages
Search aliases: ∑_r=1^q sum
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5038
[Competing risk packets] Suppose the exit row splits into nonnegative cause rows a^T = ∑_r=1^q a_r^T. Define f_r(t)=a_r^T e^tAp_0. Then f(t)=∑_r=1^qf_r(t). The cause specific hazard is h_r(t)=fracf_r(t)S(t), and h(t)=∑_r=1^qh_r(t). If the spectrum of A lies in the open left half-plane, the eventual probability of cause r is ∫_0^∞ f_r(t) d t = a_r^T(-A)^-1p_0.
Exact mathematical passage
a^T = \sum_{r=1}^q a_r^T.
\sum_{r\in B_\alpha}Use in source · 4 passages
Search aliases: ∑_r∈ B_α sum in alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2897
[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.
Exact mathematical passage
\Ob_C(K)(e_i-e_j) = \left( \sum_{r\in B_\alpha}(K_{ri}-K_{rj}) \right)_{\alpha=1}^m.
\sum_{r\ge0}Use in source · 1 passage
Search aliases: ∑_r≥0 sum ge
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 342
Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.
Exact mathematical passage
e^{sA}e^{tA} =\sum_{m,n\ge0}\frac{s^mt^n}{m!n!}A^{m+n} =\sum_{r\ge0}\frac{(s+t)^r}{r!}A^r.
\sum_\alphaUse in source · 2 passages
Search aliases: ∑_α sum alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 430
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
I=\sum_\alpha P_\alpha
\sum_{\alpha,\beta}Use in source · 1 passage
Search aliases: ∑_α,β sum alpha beta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 378
Let P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For any operator B, define B_βα=P_β BP_α. Then B=∑_α,βB_βα. For the semigroup, write T_βα(t)=P_β e^tAP_α.
Exact mathematical passage
B=\sum_{\alpha,\beta}B_{\beta\alpha}.
\sum_{\alpha,\beta\in I_0}Use in source · 1 passage
Search aliases: ∑_α,β∈ I_0 sum alpha beta in
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5191
X). The general Laplace formula is [source reference]. [Packet convolution] For all s,t≥0 and all packets α,β, T_βα(s+t) = ∑_γ∈ I_0 T_βγ(s)T_γα(t). The original semigroup is reconstructed from its blocks by T(t)=∑_α,β∈ I_0T_βα(t).
Exact mathematical passage
T(t)=\sum_{\alpha,\beta\in I_0}T_{\beta\alpha}(t).
\sum_{\alpha:n_\alpha>0}Use in source · 1 passage
Search aliases: ∑_α:n_α>0 sum alpha alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3654
on decomposition] Let p and q be probability vectors on a finite partition (I_α)_α, and assume q_i=0 whenever p_i=0. Let m=Cp, n=Cq, and let p_α and q_α be the conditional distributions inside packet I_α whenever the corresponding mass is positive. Then D(qVert p) = D(nVert m) + ∑_α:n_α>0 n_α D(q_αVert p_α). The first term is the information retained by the coarse masses. The second term is the exact information carried by changes inside the packets.
Exact mathematical passage
D(q\Vert p) = D(n\Vert m) + \sum_{\alpha:n_\alpha>0} n_\alpha D(q_\alpha\Vert p_\alpha).
\sum_{\alpha:Z_\alpha>0}Use in source · 1 passage
Search aliases: ∑_α:Z_α>0 sum alpha alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3384
[Bayesian packet reconstruction] The posterior is the disjoint packet synthesis B_λ(p) = ∑_α:Z_α>0 m_α q_α. The local shape q_α depends only on the prior and likelihood inside packet I_α. Coupling between different packets occurs through the normalization of the packet evidences.
Exact mathematical passage
\Bcal_\lambda(p) = \sum_{\alpha:Z_\alpha>0} m_\alpha q_\alpha.
\sum_{\alpha_1,\ldots,\alpha_{m-1}}Use in source · 1 passage
Search aliases: ∑_α_1,…,α_m-1 sum alpha ldots alpha
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 431
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
\sum_{\alpha\in I}Use in source · 3 passages
Search aliases: ∑_α∈ I sum alpha in
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 368
Let P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For any operator B, define B_βα=P_β BP_α. Then B=∑_α,βB_βα. For the semigroup, write T_βα(t)=P_β e^tAP_α.
Exact mathematical passage
P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I.
\sum_\betaUse in source · 5 passages
Search aliases: ∑_β sum beta
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 762
Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.
Exact mathematical passage
Z_\alpha=\sum_{i\in I_\alpha}\lambda_ip_i, \qquad m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta}, \qquad q_{\alpha,i}=\begin{cases} \lambda_ip_i/Z_\alpha,&Z_\alpha>0,\ i\in I_\alpha,\\ 0,&\text{otherwise}. \end{cases}
\sum_\gammaUse in source · 4 passages
Search aliases: ∑_γ sum gamma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 399
Insert the resolution of the identity between the two semigroup factors: P_β e^(s+t)AP_α · =P_β e^sA(∑_γ P_γ)e^tAP_α · =∑_γ (P_β e^sAP_γ)(P_γ e^tAP_α).
Exact mathematical passage
\begin{aligned} P_\beta e^{(s+t)A}P_\alpha &=P_\beta e^{sA}\left(\sum_\gamma P_\gamma\right)e^{tA}P_\alpha\\ &=\sum_\gamma (P_\beta e^{sA}P_\gamma)(P_\gamma e^{tA}P_\alpha). \end{aligned}
\sum_{\gamma\in I}Use in source · 3 passages
Search aliases: ∑_γ∈ I sum gamma in
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 388
[Packet semigroup convolution] For every s,t and every pair of packets α,β, T_βα(s+t) =∑_γ∈ IT_βγ(s)T_γα(t). Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.
Exact mathematical passage
\boxed{ T_{\beta\alpha}(s+t) =\sum_{\gamma\in I}T_{\beta\gamma}(s)T_{\gamma\alpha}(t). }
\sum_{\gamma\in I_0}Use in source · 1 passage
Search aliases: ∑_γ∈ I_0 sum gamma in
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5182
apply the integrand to each x∈ X and take its Bochner integral in X. It is not asserted to be a Bochner integral in the operator-norm space B(X). The general Laplace formula is [source reference]. [Packet convolution] For all s,t≥0 and all packets α,β, T_βα(s+t) = ∑_γ∈ I_0 T_βγ(s)T_γα(t). The original semigroup is reconstructed from its blocks by T(t)=∑_α,β∈ I_0T_βα(t).
Exact mathematical passage
\boxed{ T_{\beta\alpha}(s+t) = \sum_{\gamma\in I_0} T_{\beta\gamma}(s)T_{\gamma\alpha}(t). }
\sum_\ellUse in source · 2 passages
Search aliases: ∑_ℓ sum ell
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1329
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces; no operator sum across unrelated spaces…
Exact mathematical passage
\sum_\ell\norm{D_\ell}^{\alpha}<\infty,
\sum_{\ell=0}^{\infty}Use in source · 2 passages
Search aliases: ∑_ℓ=0^∞ sum ell infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2599
[Contractive summable defect completion] Assume normC_ℓ≤1, normR_ℓ≤1, normT_ℓ(t)≤1 for every ℓ and t≥0. If ∑_ℓ=0^∞ sup_0≤ t≤ TnormE_ℓ(t)<∞, then the conclusion of [source reference] holds, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_ℓ=n^∞ sup_0≤ t≤ TnormE_ℓ(t).
Exact mathematical passage
\sum_{\ell=0}^{\infty} \sup_{0\leq t\leq T}\norm{E_\ell(t)}<\infty,
\sum_{\ell=n}^{\infty}Use in source · 2 passages
Search aliases: ∑_ℓ=n^∞ sum ell infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2604
ontractive summable defect completion] Assume normC_ℓ≤1, normR_ℓ≤1, normT_ℓ(t)≤1 for every ℓ and t≥0. If ∑_ℓ=0^∞ sup_0≤ t≤ TnormE_ℓ(t)<∞, then the conclusion of [source reference] holds, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_ℓ=n^∞ sup_0≤ t≤ TnormE_ℓ(t).
Exact mathematical passage
\sup_{0\leq t\leq T} \norm{U_m(t)-U_{m,n}(t)} \leq \sum_{\ell=n}^{\infty} \sup_{0\leq t\leq T}\norm{E_\ell(t)}.
\circUse in source · 5 passages
Search aliases: ∘ circ
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1040
[Bayesian nonlinear semigroup] Φ_s+t=Φ_s∘Φ_t. Its generator is the centered multiplicative field fracddtp_i(t) = (g_i-⟨ g,p(t)⟩)p_i(t).
Exact mathematical passage
\boxed{ \Phi_{s+t}=\Phi_s\circ\Phi_t. }
\intUse in source · 75 passages
Search aliases: ∫ int
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 462
[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.
Exact mathematical passage
\boxed{ Ce^{tA_f}-e^{tA_c}C = \int_0^t e^{(t-r)A_c}\Delta e^{rA_f}\,dr. }
\int_{0<s_1<\cdots<s_m<t}Use in source · 1 passage
Search aliases: ∫_0<s_1<⋯<s_m<t int cdots
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2235
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
\int_0^1Use in source · 3 passages
Search aliases: ∫_0^1 int
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1362
On the interval (0,1) with Dirichlet boundary conditions, the Green kernel for -u''=f is G(x,y)=min(x,y)-xy. Thus the point source δ_y generates the solution kernel, and a general source is reconstructed by u(x)=∫_0^1G(x,y)f(y) dy.
Exact mathematical passage
u(x)=\int_0^1G(x,y)f(y)\,dy.
\int_0^rUse in source · 1 passage
Search aliases: ∫_0^r int
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1264
For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D^α. The zero cases are immediate.
Exact mathematical passage
\begin{aligned} \norm{D_\alpha} &\le \frac{\sin(\pi\alpha)}{\pi} \left( 2\norm C\int_0^r t^{\alpha-1}\,dt + \norm D\int_r^\infty t^{\alpha-2}\,dt \right)\\ &= K_\alpha\norm C^{1-\alpha}\norm D^\alpha. \end{aligned}
\int_0^sUse in source · 2 passages
Search aliases: ∫_0^s int
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1296
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ Ce^{-sA^\alpha}-e^{-sB^\alpha}C = -\int_0^s e^{-(s-r)B^\alpha}D_\alpha e^{-rA^\alpha}\,dr. }
\int_0^tUse in source · 27 passages
Search aliases: ∫_0^t int
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 462
[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.
Exact mathematical passage
\boxed{ Ce^{tA_f}-e^{tA_c}C = \int_0^t e^{(t-r)A_c}\Delta e^{rA_f}\,dr. }
\int_0^\inftyUse in source · 32 passages
Search aliases: ∫_0^∞ int infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1166
[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.
Exact mathematical passage
\boxed{ e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)}e^{-sL}\,ds. }
\int_0^{\infty}Use in source · 2 passages
Search aliases: ∫_0^∞ int infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 5211
[Packet resolvent convolution] Let z belong to the resolvent set of A. If Rez is larger than a growth bound for the semigroup, then P_β(zI-A)^-1P_α = ∫_0^∞ e^-ztT_βα(t) d t.
Exact mathematical passage
P_\beta(zI-A)^{-1}P_\alpha = \int_0^{\infty}\e^{-zt}T_{\beta\alpha}(t)\dd t.
\int_r^\inftyUse in source · 1 passage
Search aliases: ∫_r^∞ int infty
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1264
For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D^α. The zero cases are immediate.
Exact mathematical passage
\begin{aligned} \norm{D_\alpha} &\le \frac{\sin(\pi\alpha)}{\pi} \left( 2\norm C\int_0^r t^{\alpha-1}\,dt + \norm D\int_r^\infty t^{\alpha-2}\,dt \right)\\ &= K_\alpha\norm C^{1-\alpha}\norm D^\alpha. \end{aligned}
\int_\RRUse in source · 4 passages
Search aliases: ∫_RR ∫_ R int RR int mathbb
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3120
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
(\Ocal_t f)(x) = \int_\RR f\left( \e^{-t}x+\sqrt{1-\e^{-2t}}\,y \right) \gamma(\dd y).
\int_\GammaUse in source · 3 passages
Search aliases: ∫_Γ int Gamma
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2379
elextended:thm:holomorphic_defect Let Γ be the positively oriented rectifiable boundary of a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that Γ lies in the common resolvent set. Thus its winding number is one about every spectral point. If g is holomorphic on a neighborhood of the closure of the union, then Cg(A_f)-g(A_c)C = frac12π i ∫_Γ g(z)R_c(z)Δ R_f(z) d z. In particular, normCg(A_f)-g(A_c)C ≤ · fraclength(Γ)2π sup_z∈Γabsg(z) …
Exact mathematical passage
Cg(A_f)-g(A_c)C = \frac{1}{2\pi\ii} \int_\Gamma g(z)R_c(z)\Delta R_f(z) \dd z.
\simUse in source · 2 passages
Search aliases: ∼ sim
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 989
[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its evidence is positive. If all likelihoods are strictly positive and there are at least two profile classes, this is also the unique coarsest nontrivia…
Exact mathematical passage
i\sim j \quad\Longleftrightarrow\quad \lambda_i^{(y)}=\lambda_j^{(y)} \text{ for every }y\in Y.
\sim_{\mathcal L}Use in source · 1 passage
Search aliases: ∼_ L sim mathcal
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3544
[Coarsest fixed sufficient partition] Let L be a nonempty family of strictly positive likelihood vectors. Define i∼_ Lj Longleftrightarrow λ_i=λ_j hboxfor every λ∈ L. Let Q_ L be the partition into equivalence classes. A partition Q gives a fixed coarse sufficient description in the sense of [source reference] for every λ∈ L if and only if every block of Q is contained in a block of Q_ L. Hence Q_ L is the unique coarsest partition with this property.
Exact mathematical passage
i\sim_{\mathcal L}j \quad\Longleftrightarrow\quad \lambda_i=\lambda_j \quad\hbox{for every }\lambda\in\mathcal L.
\neUse in source · 7 passages
Search aliases: ≠ ne
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 410
[Reducing packets] If P_β AP_α=0 for every β≠α, then P_β e^tAP_α=0 (β≠α) for every t. Each packet evolves independently under P_α AP_α.
Exact mathematical passage
\beta\ne\alpha
\neqUse in source · 7 passages
Search aliases: ≠ neq
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 2184
[Reducing packet criterion] The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.
Exact mathematical passage
\beta\neq\alpha
\equivUse in source · 1 passage
Search aliases: ≡ equiv
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 578
[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.
Exact mathematical passage
k\equiv n\pmod2
\leUse in source · 33 passages
Search aliases: ≤ le
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 486
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{e^{tA_f}}\le M_f e^{\omega_f t}, \qquad \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
\geUse in source · 36 passages
Search aliases: ≥ ge
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 299
Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usually written with e^tA; positive diffusion and elliptic operators are usually written with e^-tL.
Exact mathematical passage
T(t)=e^{tA},\qquad t\ge 0,
\subsetUse in source · 1 passage
Search aliases: ⊂ subset
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1665
.F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ)⊂ U, and Cauchy's formula gives fracF^(k)(μ)k! =frac12π ioint_|z-μ|=r_μ fracF(z)(z-μ)^k+1 dz. Consequently the same canonical operator is obtained fr…
Exact mathematical passage
\overline{D(\mu,r_\mu)}\subset U
\subseteqUse in source · 2 passages
Search aliases: ⊆ subseteq
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 3052
[Tower law and martingale packets] Let P_0,P_1,…,P_r come from increasingly fine partitions, so that Ran P_0⊆Ran P_1⊆⋯⊆Ran P_r. Define D_0=P_0, D_j=P_j-P_j-1 (1≤ j≤ r). Then P_iP_j=P_min(i,j), D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j. Every D_j is an orthogonal projector, and normP_rf_π^2 = ∑_j=0^rnormD_jf_π^2 and norm f_π^2 = ∑_j=0^rnormD_jf_π^2 + normf-P_rf_π^2.
Exact mathematical passage
\Ran P_0\subseteq\Ran P_1\subseteq\cdots\subseteq\Ran P_r.
\oplusUse in source · 2 passages
Search aliases: ⊕ oplus
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1485
Let X=P X⊕ Q X and write A= A_PP · A_PQ
Exact mathematical passage
X=P X\oplus Q X
\cdotsUse in source · 12 passages
Search aliases: ⋯ cdots
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 431
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
\langleUse in source · 6 passages
Search aliases: ⟨ langle
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 658
[Finite conditional-expectation projector] With the weighted inner product ⟨ f,g⟩_π=∑_iπ_if_ioverlineg_i, the operator P_ Q satisfies P_ Q^2=P_ Q, ⟨ P_ Qf,g⟩_π = ⟨ f,P_ Qg⟩_π. Its range is the space of functions constant on each partition block.
Exact mathematical passage
\langle f,g\rangle_\pi=\sum_i\pi_if_i\overline{g_i},
\rangleUse in source · 6 passages
Search aliases: ⟩ rangle
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 658
[Finite conditional-expectation projector] With the weighted inner product ⟨ f,g⟩_π=∑_iπ_if_ioverlineg_i, the operator P_ Q satisfies P_ Q^2=P_ Q, ⟨ P_ Qf,g⟩_π = ⟨ f,P_ Qg⟩_π. Its range is the space of functions constant on each partition block.
Exact mathematical passage
\langle f,g\rangle_\pi=\sum_i\pi_if_i\overline{g_i},
\rangle_\piUse in source · 2 passages
Search aliases: ⟩_π rangle pi
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 658
[Finite conditional-expectation projector] With the weighted inner product ⟨ f,g⟩_π=∑_iπ_if_ioverlineg_i, the operator P_ Q satisfies P_ Q^2=P_ Q, ⟨ P_ Qf,g⟩_π = ⟨ f,P_ Qg⟩_π. Its range is the space of functions constant on each partition block.
Exact mathematical passage
\langle f,g\rangle_\pi=\sum_i\pi_if_i\overline{g_i},
\longrightarrowUse in source · 5 passages
Search aliases: ⟶ longrightarrow
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 309
The common operator chain is seed⟶transfer⟶project ⟶condition⟶refine ⟶complete⟶reconstruct.
Exact mathematical passage
\boxed{ \text{seed}\longrightarrow\text{transfer}\longrightarrow\text{project} \longrightarrow\text{condition}\longrightarrow\text{refine} \longrightarrow\text{complete}\longrightarrow\text{reconstruct}. }
\longmapstoUse in source · 2 passages
Search aliases: ⟼ longmapsto
Use in source · Read passage · Paper_VI_Traditional_Final.tex, line 1321
Differentiate r⟼ e^-(s-r)B^αCe^-rA^α and integrate from 0 to s. Positive fractional semigroups are contractions; under the gap, the product of their two norm bounds is e^-sm^α.
Exact mathematical passage
r\longmapsto e^{-(s-r)B^\alpha}Ce^{-rA^\alpha}
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