Notation by work
Paper II: Refinement Towers, Weighted Transfer, and Integral Separation of Operator Packets
All notation · Glossary · Source coverage
674 symbol spellings in 2177 mathematical passages across 36 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 674 entries are available below.
+Notation table · 127 passages
Search aliases: +
Notation table · Read passage · 06_models.tex, line 182
J_ℓ+1,ℓ · exact coordinate refinement map
Exact mathematical passage
J_{\ell+1,\ell}
ANotation table · 114 passages
Search aliases: A
Notation table · Read passage · 06_models.tex, line 188
A(t) · polynomial weighted adjacency matrix
Exact mathematical passage
A(t)
CNotation table · 55 passages
Search aliases: C
Notation table · Read passage · 06_models.tex, line 183
C_ℓ,ℓ+1 · exact coordinate regrouping from fine to coarse
Exact mathematical passage
C_{\ell,\ell+1}
cNotation table · 50 passages
Search aliases: c
Notation table · Read passage · 06_models.tex, line 191
Δ · linear refinement defect for one step CT_f-T_cC
Exact mathematical passage
CT_f-T_cC
C_{\ell,\ell+1}Notation table · 1 passage
Search aliases: C_ℓ,ℓ+1 ell ell
Notation table · Read passage · 06_models.tex, line 183
C_ℓ,ℓ+1 · exact coordinate regrouping from fine to coarse
Exact mathematical passage
C_{\ell,\ell+1}
\cANotation table · 2 passages
Search aliases: cA A mathcal
Notation table · Read passage · 06_models.tex, line 181
A_ℓ, S_ℓ · packet analysis and synthesis maps
Exact mathematical passage
\cA_\ell,\cS_\ell
\cA_\ellNotation table · 1 passage
Search aliases: cA_ℓ A_ℓ cA ell mathcal ell
Notation table · Read passage · 06_models.tex, line 181
A_ℓ, S_ℓ · packet analysis and synthesis maps
Exact mathematical passage
\cA_\ell,\cS_\ell
\cSNotation table · 2 passages
Search aliases: cS S mathcal
Notation table · Read passage · 06_models.tex, line 181
A_ℓ, S_ℓ · packet analysis and synthesis maps
Exact mathematical passage
\cA_\ell,\cS_\ell
\cS_\ellNotation table · 1 passage
Search aliases: cS_ℓ S_ℓ cS ell mathcal ell
Notation table · Read passage · 06_models.tex, line 181
A_ℓ, S_ℓ · packet analysis and synthesis maps
Exact mathematical passage
\cA_\ell,\cS_\ell
ENotation table · 111 passages
Search aliases: E
Notation table · Read passage · 06_models.tex, line 190
s_π,Λ(E) · separation weight of a rational projector on the chosen lattice, when a clearing exponent exists
Exact mathematical passage
s_{\pi,\Lambda}(E)
eNotation table · 55 passages
Search aliases: e
Notation table · Read passage · 06_models.tex, line 186
w(e) · positive filtration weight of a transfer edge
Exact mathematical passage
w(e)
fNotation table · 77 passages
Search aliases: f
Notation table · Read passage · 06_models.tex, line 191
Δ · linear refinement defect for one step CT_f-T_cC
Exact mathematical passage
CT_f-T_cC
INotation table · 86 passages
Search aliases: I
Notation table · Read passage · 06_models.tex, line 179
I_ℓ · packet address set at level ℓ
Exact mathematical passage
I_\ell
I_\ellNotation table · 3 passages
Search aliases: I_ℓ ell
Notation table · Read passage · 06_models.tex, line 179
I_ℓ · packet address set at level ℓ
Exact mathematical passage
I_\ell
JNotation table · 18 passages
Search aliases: J
Notation table · Read passage · 06_models.tex, line 182
J_ℓ+1,ℓ · exact coordinate refinement map
Exact mathematical passage
J_{\ell+1,\ell}
jNotation table · 59 passages
Search aliases: j
Notation table · Read passage · 06_models.tex, line 185
P_N,j · residue or phase selector for address jpmod N
Exact mathematical passage
P_{N,j}
J_{\ell+1,\ell}Notation table · 1 passage
Search aliases: J_ℓ+1,ℓ ell ell
Notation table · Read passage · 06_models.tex, line 182
J_ℓ+1,ℓ · exact coordinate refinement map
Exact mathematical passage
J_{\ell+1,\ell}
NNotation table · 140 passages
Search aliases: N
Notation table · Read passage · 06_models.tex, line 185
P_N,j · residue or phase selector for address jpmod N
Exact mathematical passage
P_{N,j}
PNotation table · 92 passages
Search aliases: P
Notation table · Read passage · 06_models.tex, line 178
P_ℓ,α · packet projector at refinement level ℓ
Exact mathematical passage
P_{\ell,\alpha}
P_{N,j}Notation table · 16 passages
Search aliases: P_N,j
Notation table · Read passage · 06_models.tex, line 185
P_N,j · residue or phase selector for address jpmod N
Exact mathematical passage
P_{N,j}
P_{\ell,\alpha}Notation table · 2 passages
Search aliases: P_ℓ,α ell alpha
Notation table · Read passage · 06_models.tex, line 178
P_ℓ,α · packet projector at refinement level ℓ
Exact mathematical passage
P_{\ell,\alpha}
P_{\ell,\beta}Notation table · 1 passage
Search aliases: P_ℓ,β ell beta
Notation table · Read passage · 06_models.tex, line 184
T^(ℓ)_βα · transfer block P_ℓ,βTP_ℓ,α
Exact mathematical passage
P_{\ell,\beta}TP_{\ell,\alpha}
\pmodNotation table · 31 passages
Search aliases: pmod
Notation table · Read passage · 06_models.tex, line 185
P_N,j · residue or phase selector for address jpmod N
Exact mathematical passage
j\pmod N
sNotation table · 118 passages
Search aliases: s
Notation table · Read passage · 06_models.tex, line 189
Z_s,t(t) · weighted transfer-zeta function from vertex s to vertex t
Exact mathematical passage
Z_{s,t}(t)
s_{\pi,\Lambda}Notation table · 3 passages
Search aliases: s_π,Λ pi Lambda
Notation table · Read passage · 06_models.tex, line 190
s_π,Λ(E) · separation weight of a rational projector on the chosen lattice, when a clearing exponent exists
Exact mathematical passage
s_{\pi,\Lambda}(E)
\succNotation table · 2 passages
Search aliases: succ
Notation table · Read passage · 06_models.tex, line 180
βsuccα · fine packet β is a child of coarse packet α
Exact mathematical passage
\beta\succ\alpha
TNotation table · 99 passages
Search aliases: T
Notation table · Read passage · 06_models.tex, line 184
T^(ℓ)_βα · transfer block P_ℓ,βTP_ℓ,α
Exact mathematical passage
T^{(\ell)}_{\beta\alpha}
tNotation table · 55 passages
Search aliases: t
Notation table · Read passage · 06_models.tex, line 188
A(t) · polynomial weighted adjacency matrix
Exact mathematical passage
A(t)
T^{(\ell)}_{\beta\alpha}Notation table · 1 passage
Search aliases: T^(ℓ)_βα ell beta alpha
Notation table · Read passage · 06_models.tex, line 184
T^(ℓ)_βα · transfer block P_ℓ,βTP_ℓ,α
Exact mathematical passage
T^{(\ell)}_{\beta\alpha}
T_cNotation table · 29 passages
Search aliases: T_c
Notation table · Read passage · 06_models.tex, line 191
Δ · linear refinement defect for one step CT_f-T_cC
Exact mathematical passage
CT_f-T_cC
T_fNotation table · 30 passages
Search aliases: T_f
Notation table · Read passage · 06_models.tex, line 191
Δ · linear refinement defect for one step CT_f-T_cC
Exact mathematical passage
CT_f-T_cC
WNotation table · 21 passages
Search aliases: W
Notation table · Read passage · 06_models.tex, line 187
W(γ) · cumulative weight of a transfer word or graph walk
Exact mathematical passage
W(\gamma)
wNotation table · 27 passages
Search aliases: w
Notation table · Read passage · 06_models.tex, line 186
w(e) · positive filtration weight of a transfer edge
Exact mathematical passage
w(e)
ZNotation table · 24 passages
Search aliases: Z
Notation table · Read passage · 06_models.tex, line 189
Z_s,t(t) · weighted transfer-zeta function from vertex s to vertex t
Exact mathematical passage
Z_{s,t}(t)
Z_{s,t}Notation table · 4 passages
Search aliases: Z_s,t
Notation table · Read passage · 06_models.tex, line 189
Z_s,t(t) · weighted transfer-zeta function from vertex s to vertex t
Exact mathematical passage
Z_{s,t}(t)
\alphaNotation table · 95 passages
Search aliases: α alpha
Notation table · Read passage · 06_models.tex, line 178
P_ℓ,α · packet projector at refinement level ℓ
Exact mathematical passage
P_{\ell,\alpha}
\betaNotation table · 62 passages
Search aliases: β beta
Notation table · Read passage · 06_models.tex, line 180
βsuccα · fine packet β is a child of coarse packet α
Exact mathematical passage
\beta\succ\alpha
\gammaNotation table · 34 passages
Search aliases: γ gamma
Notation table · Read passage · 06_models.tex, line 187
W(γ) · cumulative weight of a transfer word or graph walk
Exact mathematical passage
W(\gamma)
\DeltaNotation table · 28 passages
Search aliases: Δ Delta
Notation table · Read passage · 06_models.tex, line 191
Δ · linear refinement defect for one step CT_f-T_cC
Exact mathematical passage
\Delta
\LambdaNotation table · 103 passages
Search aliases: Λ Lambda
Notation table · Read passage · 06_models.tex, line 190
s_π,Λ(E) · separation weight of a rational projector on the chosen lattice, when a clearing exponent exists
Exact mathematical passage
s_{\pi,\Lambda}(E)
\piNotation table · 80 passages
Search aliases: π pi
Notation table · Read passage · 06_models.tex, line 190
s_π,Λ(E) · separation weight of a rational projector on the chosen lattice, when a clearing exponent exists
Exact mathematical passage
s_{\pi,\Lambda}(E)
\ellNotation table · 32 passages
Search aliases: ℓ ell
Notation table · Read passage · 06_models.tex, line 178
P_ℓ,α · packet projector at refinement level ℓ
Exact mathematical passage
P_{\ell,\alpha}
=Definition passage · 505 passages
Search aliases: =
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 7
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Pcal=(P_\alpha)_{\alpha\in I} \subseteq\End(X)
aDefinition passage · 87 passages
Search aliases: a
Definition passage: Residue selector · Read passage · 02_coordinates.tex, line 370
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
F(z)=\sum_{n\geq0}a_nz^n\in R[[z]].
a_nDefinition passage · 18 passages
Search aliases: a_n
Definition passage: Residue selector · Read passage · 02_coordinates.tex, line 370
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
F(z)=\sum_{n\geq0}a_nz^n\in R[[z]].
\AcalDefinition passage · 7 passages
Search aliases: Acal A mathcal
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 25
ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Acal_{\Pcal}x=(P_\alpha x)_\alpha, \qquad \Scal_{\Pcal}(x_\alpha)_\alpha=\sum_\alpha x_\alpha.
\Acal_{\Pcal}Definition passage · 4 passages
Search aliases: Acal_Pcal A_ P Acal Pcal mathcal mathcal
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 25
ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Acal_{\Pcal}x=(P_\alpha x)_\alpha, \qquad \Scal_{\Pcal}(x_\alpha)_\alpha=\sum_\alpha x_\alpha.
\Acal_{\Qcal}Definition passage · 4 passages
Search aliases: Acal_Qcal A_ Q Acal Qcal mathcal mathcal
Definition passage: Coordinate change across levels · Read passage · 02_coordinates.tex, line 84
[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.
Exact mathematical passage
J_{\Qcal\leftarrow\Pcal} = \Acal_{\Qcal}\Scal_{\Pcal}:X^{\Pcal}\longrightarrow X^{\Qcal}
\bbkDefinition passage · 7 passages
Search aliases: bbk
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 373
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
\bbk
\BcalDefinition passage · 34 passages
Search aliases: Bcal B mathcal
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 6
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal
C_{\Pcal\leftarrow\Qcal}Definition passage · 6 passages
Search aliases: C_Pcal←Qcal C_ P← Q Pcal leftarrow Qcal mathcal leftarrow mathcal
Definition passage: Coordinate change across levels · Read passage · 02_coordinates.tex, line 90
[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.
Exact mathematical passage
C_{\Pcal\leftarrow\Qcal} = \Acal_{\Pcal}\Scal_{\Qcal}:X^{\Qcal}\longrightarrow X^{\Pcal}.
e_1Definition passage · 13 passages
Search aliases: e_1
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 390
W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.
Exact mathematical passage
\gamma=(e_1,\ldots,e_{|\gamma|})
e_{|\gamma|}Definition passage · 1 passage
Search aliases: e_|γ| gamma
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 390
W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.
Exact mathematical passage
\gamma=(e_1,\ldots,e_{|\gamma|})
E_\alphaDefinition passage · 13 passages
Search aliases: E_α alpha
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 239
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
E_\alpha
e_\nuDefinition passage · 1 passage
Search aliases: e_ν nu
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\EndDefinition passage · 10 passages
Search aliases: End operatorname
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 7
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Pcal=(P_\alpha)_{\alpha\in I} \subseteq\End(X)
\expnDefinition passage · 4 passages
Search aliases: expn exp operatorname
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 261
mbda_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
\expn_\pi(G)
\expn_\piDefinition passage · 4 passages
Search aliases: expn_π exp_π expn pi operatorname pi
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 261
mbda_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
\expn_\pi(G)
FDefinition passage · 57 passages
Search aliases: F
Definition passage: Residue selector · Read passage · 02_coordinates.tex, line 370
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
F(z)=\sum_{n\geq0}a_nz^n\in R[[z]].
F^0Definition passage · 5 passages
Search aliases: F^0
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 7
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal=F^0\Bcal\supseteq F^1\Bcal\supseteq F^2\Bcal\supseteq\cdots
F^1Definition passage · 2 passages
Search aliases: F^1
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 7
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal=F^0\Bcal\supseteq F^1\Bcal\supseteq F^2\Bcal\supseteq\cdots
F^2Definition passage · 1 passage
Search aliases: F^2
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 7
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal=F^0\Bcal\supseteq F^1\Bcal\supseteq F^2\Bcal\supseteq\cdots
F^LDefinition passage · 16 passages
Search aliases: F^L
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
F^rDefinition passage · 3 passages
Search aliases: F^r
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
F^{r+s}Definition passage · 1 passage
Search aliases: F^r+s
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
F^sDefinition passage · 1 passage
Search aliases: F^s
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
F^{w(e)}Definition passage · 5 passages
Search aliases: F^w(e)
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 21
with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
N_e\in F^{w(e)}\Bcal.
GDefinition passage · 38 passages
Search aliases: G
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 373
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
G
gDefinition passage · 25 passages
Search aliases: g
Definition passage: Separation weight · Read passage · 04_arithmetic.tex, line 186
[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero s…
Exact mathematical passage
v,g
G_{\Lambda,\Pcal}Definition passage · 12 passages
Search aliases: G_Λ,Pcal G_Λ, P Lambda Pcal Lambda mathcal
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 246
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
G_{\Lambda,\Pcal} = \Lambda/\Lambda_{\mathrm{split}}.
iDefinition passage · 46 passages
Search aliases: i
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 374
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
A(t)_{ij} = \sum_{e:i\to j}m(e)t^{w(e)}.
I_XDefinition passage · 6 passages
Search aliases: I_X
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 12
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
P_\alpha P_\beta=0 \quad\text{when }\alpha\ne\beta, \qquad \sum_{\alpha\in I}P_\alpha=I_X.
J_{\Qcal\leftarrow\Pcal}Definition passage · 6 passages
Search aliases: J_Qcal←Pcal J_ Q← P Qcal leftarrow Pcal mathcal leftarrow mathcal
Definition passage: Coordinate change across levels · Read passage · 02_coordinates.tex, line 84
[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.
Exact mathematical passage
J_{\Qcal\leftarrow\Pcal} = \Acal_{\Qcal}\Scal_{\Pcal}:X^{\Pcal}\longrightarrow X^{\Qcal}
KDefinition passage · 23 passages
Search aliases: K
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 239
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
K
LDefinition passage · 111 passages
Search aliases: L
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
\leadstoDefinition passage · 2 passages
Search aliases: leadsto
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
mDefinition passage · 59 passages
Search aliases: m
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 373
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
m(e)
\maxDefinition passage · 4 passages
Search aliases: max
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 258
et Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
s_\Lambda(\Pcal)=\max\bigl(\{0\}\cup\{s_\Lambda(E_\alpha):\alpha\in I\}\bigr).
\minDefinition passage · 14 passages
Search aliases: min
Definition passage: Separation weight · Read passage · 04_arithmetic.tex, line 181
[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero s…
Exact mathematical passage
s_{\pi,\Lambda}(E) = \min\{s\ge0:\pi^sE(\Lambda)\subseteq\Lambda\}.
nDefinition passage · 80 passages
Search aliases: n
Definition passage: Residue selector · Read passage · 02_coordinates.tex, line 370
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
F(z)=\sum_{n\geq0}a_nz^n\in R[[z]].
N_eDefinition passage · 7 passages
Search aliases: N_e
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 20
ef:filtered_algebra A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
N_e
\OODefinition passage · 55 passages
Search aliases: OO O mathcal
Definition passage: Separation weight · Read passage · 04_arithmetic.tex, line 179
[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero s…
Exact mathematical passage
\OO
P_\alphaDefinition passage · 37 passages
Search aliases: P_α alpha
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 7
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Pcal=(P_\alpha)_{\alpha\in I} \subseteq\End(X)
P_\betaDefinition passage · 9 passages
Search aliases: P_β beta
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 12
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
P_\alpha P_\beta=0 \quad\text{when }\alpha\ne\beta, \qquad \sum_{\alpha\in I}P_\alpha=I_X.
\PcalDefinition passage · 56 passages
Search aliases: Pcal P mathcal
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 7
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Pcal=(P_\alpha)_{\alpha\in I} \subseteq\End(X)
QDefinition passage · 55 passages
Search aliases: Q
Definition passage: Refinement · Read passage · 02_coordinates.tex, line 34
[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).
Exact mathematical passage
\Qcal=(Q_\beta)_{\beta\in J}
Q_\betaDefinition passage · 27 passages
Search aliases: Q_β beta
Definition passage: Refinement · Read passage · 02_coordinates.tex, line 34
[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).
Exact mathematical passage
\Qcal=(Q_\beta)_{\beta\in J}
\QcalDefinition passage · 32 passages
Search aliases: Qcal Q mathcal
Definition passage: Refinement · Read passage · 02_coordinates.tex, line 34
[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).
Exact mathematical passage
\Qcal=(Q_\beta)_{\beta\in J}
RDefinition passage · 42 passages
Search aliases: R
Definition passage: Residue selector · Read passage · 02_coordinates.tex, line 369
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
R
rDefinition passage · 98 passages
Search aliases: r
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
s_\LambdaDefinition passage · 11 passages
Search aliases: s_Λ Lambda
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 252
et V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
s_\Lambda(E_\alpha) = \min\{s\geq0:\pi^sE_\alpha\Lambda\subseteq\Lambda\}.
\ScalDefinition passage · 7 passages
Search aliases: Scal S mathcal
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 25
ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Acal_{\Pcal}x=(P_\alpha x)_\alpha, \qquad \Scal_{\Pcal}(x_\alpha)_\alpha=\sum_\alpha x_\alpha.
\Scal_{\Pcal}Definition passage · 4 passages
Search aliases: Scal_Pcal S_ P Scal Pcal mathcal mathcal
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 25
ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Acal_{\Pcal}x=(P_\alpha x)_\alpha, \qquad \Scal_{\Pcal}(x_\alpha)_\alpha=\sum_\alpha x_\alpha.
\Scal_{\Qcal}Definition passage · 4 passages
Search aliases: Scal_Qcal S_ Q Scal Qcal mathcal mathcal
Definition passage: Coordinate change across levels · Read passage · 02_coordinates.tex, line 90
[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.
Exact mathematical passage
C_{\Pcal\leftarrow\Qcal} = \Acal_{\Pcal}\Scal_{\Qcal}:X^{\Qcal}\longrightarrow X^{\Pcal}.
\mathrm{split}Definition passage · 7 passages
Search aliases: split mathrm
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 240
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
\Lambda_\alpha=\Lambda\cap V_\alpha, \qquad \Lambda_{\mathrm{split}}=\bigoplus_\alpha\Lambda_\alpha,
t^nDefinition passage · 3 passages
Search aliases: t^n
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 380
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
Z_{u,v}(t) = \sum_{n\geq0}a_nt^n,
t^{w(e)}Definition passage · 3 passages
Search aliases: t^w(e)
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 374
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
A(t)_{ij} = \sum_{e:i\to j}m(e)t^{w(e)}.
uDefinition passage · 20 passages
Search aliases: u
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 379
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
u
u^{-1}Definition passage · 4 passages
Search aliases: u^-1
Definition passage: Separation weight · Read passage · 04_arithmetic.tex, line 187
[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero s…
Exact mathematical passage
u^{-1}
u_iDefinition passage · 2 passages
Search aliases: u_i
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
VDefinition passage · 45 passages
Search aliases: V
Definition passage: Separation weight · Read passage · 04_arithmetic.tex, line 193
rojector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero scalar in the fraction field need not…
Exact mathematical passage
\boxed{\pi^sE=u^{-1}VG}
vDefinition passage · 25 passages
Search aliases: v
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 379
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
v
v_jDefinition passage · 1 passage
Search aliases: v_j
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
V_\alphaDefinition passage · 7 passages
Search aliases: V_α alpha
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 236
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
V=\bigoplus_{\alpha\in I}V_\alpha
\valDefinition passage · 2 passages
Search aliases: val operatorname
Definition passage: Separation weight · Read passage · 04_arithmetic.tex, line 189
ight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero scalar in the f…
Exact mathematical passage
s_{\pi,\Lambda}(E)\le s=\val_\pi(r).
\val_\piDefinition passage · 2 passages
Search aliases: val_π val pi operatorname pi
Definition passage: Separation weight · Read passage · 04_arithmetic.tex, line 189
ight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero scalar in the f…
Exact mathematical passage
s_{\pi,\Lambda}(E)\le s=\val_\pi(r).
\varprojlimDefinition passage · 1 passage
Search aliases: varprojlim
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 17
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal\longrightarrow\varprojlim_L\Bcal/F^L\Bcal
\varprojlim_LDefinition passage · 1 passage
Search aliases: varprojlim_L varprojlim
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 17
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal\longrightarrow\varprojlim_L\Bcal/F^L\Bcal
XDefinition passage · 50 passages
Search aliases: X
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 6
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
X
xDefinition passage · 120 passages
Search aliases: x
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 25
ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Acal_{\Pcal}x=(P_\alpha x)_\alpha, \qquad \Scal_{\Pcal}(x_\alpha)_\alpha=\sum_\alpha x_\alpha.
X^{\Pcal}Definition passage · 4 passages
Search aliases: X^Pcal X^ P Pcal mathcal
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 19
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
X_\alpha=P_\alpha X, \qquad X^{\Pcal}=\bigoplus_{\alpha\in I}X_\alpha.
X^{\Qcal}Definition passage · 3 passages
Search aliases: X^Qcal X^ Q Qcal mathcal
Definition passage: Coordinate change across levels · Read passage · 02_coordinates.tex, line 84
[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.
Exact mathematical passage
J_{\Qcal\leftarrow\Pcal} = \Acal_{\Qcal}\Scal_{\Pcal}:X^{\Pcal}\longrightarrow X^{\Qcal}
X_\alphaDefinition passage · 1 passage
Search aliases: X_α alpha
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 19
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
X_\alpha=P_\alpha X, \qquad X^{\Pcal}=\bigoplus_{\alpha\in I}X_\alpha.
x_\alphaDefinition passage · 5 passages
Search aliases: x_α alpha
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 25
ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Acal_{\Pcal}x=(P_\alpha x)_\alpha, \qquad \Scal_{\Pcal}(x_\alpha)_\alpha=\sum_\alpha x_\alpha.
\mathbb ZDefinition passage · 10 passages
Search aliases: Z mathbb
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 20
bra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
w(e)\in\mathbb Z_{>0}
zDefinition passage · 53 passages
Search aliases: z
Definition passage: Residue selector · Read passage · 02_coordinates.tex, line 370
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
F(z)=\sum_{n\geq0}a_nz^n\in R[[z]].
z^nDefinition passage · 6 passages
Search aliases: z^n
Definition passage: Residue selector · Read passage · 02_coordinates.tex, line 370
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
F(z)=\sum_{n\geq0}a_nz^n\in R[[z]].
\mathbb Z_{>0}Definition passage · 1 passage
Search aliases: Z_>0 mathbb
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 20
bra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
w(e)\in\mathbb Z_{>0}
Z_{u,v}Definition passage · 5 passages
Search aliases: Z_u,v
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 380
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
Z_{u,v}(t) = \sum_{n\geq0}a_nt^n,
|Definition passage · 14 passages
Search aliases: |
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\Lambda_{\mathrm{split}}Definition passage · 7 passages
Search aliases: Λ_split Lambda mathrm
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 240
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
\Lambda_\alpha=\Lambda\cap V_\alpha, \qquad \Lambda_{\mathrm{split}}=\bigoplus_\alpha\Lambda_\alpha,
\Lambda_\alphaDefinition passage · 7 passages
Search aliases: Λ_α Lambda alpha
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 240
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
\Lambda_\alpha=\Lambda\cap V_\alpha, \qquad \Lambda_{\mathrm{split}}=\bigoplus_\alpha\Lambda_\alpha,
\nuDefinition passage · 1 passage
Search aliases: ν nu
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\pi^sDefinition passage · 24 passages
Search aliases: π^s pi
Definition passage: Separation weight · Read passage · 04_arithmetic.tex, line 181
[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero s…
Exact mathematical passage
s_{\pi,\Lambda}(E) = \min\{s\ge0:\pi^sE(\Lambda)\subseteq\Lambda\}.
\rhoDefinition passage · 20 passages
Search aliases: ρ rho
Definition passage: Refinement · Read passage · 02_coordinates.tex, line 35
[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).
Exact mathematical passage
\rho:J\longrightarrow I
\ldotsDefinition passage · 17 passages
Search aliases: … ldots
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 390
W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.
Exact mathematical passage
\gamma=(e_1,\ldots,e_{|\gamma|})
\leftarrowDefinition passage · 8 passages
Search aliases: ← leftarrow
Definition passage: Coordinate change across levels · Read passage · 02_coordinates.tex, line 84
[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.
Exact mathematical passage
J_{\Qcal\leftarrow\Pcal} = \Acal_{\Qcal}\Scal_{\Pcal}:X^{\Pcal}\longrightarrow X^{\Qcal}
\toDefinition passage · 19 passages
Search aliases: → to
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 374
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
A(t)_{ij} = \sum_{e:i\to j}m(e)t^{w(e)}.
\inDefinition passage · 81 passages
Search aliases: ∈ in
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 7
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Pcal=(P_\alpha)_{\alpha\in I} \subseteq\End(X)
\prodDefinition passage · 3 passages
Search aliases: ∏ prod
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\prod_{\nu=1}^{|\gamma|}Definition passage · 1 passage
Search aliases: ∏_ν=1^|γ| prod nu gamma
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\sumDefinition passage · 78 passages
Search aliases: ∑ sum
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 12
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
P_\alpha P_\beta=0 \quad\text{when }\alpha\ne\beta, \qquad \sum_{\alpha\in I}P_\alpha=I_X.
\sum_{e:i\to j}Definition passage · 2 passages
Search aliases: ∑_e:i→ j sum to
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 374
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
A(t)_{ij} = \sum_{e:i\to j}m(e)t^{w(e)}.
\sum_{i,j}Definition passage · 1 passage
Search aliases: ∑_i,j sum
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\sum_{n\equiv j\pmod N}Definition passage · 1 passage
Search aliases: ∑_n≡ jpmod N sum equiv pmod
Definition passage: Residue selector · Read passage · 02_coordinates.tex, line 374
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
(P_{N,j}F)(z) = \sum_{n\equiv j\pmod N}a_nz^n.
\sum_{n\geq0}Definition passage · 2 passages
Search aliases: ∑_n≥0 sum geq
Definition passage: Residue selector · Read passage · 02_coordinates.tex, line 370
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
F(z)=\sum_{n\geq0}a_nz^n\in R[[z]].
\sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}}Definition passage · 1 passage
Search aliases: ∑_substackγ:ileadsto j W(γ)=n sum substack gamma leadsto W gamma
Definition passage: Weighted transfer series · Read passage · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\sum_\alphaDefinition passage · 11 passages
Search aliases: ∑_α sum alpha
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 25
ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Acal_{\Pcal}x=(P_\alpha x)_\alpha, \qquad \Scal_{\Pcal}(x_\alpha)_\alpha=\sum_\alpha x_\alpha.
\sum_{\alpha\in I}Definition passage · 1 passage
Search aliases: ∑_α∈ I sum alpha in
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 12
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
P_\alpha P_\beta=0 \quad\text{when }\alpha\ne\beta, \qquad \sum_{\alpha\in I}P_\alpha=I_X.
\sum_{\rho(\beta)=\alpha}Definition passage · 2 passages
Search aliases: ∑_ρ(β)=α sum rho beta alpha
Definition passage: Refinement · Read passage · 02_coordinates.tex, line 39
[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).
Exact mathematical passage
P_\alpha=\sum_{\rho(\beta)=\alpha}Q_\beta
\capDefinition passage · 7 passages
Search aliases: ∩ cap
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 240
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
\Lambda_\alpha=\Lambda\cap V_\alpha, \qquad \Lambda_{\mathrm{split}}=\bigoplus_\alpha\Lambda_\alpha,
\cupDefinition passage · 4 passages
Search aliases: ∪ cup
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 258
et Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
s_\Lambda(\Pcal)=\max\bigl(\{0\}\cup\{s_\Lambda(E_\alpha):\alpha\in I\}\bigr).
\neDefinition passage · 23 passages
Search aliases: ≠ ne
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 12
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
P_\alpha P_\beta=0 \quad\text{when }\alpha\ne\beta, \qquad \sum_{\alpha\in I}P_\alpha=I_X.
\equivDefinition passage · 23 passages
Search aliases: ≡ equiv
Definition passage: Residue selector · Read passage · 02_coordinates.tex, line 374
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
(P_{N,j}F)(z) = \sum_{n\equiv j\pmod N}a_nz^n.
\leDefinition passage · 17 passages
Search aliases: ≤ le
Definition passage: Separation weight · Read passage · 04_arithmetic.tex, line 189
ight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero scalar in the f…
Exact mathematical passage
s_{\pi,\Lambda}(E)\le s=\val_\pi(r).
\leqDefinition passage · 23 passages
Search aliases: ≤ leq
Definition passage: Residue selector · Read passage · 02_coordinates.tex, line 373
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
0\leq j<N
\geDefinition passage · 10 passages
Search aliases: ≥ ge
Definition passage: Separation weight · Read passage · 04_arithmetic.tex, line 181
[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero s…
Exact mathematical passage
s_{\pi,\Lambda}(E) = \min\{s\ge0:\pi^sE(\Lambda)\subseteq\Lambda\}.
\geqDefinition passage · 21 passages
Search aliases: ≥ geq
Definition passage: Residue selector · Read passage · 02_coordinates.tex, line 370
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
F(z)=\sum_{n\geq0}a_nz^n\in R[[z]].
\subseteqDefinition passage · 12 passages
Search aliases: ⊆ subseteq
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 7
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Pcal=(P_\alpha)_{\alpha\in I} \subseteq\End(X)
\supseteqDefinition passage · 1 passage
Search aliases: ⊇ supseteq
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 7
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal=F^0\Bcal\supseteq F^1\Bcal\supseteq F^2\Bcal\supseteq\cdots
\bigoplusDefinition passage · 12 passages
Search aliases: ⊕ bigoplus
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 19
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
X_\alpha=P_\alpha X, \qquad X^{\Pcal}=\bigoplus_{\alpha\in I}X_\alpha.
\bigoplus_\alphaDefinition passage · 4 passages
Search aliases: ⊕_α bigoplus alpha
Definition passage: Lattice packet data · Read passage · 04_arithmetic.tex, line 240
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
\Lambda_\alpha=\Lambda\cap V_\alpha, \qquad \Lambda_{\mathrm{split}}=\bigoplus_\alpha\Lambda_\alpha,
\bigoplus_{\alpha\in I}Definition passage · 3 passages
Search aliases: ⊕_α∈ I bigoplus alpha in
Definition passage: Packet system · Read passage · 02_coordinates.tex, line 19
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
X_\alpha=P_\alpha X, \qquad X^{\Pcal}=\bigoplus_{\alpha\in I}X_\alpha.
\bigcapDefinition passage · 2 passages
Search aliases: ⋂ bigcap
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
\bigcap_{L\geq0}Definition passage · 2 passages
Search aliases: ⋂_L≥0 bigcap geq
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
\cdotsDefinition passage · 14 passages
Search aliases: ⋯ cdots
Definition passage: Complete multiplicative filtration · Read passage · 03_weighted.tex, line 7
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal=F^0\Bcal\supseteq F^1\Bcal\supseteq F^2\Bcal\supseteq\cdots
\longrightarrowDefinition passage · 13 passages
Search aliases: ⟶ longrightarrow
Definition passage: Refinement · Read passage · 02_coordinates.tex, line 35
[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).
Exact mathematical passage
\rho:J\longrightarrow I
\bmodIntroduction passage · 8 passages
Search aliases: bmod
Introduction passage · visual_atlas.tex, line 103
vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R_N for the permutation that groups coordinates in increasing residue order modulo N.
Exact mathematical passage
Se_j=e_{j+1\bmod8}
dIntroduction passage · 30 passages
Search aliases: d
Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
Exact mathematical passage
\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
e_jIntroduction passage · 3 passages
Search aliases: e_j
Introduction passage · visual_atlas.tex, line 103
vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R_N for the permutation that groups coordinates in increasing residue order modulo N.
Exact mathematical passage
Se_j=e_{j+1\bmod8}
e_{j+1\bmod8}Introduction passage · 1 passage
Search aliases: e_j+1bmod8 bmod
Introduction passage · visual_atlas.tex, line 103
vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R_N for the permutation that groups coordinates in increasing residue order modulo N.
Exact mathematical passage
Se_j=e_{j+1\bmod8}
HIntroduction passage · 20 passages
Search aliases: H
Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
Exact mathematical passage
\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
hIntroduction passage · 9 passages
Search aliases: h
Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
Exact mathematical passage
\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
H_{\rm phase}Introduction passage · 1 passage
Search aliases: H_rm phase rm
Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
Exact mathematical passage
\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
oIntroduction passage · 5 passages
Search aliases: o
Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
Exact mathematical passage
\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
pIntroduction passage · 48 passages
Search aliases: p
Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
Exact mathematical passage
\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
\QQIntroduction passage · 2 passages
Search aliases: QQ
Introduction passage · visual_atlas.tex, line 103
vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R_N for the permutation that groups coordinates in increasing residue order modulo N.
Exact mathematical passage
X=\QQ^8
\QQ^8Introduction passage · 1 passage
Search aliases: QQ^8 QQ
Introduction passage · visual_atlas.tex, line 103
vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R_N for the permutation that groups coordinates in increasing residue order modulo N.
Exact mathematical passage
X=\QQ^8
R_NIntroduction passage · 2 passages
Search aliases: R_N
Introduction passage · visual_atlas.tex, line 107
vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R_N for the permutation that groups coordinates in increasing residue order modulo N.
Exact mathematical passage
R_N
\rmIntroduction passage · 6 passages
Search aliases: rm
Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
Exact mathematical passage
\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
SIntroduction passage · 14 passages
Search aliases: S
Introduction passage · visual_atlas.tex, line 103
vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R_N for the permutation that groups coordinates in increasing residue order modulo N.
Exact mathematical passage
Se_j=e_{j+1\bmod8}
S^2Introduction passage · 1 passage
Search aliases: S^2
Introduction passage · visual_atlas.tex, line 104
vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R_N for the permutation that groups coordinates in increasing residue order modulo N.
Exact mathematical passage
T=I+2S+S^2.
yIntroduction passage · 36 passages
Search aliases: y
Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
Exact mathematical passage
\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
y_{\rm even}Introduction passage · 3 passages
Search aliases: y_rm even rm
Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
Exact mathematical passage
\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
y_{\rm odd}Introduction passage · 3 passages
Search aliases: y_rm odd rm
Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
Exact mathematical passage
\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
*Use in source · 1 passage
Search aliases: *
Use in source · finite_polyphase_filter.tex, line 15
5000,3.60000) rectangle ++(0.85000,0.62000); node[] at (2.97500,3.91000) 4; draw[fill=black!10,draw=black!45,line width=.35pt,] (3.40000,3.60000) rectangle ++(0.85000,0.62000); node[] at (3.82500,3.91000) 5; draw[fill=black!25,draw=black!45,line width=.35pt,] (4.25000,3.60000) rectangle ++(0.85000,0.62000); node[] at (4.67500,3.91000) 6; node[anchor=east] at (-0.15000,2.97000) h*x; draw[fill=black!25,draw=black!45,line width=.35pt,] (0.00000,2.67000) rectangle ++(0.85000,0.62000); node[] at (0.4…
Exact mathematical passage
h*x
\mathbf1Use in source · 2 passages
Search aliases: 1 mathbf
Use in source · Read passage · 03_weighted.tex, line 499
[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_n(s,t) = 1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, henc…
Exact mathematical passage
a_n(s,t) = \mathbf 1_{s=t,\,n=0} +\sum_{\substack{e:\,\operatorname{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\operatorname{target}(e),t).
\mathbf1_{s=t,\,n=0}Use in source · 2 passages
Search aliases: 1_s=t, n=0 mathbf
Use in source · Read passage · 03_weighted.tex, line 499
[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_n(s,t) = 1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, henc…
Exact mathematical passage
a_n(s,t) = \mathbf 1_{s=t,\,n=0} +\sum_{\substack{e:\,\operatorname{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\operatorname{target}(e),t).
\mathcal AUse in source · 3 passages
Search aliases: A mathcal
Use in source · visual_atlas.tex, line 27
A_N, S_N · Split into range-restricted packet vectors, then sum them. · A permutation/interleaving of eight coordinates.
Exact mathematical passage
\mathcal A_N,\mathcal S_N
A^2Use in source · 4 passages
Search aliases: A^2
Use in source · Read passage · 04_arithmetic.tex, line 610
Consider the integral involution A= 1 · 1 0 · -1 , A^2=I. Over a field of characteristic different from two, its sign projectors are E_+=fracI+A2 = 1 · 1/2
Exact mathematical passage
A= \begin{pmatrix} 1&1\\ 0&-1 \end{pmatrix}, \qquad A^2=I.
a^2Use in source · 1 passage
Search aliases: a^2
Use in source · visual_atlas.tex, line 207
vfig[.97]weighted_word_cutoff platecaptionAll nonzero retained terms of the signed inverse sum, followed by the first discarded nonzero word. The empty word contributes I. Other words of weight below five can still be zero because a^2=b^2=0.
Exact mathematical passage
a^2=b^2=0
A^{2k}Use in source · 1 passage
Search aliases: A^2k
Use in source · visual_atlas.tex, line 260
In this particular model, A^2=8I. Therefore v_2(A^2k)=3k, v_2(A^2k+1)=3k+1, where the matrix valuation means the minimum entry valuation. Three geometric terms leave a residual of valuation four and do not suffice modulo 32. Four terms leave valuation six and do suffice.
Exact mathematical passage
v_2(A^{2k})=3k,\qquad v_2(A^{2k+1})=3k+1,
A^{2k+1}Use in source · 1 passage
Search aliases: A^2k+1
Use in source · visual_atlas.tex, line 260
In this particular model, A^2=8I. Therefore v_2(A^2k)=3k, v_2(A^2k+1)=3k+1, where the matrix valuation means the minimum entry valuation. Three geometric terms leave a residual of valuation four and do not suffice modulo 32. Four terms leave valuation six and do suffice.
Exact mathematical passage
v_2(A^{2k})=3k,\qquad v_2(A^{2k+1})=3k+1,
A^NUse in source · 2 passages
Search aliases: A^N
Use in source · Read passage · 03_weighted.tex, line 328
W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is a two-sided inverse of q_L(I+A), where A=∑_eA_e=D^-1H. With the supplied two-sided inverse D^-1 of D, this gives the explicit two-sided inverse q_L (S^pr_L,ND^-1) of q_L(D+H). If every edge weight is at least m and L≤ mN, then the nilpotence condition follows from the filtration data because m|γ|≤ W(γ). For m>0, the executable choice N=lceil L/mrceil satisfies L≤ mN. This elem…
Exact mathematical passage
q_L(A^N)=0
A^rUse in source · 1 passage
Search aliases: A^r
Use in source · Read passage · 03_weighted.tex, line 277
[Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.
Exact mathematical passage
A^r=0
a_0Use in source · 1 passage
Search aliases: a_0
Use in source · visual_atlas.tex, line 311
From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.
Exact mathematical passage
a_0=a_1=0,\quad a_2=2,\qquad a_n=a_{n-1}+2a_{n-3}\quad(n\ge3).
a_1Use in source · 3 passages
Search aliases: a_1
Use in source · Read passage · 04_arithmetic.tex, line 289
_packet, the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).
Exact mathematical passage
a_1\leq\cdots\leq a_r
\mathcal A_2Use in source · 1 passage
Search aliases: A_2 mathcal
Use in source · visual_atlas.tex, line 87
Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.
Exact mathematical passage
J=\mathcal A_4\mathcal S_2,\qquad C=\mathcal A_2\mathcal S_4, \qquad CJ=I,\quad JC=I.
a_2Use in source · 1 passage
Search aliases: a_2
Use in source · visual_atlas.tex, line 311
From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.
Exact mathematical passage
a_0=a_1=0,\quad a_2=2,\qquad a_n=a_{n-1}+2a_{n-3}\quad(n\ge3).
\mathcal A_4Use in source · 1 passage
Search aliases: A_4 mathcal
Use in source · visual_atlas.tex, line 87
Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.
Exact mathematical passage
J=\mathcal A_4\mathcal S_2,\qquad C=\mathcal A_2\mathcal S_4, \qquad CJ=I,\quad JC=I.
a_5Use in source · 1 passage
Search aliases: a_5
Use in source · zeta_recurrence.tex, line 57
e ++(0.87000,0.65000); node[] at (10.87500,2.07500) 12; draw[fill=black!20,draw=black!45,line width=.35pt,] (10.44000,1.00000) rectangle ++(0.87000,0.65000); node[] at (10.87500,1.32500) 218; node[anchor=east] at (-0.15000,2.07500) n; node[anchor=east] at (-0.15000,1.32500) a_n; node[] at (5.65000,3.10000) Z_0,1(t)=dfrac2t^21-t-2t^3; node[] at (5.65000,0.00000) a_8=a_7+2a_5=14+2·6=26;
Exact mathematical passage
a_8=a_7+2a_5=14+2\cdot6=26
a_7Use in source · 1 passage
Search aliases: a_7
Use in source · zeta_recurrence.tex, line 57
e ++(0.87000,0.65000); node[] at (10.87500,2.07500) 12; draw[fill=black!20,draw=black!45,line width=.35pt,] (10.44000,1.00000) rectangle ++(0.87000,0.65000); node[] at (10.87500,1.32500) 218; node[anchor=east] at (-0.15000,2.07500) n; node[anchor=east] at (-0.15000,1.32500) a_n; node[] at (5.65000,3.10000) Z_0,1(t)=dfrac2t^21-t-2t^3; node[] at (5.65000,0.00000) a_8=a_7+2a_5=14+2·6=26;
Exact mathematical passage
a_8=a_7+2a_5=14+2\cdot6=26
a_8Use in source · 1 passage
Search aliases: a_8
Use in source · zeta_recurrence.tex, line 57
e ++(0.87000,0.65000); node[] at (10.87500,2.07500) 12; draw[fill=black!20,draw=black!45,line width=.35pt,] (10.44000,1.00000) rectangle ++(0.87000,0.65000); node[] at (10.87500,1.32500) 218; node[anchor=east] at (-0.15000,2.07500) n; node[anchor=east] at (-0.15000,1.32500) a_n; node[] at (5.65000,3.10000) Z_0,1(t)=dfrac2t^21-t-2t^3; node[] at (5.65000,0.00000) a_8=a_7+2a_5=14+2·6=26;
Exact mathematical passage
a_8=a_7+2a_5=14+2\cdot6=26
A_eUse in source · 5 passages
Search aliases: A_e
Use in source · Read passage · 03_weighted.tex, line 48
[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
A_e=D^{-1}N_e.
A_{e_1}Use in source · 2 passages
Search aliases: A_e_1
Use in source · Read passage · 03_weighted.tex, line 52
[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
A_\gamma=A_{e_r}\cdots A_{e_1}, \qquad W(\gamma)=\sum_{j=1}^rw(e_j).
A_{e_r}Use in source · 2 passages
Search aliases: A_e_r
Use in source · Read passage · 03_weighted.tex, line 52
[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
A_\gamma=A_{e_r}\cdots A_{e_1}, \qquad W(\gamma)=\sum_{j=1}^rw(e_j).
a_jUse in source · 5 passages
Search aliases: a_j
Use in source · Read passage · 04_arithmetic.tex, line 284
ght] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).
Exact mathematical passage
G_{\Lambda,\Pcal} \cong \bigoplus_{j=1}^r\OO/\pi^{a_j}\OO
a_kUse in source · 1 passage
Search aliases: a_k
Use in source · Read passage · 03_weighted.tex, line 441
In the recurrence, set a_k=0 when k<0. This is the coefficient convention for the formal power series Z_u,v(t) and makes the first recurrence equations meaningful even when n<d.
Exact mathematical passage
a_k=0
\mathcal A_NUse in source · 2 passages
Search aliases: A_N mathcal
Use in source · visual_atlas.tex, line 27
A_N, S_N · Split into range-restricted packet vectors, then sum them. · A permutation/interleaving of eight coordinates.
Exact mathematical passage
\mathcal A_N,\mathcal S_N
a_{n-1}Use in source · 2 passages
Search aliases: a_n-1
Use in source · Read passage · 03_weighted.tex, line 428
[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.
Exact mathematical passage
a_n+q_1a_{n-1}+\cdots+q_da_{n-d}=0.
a_{n-3}Use in source · 1 passage
Search aliases: a_n-3
Use in source · visual_atlas.tex, line 311
From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.
Exact mathematical passage
a_0=a_1=0,\quad a_2=2,\qquad a_n=a_{n-1}+2a_{n-3}\quad(n\ge3).
a_{n-d}Use in source · 1 passage
Search aliases: a_n-d
Use in source · Read passage · 03_weighted.tex, line 428
[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.
Exact mathematical passage
a_n+q_1a_{n-1}+\cdots+q_da_{n-d}=0.
a_{n-k}Use in source · 1 passage
Search aliases: a_n-k
Use in source · Read passage · 03_weighted.tex, line 564
[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).
Exact mathematical passage
\boxed{ \sum_{k=0}^{n}q_k a_{n-k}(s,t)=0 \qquad\bigl(n>\deg P_{s,t}\bigr). }
a_{n-w(e)}Use in source · 2 passages
Search aliases: a_n-w(e)
Use in source · Read passage · 03_weighted.tex, line 499
[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_n(s,t) = 1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, henc…
Exact mathematical passage
a_n(s,t) = \mathbf 1_{s=t,\,n=0} +\sum_{\substack{e:\,\operatorname{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\operatorname{target}(e),t).
a_rUse in source · 1 passage
Search aliases: a_r
Use in source · Read passage · 04_arithmetic.tex, line 289
_packet, the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).
Exact mathematical passage
a_1\leq\cdots\leq a_r
A_\gammaUse in source · 9 passages
Search aliases: A_γ gamma
Use in source · Read passage · 03_weighted.tex, line 52
[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
A_\gamma=A_{e_r}\cdots A_{e_1}, \qquad W(\gamma)=\sum_{j=1}^rw(e_j).
\operatorname{adj}Use in source · 5 passages
Search aliases: adj operatorname
Use in source · Read passage · 03_weighted.tex, line 412
ce every entry of A(t) has zero constant term, the inverse exists in the formal power series ring and (I-A(t))^-1=∑_r≥0A(t)^r. The (i,j) entry of A(t)^r is the sum of the monomials attached to all paths of length r from i to j. Summing over r records paths of every length. Multiplication by the endpoint vectors gives the stated series. The adjugate formula gives (I-A(t))^-1 = fracadj(I-A(t))det(I-A(t)), so every entry is rational.
Exact mathematical passage
(I-A(t))^{-1} = \frac{\operatorname{adj}(I-A(t))}{\det(I-A(t))},
\operatorname{analyze}Use in source · 1 passage
Search aliases: analyze operatorname
Use in source · Read passage · 06_models.tex, line 214
The verified contract is synthesize(analyze(x,N))=x. The Lean counterpart uses packet coordinates restricted to projector ranges and proves both inverse laws. The finite Python routine checks the concrete array realization.
Exact mathematical passage
\operatorname{synthesize}(\operatorname{analyze}(x,N))=x
BUse in source · 17 passages
Search aliases: B
Use in source · Read passage · 02_coordinates.tex, line 499
[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.
Exact mathematical passage
B
bUse in source · 59 passages
Search aliases: b
Use in source · Read passage · 02_coordinates.tex, line 176
Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children to their parents. Coordinate coherence itself does not require them.
Exact mathematical passage
a,b,c
B^{-1}Use in source · 3 passages
Search aliases: B^-1
Use in source · visual_atlas.tex, line 356
Write B=1 · -1 0 · 2, AB=Bdiag(1,-1), E_+=Bdiag(1,0)B^-1. The index-two eigenlattice explains the denominator in B^-1 and in the projector. Over the localized lattice Λ_p=ZZ_(p)^2, the number 2 is a unit for every odd prime p. Hence the local denominator weight is zero there and one at p=2.
Exact mathematical passage
B=\begin{pmatrix}1&-1\\0&2\end{pmatrix},\qquad AB=B\diag(1,-1),\qquad E_+=B\diag(1,0)B^{-1}.
b^2Use in source · 1 passage
Search aliases: b^2
Use in source · visual_atlas.tex, line 207
vfig[.97]weighted_word_cutoff platecaptionAll nonzero retained terms of the signed inverse sum, followed by the first discarded nonzero word. The empty word contributes I. Other words of weight below five can still be zero because a^2=b^2=0.
Exact mathematical passage
a^2=b^2=0
b_iUse in source · 2 passages
Search aliases: b_i
Use in source · Read passage · 04_arithmetic.tex, line 838
[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
p^{\min(s_i,L)}\mid (b_i)_{\mathrm{val}}
b_{\mathrm{val}}Use in source · 2 passages
Search aliases: b_val mathrm
Use in source · Read passage · 04_arithmetic.tex, line 856
Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fibre has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global count. If one coordinate fibre is empty, the global fibre is empty. Lean proves both the global solvability equivalence and the conditional count, inc…
Exact mathematical passage
\bigl(\exists x\in\mathbb Z/p^L\mathbb Z:\quad p^s x=b\pmod{p^L}\bigr) \quad\Longleftrightarrow\quad p^{\min(s,L)}\mid b_{\mathrm{val}},
\bbk^2Use in source · 1 passage
Search aliases: bbk^2 bbk
Use in source · Read passage · 05_defects.tex, line 262
[A visible one step defect] Let X_f=bbk^2, let X_c=bbk, and let C(x_1,x_2)=x_1. Take T_f= a · b
Exact mathematical passage
X_f=\bbk^2
\bulletUse in source · 1 passage
Search aliases: bullet
Use in source · Read passage · 03_weighted.tex, line 310
[Finite precision] Let F^bullet R be a multiplicative decreasing filtration and let q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n
Exact mathematical passage
F^\bullet R
C^{-1}Use in source · 1 passage
Search aliases: C^-1
Use in source · visual_atlas.tex, line 547
Had we chosen the transported operator T_c'=CT_fC^-1, the defect would be zero and every polynomial action would intertwine exactly. That is a different choice of coarse operator, not a stronger conclusion from the faulty one.
Exact mathematical passage
T_c'=CT_fC^{-1}
C_{0,L}Use in source · 3 passages
Search aliases: C_0,L
Use in source · Read passage · 05_defects.tex, line 228
[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .
Exact mathematical passage
C_{0,L}T_L-T_0C_{0,L} = \sum_{\ell=0}^{L-1} C_{0,\ell}\Delta_\ell C_{\ell+1,L}.
C_{0,\ell}Use in source · 4 passages
Search aliases: C_0,ℓ ell
Use in source · Read passage · 05_defects.tex, line 228
[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .
Exact mathematical passage
C_{0,L}T_L-T_0C_{0,L} = \sum_{\ell=0}^{L-1} C_{0,\ell}\Delta_\ell C_{\ell+1,L}.
c_{00}Use in source · 1 passage
Search aliases: c_00
Use in source · Read passage · 06_models.tex, line 5
Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer blocks. Refining from N to M with Nmid M splits each coarse phase into its descendants without changing the reconstructed signal. The same formulas extend…
Exact mathematical passage
x=(x_n)_{n\in\Z}\in c_{00}(\Z;R)
C_aUse in source · 1 passage
Search aliases: C_a
Use in source · Read passage · 05_defects.tex, line 221
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_{a,b}=C_aC_{a+1}\cdots C_{b-1},
C_{a+1}Use in source · 1 passage
Search aliases: C_a+1
Use in source · Read passage · 05_defects.tex, line 221
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_{a,b}=C_aC_{a+1}\cdots C_{b-1},
C_{a,a}Use in source · 1 passage
Search aliases: C_a,a
Use in source · Read passage · 05_defects.tex, line 224
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_{a,a}=I
C_{a,b}Use in source · 1 passage
Search aliases: C_a,b
Use in source · Read passage · 05_defects.tex, line 221
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_{a,b}=C_aC_{a+1}\cdots C_{b-1},
C_{b-1}Use in source · 1 passage
Search aliases: C_b-1
Use in source · Read passage · 05_defects.tex, line 221
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_{a,b}=C_aC_{a+1}\cdots C_{b-1},
C_{\Pcal\leftarrow\Rcal}Use in source · 1 passage
Search aliases: C_Pcal←Rcal C_ P← R Pcal leftarrow Rcal mathcal leftarrow mathcal
Use in source · Read passage · 02_coordinates.tex, line 115
herence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
C_{\Pcal\leftarrow\Qcal}C_{\Qcal\leftarrow\Rcal} = C_{\Pcal\leftarrow\Rcal}.
C_{\Qcal\leftarrow\Rcal}Use in source · 1 passage
Search aliases: C_Qcal←Rcal C_ Q← R Qcal leftarrow Rcal mathcal leftarrow mathcal
Use in source · Read passage · 02_coordinates.tex, line 115
herence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
C_{\Pcal\leftarrow\Qcal}C_{\Qcal\leftarrow\Rcal} = C_{\Pcal\leftarrow\Rcal}.
C_\ellUse in source · 3 passages
Search aliases: C_ℓ ell
Use in source · Read passage · 05_defects.tex, line 213
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_\ell:X_{\ell+1}\longrightarrow X_\ell
C_{\ell+1,L}Use in source · 3 passages
Search aliases: C_ℓ+1,L ell
Use in source · Read passage · 05_defects.tex, line 228
[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .
Exact mathematical passage
C_{0,L}T_L-T_0C_{0,L} = \sum_{\ell=0}^{L-1} C_{0,\ell}\Delta_\ell C_{\ell+1,L}.
C_{\ell,L}Use in source · 1 passage
Search aliases: C_ℓ,L ell
Use in source · Read passage · 05_defects.tex, line 247
Insert the intermediate terms C_0,ℓT_ℓ C_ℓ,L for 0≤ℓ≤ L. Consecutive differences telescope. The difference between the terms indexed by ℓ+1 and ℓ is C_0,ℓ(C_ℓ T_ℓ+1-T_ℓ C_ℓ)C_ℓ+1,L. Summing gives the identity. The norm estimate follows from the triangle inequality.
Exact mathematical passage
C_{0,\ell}T_\ell C_{\ell,L}
\cA_QUse in source · 1 passage
Search aliases: cA_Q A_Q cA mathcal
Use in source · Read passage · 02_coordinates.tex, line 189
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\widehat T_{Q,P}=\cA_QT\cS_P
\cBUse in source · 4 passages
Search aliases: cB B mathcal
Use in source · Read passage · 03_weighted.tex, line 237
Let O be a complete discrete valuation ring with uniformizer π, fraction field K, and filtration F^m B=π^m B on an O-algebra B. The finite quotient statements below do not require B itself to be complete. Suppose T=D+H, where D is invertible and the perturbation H raises filtration. The symbol K is reserved for the fraction field.
Exact mathematical passage
F^m\cB=\pi^m\cB
\cS_PUse in source · 1 passage
Search aliases: cS_P S_P cS mathcal
Use in source · Read passage · 02_coordinates.tex, line 189
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\widehat T_{Q,P}=\cA_QT\cS_P
DUse in source · 59 passages
Search aliases: D
Use in source · Read passage · 01_orientation.tex, line 8
The second part studies inversion in a complete filtered algebra. Suppose an operator has the form T=D+N, where D is invertible and every packet transfer in N raises the filtration by a positive amount. The ordinary Neumann series can then be expanded into weighted paths. The weight is not the number of edges. It is the total filtration gain along the path. At precision L, every path of weight at least L vanishes. A finite transfer graph also gives a rational generating function for the number o…
Exact mathematical passage
T=D+N,
D^{-1}Use in source · 32 passages
Search aliases: D^-1
Use in source · Read passage · 03_weighted.tex, line 36
[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
D,D^{-1}\in F^0\Bcal.
D_0Use in source · 1 passage
Search aliases: D_0
Use in source · Read passage · 01_orientation.tex, line 56
A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operator on one selected summand may also fail to intertwine with the full matrix. The later sections isolate these four effects one at a time.
Exact mathematical passage
X=X_0\oplus X_1, \qquad D=\begin{pmatrix}D_0&0\\0&D_1\end{pmatrix}, \qquad N=\begin{pmatrix}0&N_{01}\\N_{10}&0\end{pmatrix}.
D_1Use in source · 1 passage
Search aliases: D_1
Use in source · Read passage · 01_orientation.tex, line 56
A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operator on one selected summand may also fail to intertwine with the full matrix. The later sections isolate these four effects one at a time.
Exact mathematical passage
X=X_0\oplus X_1, \qquad D=\begin{pmatrix}D_0&0\\0&D_1\end{pmatrix}, \qquad N=\begin{pmatrix}0&N_{01}\\N_{10}&0\end{pmatrix}.
D_\alphaUse in source · 2 passages
Search aliases: D_α alpha
Use in source · Read passage · 03_weighted.tex, line 94
[Block path formula] Let P=(P_α)_α∈ I be a finite packet system in a complete filtered algebra of operators. Suppose D=∑_α P_α D P_α belongs to B and is invertible on every packet. Assume that the assembled inverse D^-1=∑_α D_α^-1P_α∈ B=F^0 B, D_α=P_α DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be finite, assume w(e)≥1 for every edge, and write N=∑_e:α→βN_e, N_e=P_β N_eP_α, N_e∈ F^w(e) B. For an edge e:α→β, put A_e=(P_β DP_β)^-1N_e. Then the block f…
Exact mathematical passage
D^{-1}=\sum_\alpha D_\alpha^{-1}P_\alpha\in\Bcal=F^0\Bcal, \qquad D_\alpha=P_\alpha DP_\alpha,
D_\alpha^{-1}Use in source · 3 passages
Search aliases: D_α^-1 alpha
Use in source · Read passage · 03_weighted.tex, line 94
[Block path formula] Let P=(P_α)_α∈ I be a finite packet system in a complete filtered algebra of operators. Suppose D=∑_α P_α D P_α belongs to B and is invertible on every packet. Assume that the assembled inverse D^-1=∑_α D_α^-1P_α∈ B=F^0 B, D_α=P_α DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be finite, assume w(e)≥1 for every edge, and write N=∑_e:α→βN_e, N_e=P_β N_eP_α, N_e∈ F^w(e) B. For an edge e:α→β, put A_e=(P_β DP_β)^-1N_e. Then the block f…
Exact mathematical passage
D^{-1}=\sum_\alpha D_\alpha^{-1}P_\alpha\in\Bcal=F^0\Bcal, \qquad D_\alpha=P_\alpha DP_\alpha,
\degUse in source · 4 passages
Search aliases: deg
Use in source · Read passage · 03_weighted.tex, line 427
[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.
Exact mathematical passage
\deg 0=-\infty
\detUse in source · 7 passages
Search aliases: det
Use in source · Read passage · 03_weighted.tex, line 401
[Rationality of the transfer series] The weighted transfer series satisfies Z_u,v(t) = u^ T(I-A(t))^-1v. It is a rational function whose denominator divides det(I-A(t)).
Exact mathematical passage
\det(I-A(t)).
\diagUse in source · 9 passages
Search aliases: diag operatorname
Use in source · Read passage · 04_arithmetic.tex, line 100
[A resultant bound that is not exact] Let A=diag(0,π^s) on the standard lattice Λ= O^2. The factors x and x-π^s have resultant -π^s, up to the harmless choice of resultant order, but the two spectral projectors are the integral coordinate projectors. Their exact separation weight is zero. The resultant sees the closeness of the two eigenvalues. It does not see that the chosen lattice already splits along their eigenspaces.
Exact mathematical passage
A=\diag(0,\pi^s)
e^{(t-s)T_c}Use in source · 4 passages
Search aliases: e^(t-s)T_c
Use in source · Read passage · 05_defects.tex, line 182
[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t is complex. For real t≥0, Ce^tT_f-e^tT_cC ≤ Δ ∫_0^t e^(t-s)T_c e^sT_f ds.
Exact mathematical passage
Ce^{tT_f}-e^{tT_c}C = \int_0^t e^{(t-s)T_c}\Delta e^{sT_f}\,ds,
E^2Use in source · 1 passage
Search aliases: E^2
Use in source · Read passage · 04_arithmetic.tex, line 156
The B ezout identity gives N=rI-UF. The two stated commutation relations and FG=0 give N(UF)=VGUF=VUGF=VUFG=0. Therefore N^2=N(rI-UF)=rN. Since r≠0, E^2=r^-2N^2=E. If Fx=0, the evaluated B ezout identity gives VGx=rx, hence Ex=x. If Gx=0, then Ex=0. The lattice assertion follows directly from rE=VG.
Exact mathematical passage
E^2=r^{-2}N^2=E
e^{ik\pi/4}Use in source · 2 passages
Search aliases: e^ikπ/4 pi
Use in source · visual_atlas.tex, line 510
AR PHASE ORBITTraditional paper: Section 12.3 (p. 42), Section 12.3, Smith-phase identity (p. 43) Smith normal form concerns linear congruences. The Smith impedance chart uses the nonlinear coordinate change Γ=fracz-1z+1, z=frac1+Γ1-Γ. They share a name, not a mathematical definition. Sample the reflection circle at the exact algebraic points Γ_k=tfrac12 e^ikπ/4.
Exact mathematical passage
\Gamma_k=\tfrac12 e^{ik\pi/4}
E^nUse in source · 1 passage
Search aliases: E^n
Use in source · Read passage · 03_weighted.tex, line 321
et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is…
Exact mathematical passage
S^{\mathrm{pr}}_{L,N} = \sum_{n=0}^{N-1}(-1)^n \sum_{\substack{\gamma\in E^n\\W(\gamma)<L}}A_\gamma.
e^{sT_f}Use in source · 4 passages
Search aliases: e^sT_f
Use in source · Read passage · 05_defects.tex, line 182
[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t is complex. For real t≥0, Ce^tT_f-e^tT_cC ≤ Δ ∫_0^t e^(t-s)T_c e^sT_f ds.
Exact mathematical passage
Ce^{tT_f}-e^{tT_c}C = \int_0^t e^{(t-s)T_c}\Delta e^{sT_f}\,ds,
e^{tT_c}Use in source · 2 passages
Search aliases: e^tT_c
Use in source · Read passage · 05_defects.tex, line 182
[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t is complex. For real t≥0, Ce^tT_f-e^tT_cC ≤ Δ ∫_0^t e^(t-s)T_c e^sT_f ds.
Exact mathematical passage
Ce^{tT_f}-e^{tT_c}C = \int_0^t e^{(t-s)T_c}\Delta e^{sT_f}\,ds,
e^{tT_f}Use in source · 2 passages
Search aliases: e^tT_f
Use in source · Read passage · 05_defects.tex, line 182
[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t is complex. For real t≥0, Ce^tT_f-e^tT_cC ≤ Δ ∫_0^t e^(t-s)T_c e^sT_f ds.
Exact mathematical passage
Ce^{tT_f}-e^{tT_c}C = \int_0^t e^{(t-s)T_c}\Delta e^{sT_f}\,ds,
E_+Use in source · 22 passages
Search aliases: E_+
Use in source · Read passage · 04_arithmetic.tex, line 563
0 · -1 act on Z_p^2. Over Q_p, the sign projectors are E_+=fracI+A2 = 1 · 1/2 0 · 0 , E_-=fracI-A2 = 0 · -1/2
Exact mathematical passage
E_+=\frac{I+A}{2} = \begin{pmatrix} 1&1/2\\ 0&0 \end{pmatrix},
E_+^2Use in source · 1 passage
Search aliases: E_+^2
Use in source · bezout_normalization.tex, line 30
000); node[] at (8.25000,2.15000) frac12; draw[fill=black!0,draw=black!45,line width=.35pt,] (7.20000,1.10000) rectangle ++(0.70000,0.70000); node[] at (7.55000,1.45000) 0; draw[fill=black!0,draw=black!45,line width=.35pt,] (7.90000,1.10000) rectangle ++(0.70000,0.70000); node[] at (8.25000,1.45000) 0; node[] at (11.00000,2.10000) N^2=2N; node[] at (11.00000,1.20000) E_+^2=E_+; node[] at (6.00000,-0.05000) E_+(1,0)=(1,0), E_+(-1,2)=(0,0);
Exact mathematical passage
E_+^2=E_+
E_-Use in source · 7 passages
Search aliases: E_-
Use in source · Read passage · 04_arithmetic.tex, line 571
0 · 0 , E_-=fracI-A2 = 0 · -1/2 0 · 1 . For every odd prime p, both projectors preserve Z_p^2. At p=2, s_2(E_+)=s_2(E_-)=1. The factors x-1 and x+1 have resultant 2.
Exact mathematical passage
E_-=\frac{I-A}{2} = \begin{pmatrix} 0&-1/2\\ 0&1 \end{pmatrix}.
E_{00}Use in source · 2 passages
Search aliases: E_00
Use in source · visual_atlas.tex, line 210
The words are written matrix products: in ab, the map b acts first. Therefore ab=8E_00, ba=8E_11, aba=16E_01. The two products ab and ba cannot be identified merely because they have equal weight. Both their order and their source/target information matter.
Exact mathematical passage
ab=8E_{00},\qquad ba=8E_{11},\qquad aba=16E_{01}.
E_{01}Use in source · 7 passages
Search aliases: E_01
Use in source · visual_atlas.tex, line 201
vpage9Which words can survive precision five?MODEL B / ORDERED TRANSFERS IN A FILTERED RINGTraditional paper: Theorem 5.10 (p. 17), Corollary 5.11 (p. 18) Let E_ij send the jth basis vector to the ith. In integral 2×2 matrices set a=2E_01, b=4E_10, A=a+b. Their certified dyadic weights are 1 and 2. At precision L=5, every word of weight at least five vanishes modulo 32.
Exact mathematical passage
a=2E_{01},\qquad b=4E_{10},\qquad A=a+b.
E_{10}Use in source · 3 passages
Search aliases: E_10
Use in source · visual_atlas.tex, line 201
vpage9Which words can survive precision five?MODEL B / ORDERED TRANSFERS IN A FILTERED RINGTraditional paper: Theorem 5.10 (p. 17), Corollary 5.11 (p. 18) Let E_ij send the jth basis vector to the ith. In integral 2×2 matrices set a=2E_01, b=4E_10, A=a+b. Their certified dyadic weights are 1 and 2. At precision L=5, every word of weight at least five vanishes modulo 32.
Exact mathematical passage
a=2E_{01},\qquad b=4E_{10},\qquad A=a+b.
E_{11}Use in source · 2 passages
Search aliases: E_11
Use in source · visual_atlas.tex, line 210
The words are written matrix products: in ab, the map b acts first. Therefore ab=8E_00, ba=8E_11, aba=16E_01. The two products ab and ba cannot be identified merely because they have equal weight. Both their order and their source/target information matter.
Exact mathematical passage
ab=8E_{00},\qquad ba=8E_{11},\qquad aba=16E_{01}.
e_2Use in source · 6 passages
Search aliases: e_2
Use in source · Read passage · 04_arithmetic.tex, line 501
[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Only the lattice has changed.
Exact mathematical passage
V=Ke_1\oplus Ke_2
E_fUse in source · 7 passages
Search aliases: E_f
Use in source · Read passage · 04_arithmetic.tex, line 19
[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).
Exact mathematical passage
E_f=r^{-1}v(A)g(A)
E_f^2Use in source · 1 passage
Search aliases: E_f^2
Use in source · Read passage · 04_arithmetic.tex, line 23
[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).
Exact mathematical passage
E_f^2=E_f, \qquad rE_f\in\End_{\OO}(\Lambda).
E_iUse in source · 4 passages
Search aliases: E_i
Use in source · Read passage · 04_arithmetic.tex, line 63
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
E_i
E_{ij}Use in source · 1 passage
Search aliases: E_ij
Use in source · visual_atlas.tex, line 200
vpage9Which words can survive precision five?MODEL B / ORDERED TRANSFERS IN A FILTERED RINGTraditional paper: Theorem 5.10 (p. 17), Corollary 5.11 (p. 18) Let E_ij send the jth basis vector to the ith. In integral 2×2 matrices set a=2E_01, b=4E_10, A=a+b. Their certified dyadic weights are 1 and 2. At precision L=5, every word of weight at least five vanishes modulo 32.
Exact mathematical passage
E_{ij}
E_jUse in source · 2 passages
Search aliases: E_j
Use in source · Read passage · 04_arithmetic.tex, line 64
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
\im E_i=\Ker f_i(A), \qquad \sum_iE_i=I, \qquad E_iE_j=0 \quad(i\ne j).
e_mUse in source · 1 passage
Search aliases: e_m
Use in source · visual_atlas.tex, line 481
vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONTraditional paper: Section B.1 (p. 47) Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.
Exact mathematical passage
e_m=x_{2m}
e_rUse in source · 5 passages
Search aliases: e_r
Use in source · Read passage · 03_weighted.tex, line 51
[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
\gamma=(e_1,\ldots,e_r)
E_\pmUse in source · 1 passage
Search aliases: E_± pm
Use in source · visual_atlas.tex, line 366
headThe special fiber does not retain the rational splitting Modulo two, x-1=x+1 and A becomes a nontrivial unipotent matrix. The entries 1/2 in E_± cannot be reduced modulo two. By contrast, the coefficient selectors on V6 use only zero and one and remain available.
Exact mathematical passage
E_\pm
\End_KUse in source · 1 passage
Search aliases: End_K End operatorname
Use in source · Read passage · 04_arithmetic.tex, line 121
[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).
Exact mathematical passage
F,G,U,V\in\End_K(\Lambda\otimes_\OO K)
\End_\OOUse in source · 2 passages
Search aliases: End_OO End_ O End OO operatorname mathcal
Use in source · Read passage · 04_arithmetic.tex, line 114
Let O be a DVR, K=Frac( O), and A∈End_ O(Λ) for a finite free lattice Λ. Suppose the characteristic polynomial factors over O as χ_A(x)=f(x)g(x), where f and g are coprime over K[x].
Exact mathematical passage
A\in\End_\OO(\Lambda)
\End_{\OO}Use in source · 3 passages
Search aliases: End_OO End_ O End OO operatorname mathcal
Use in source · Read passage · 04_arithmetic.tex, line 7
Let O be a discrete valuation ring with fraction field K and uniformizer π. Let Λ be a finite free O module and put V=K⊗_ OΛ. Suppose A∈End_ O(Λ).
Exact mathematical passage
A\in\End_{\OO}(\Lambda)
\FUse in source · 1 passage
Search aliases: F mathbb
Use in source · Read passage · 04_arithmetic.tex, line 668
The two characteristic factors x-1 and x+1 have the same reduction in F_2[x]. The rational matrices E_+ and E_- themselves cannot be reduced modulo 2, because their entries contain 1/2. Consequently the integral special fibre does not inherit their separate idempotent splitting, even though the two projectors remain distinct over Q_2. This is the simplest model of delayed dyadic separation.
Exact mathematical passage
\F_2[x]
\mathbb FUse in source · 7 passages
Search aliases: F mathbb
Use in source · Read passage · 02_coordinates.tex, line 443
The field or orthogonality condition is essential. A primitive element of order N in a ring with zero divisors does not by itself force the geometric sums away from the diagonal to vanish. For example, in F_3× F_3 the element (-1,1) has order two, but the second component of its nontrivial character sum is nonzero. The coefficient selector uses no division and remains valid without any Fourier certificate. This distinction becomes decisive at the dyadic prime.
Exact mathematical passage
\mathbb F_3\times\mathbb F_3
F'Use in source · 1 passage
Search aliases: F'
Use in source · Read passage · 05_defects.tex, line 204
For real t, differentiate F(s)=e^(t-s)T_cCe^sT_f. We obtain F'(s)=e^(t-s)T_cΔ e^sT_f. Integration from 0 to t gives the identity. The complex statement follows by the same calculation along the straight segment. The norm bound follows from the integral formula.
Exact mathematical passage
F'(s)=e^{(t-s)T_c}\Delta e^{sT_f}.
F^\bulletUse in source · 1 passage
Search aliases: F^bullet bullet
Use in source · Read passage · 03_weighted.tex, line 310
[Finite precision] Let F^bullet R be a multiplicative decreasing filtration and let q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n
Exact mathematical passage
F^\bullet R
F^{d(\alpha,\beta)}Use in source · 1 passage
Search aliases: F^d(α,β) alpha beta
Use in source · Read passage · 03_weighted.tex, line 130
[Minimum path weight] For packets α and β, let d(α,β) be the least total weight of a directed path from α to β, with d=∞ when no path exists. Use the convention F^∞ B=⋂_L≥0F^L B=0. If α≠β, then P_β(D+N)^-1P_α ∈ F^d(α,β) B. If there is no path from α to β, the block is zero.
Exact mathematical passage
P_\beta(D+N)^{-1}P_\alpha \in F^{d(\alpha,\beta)}\Bcal.
F^mUse in source · 1 passage
Search aliases: F^m
Use in source · Read passage · 03_weighted.tex, line 237
Let O be a complete discrete valuation ring with uniformizer π, fraction field K, and filtration F^m B=π^m B on an O-algebra B. The finite quotient statements below do not require B itself to be complete. Suppose T=D+H, where D is invertible and the perturbation H raises filtration. The symbol K is reserved for the fraction field.
Exact mathematical passage
F^m\cB=\pi^m\cB
F^{W(\gamma)}Use in source · 3 passages
Search aliases: F^W(γ) gamma
Use in source · Read passage · 03_weighted.tex, line 63
l, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
F^{W(\gamma)}\Bcal
F^\inftyUse in source · 1 passage
Search aliases: F^∞ infty
Use in source · Read passage · 03_weighted.tex, line 129
[Minimum path weight] For packets α and β, let d(α,β) be the least total weight of a directed path from α to β, with d=∞ when no path exists. Use the convention F^∞ B=⋂_L≥0F^L B=0. If α≠β, then P_β(D+N)^-1P_α ∈ F^d(α,β) B. If there is no path from α to β, the block is zero.
Exact mathematical passage
F^\infty\Bcal=\bigcap_{L\geq0}F^L\Bcal=0
\F_2Use in source · 1 passage
Search aliases: F_2 F mathbb
Use in source · Read passage · 04_arithmetic.tex, line 668
The two characteristic factors x-1 and x+1 have the same reduction in F_2[x]. The rational matrices E_+ and E_- themselves cannot be reduced modulo 2, because their entries contain 1/2. Consequently the integral special fibre does not inherit their separate idempotent splitting, even though the two projectors remain distinct over Q_2. This is the simplest model of delayed dyadic separation.
Exact mathematical passage
\F_2[x]
\mathbb F_2Use in source · 3 passages
Search aliases: F_2 mathbb
Use in source · visual_atlas.tex, line 133
vpage6Selectors survive without divisionMODEL A / RESIDUE MASKS IN CHARACTERISTIC TWOTraditional paper: Proposition 4.2 (p. 9) Work first in F_2, where 1+1=0. The eight coefficients below are just a finite display of a coefficient sequence. The even and odd projectors are diagonal masks with entries zero and one.
Exact mathematical passage
\mathbb F_2
\mathbb F_3Use in source · 3 passages
Search aliases: F_3 mathbb
Use in source · Read passage · 02_coordinates.tex, line 443
The field or orthogonality condition is essential. A primitive element of order N in a ring with zero divisors does not by itself force the geometric sums away from the diagonal to vanish. For example, in F_3× F_3 the element (-1,1) has order two, but the second component of its nontrivial character sum is nonzero. The coefficient selector uses no division and remains valid without any Fourier certificate. This distinction becomes decisive at the dyadic prime.
Exact mathematical passage
\mathbb F_3\times\mathbb F_3
\mathbb F_7Use in source · 1 passage
Search aliases: F_7 mathbb
Use in source · visual_atlas.tex, line 156
vpage7A Fourier certificate you can inspectMODEL A / AN EXACT NINE-ENTRY TABLETraditional paper: Theorem 4.3 (p. 10) Take N=3 and work in F_7. Set ω=2, ω^-1=4, η=3^-1=5. Here 2^3=1 and neither 2 nor 2^2 is one. The unnormalized geometric sums are the entries in the first matrix.
Exact mathematical passage
\mathbb F_7
f_iUse in source · 7 passages
Search aliases: f_i
Use in source · Read passage · 04_arithmetic.tex, line 56
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
\chi_A(x)=\prod_{i=1}^rf_i(x),
f_jUse in source · 2 passages
Search aliases: f_j
Use in source · Read passage · 04_arithmetic.tex, line 60
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
g_i=\prod_{j\ne i}f_j.
\FracUse in source · 1 passage
Search aliases: Frac operatorname
Use in source · Read passage · 04_arithmetic.tex, line 114
Let O be a DVR, K=Frac( O), and A∈End_ O(Λ) for a finite free lattice Λ. Suppose the characteristic polynomial factors over O as χ_A(x)=f(x)g(x), where f and g are coprime over K[x].
Exact mathematical passage
K=\Frac(\OO)
g_iUse in source · 4 passages
Search aliases: g_i
Use in source · Read passage · 04_arithmetic.tex, line 60
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
g_i=\prod_{j\ne i}f_j.
\operatorname{GL}Use in source · 1 passage
Search aliases: GL operatorname
Use in source · Read passage · 04_arithmetic.tex, line 705
[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].
Exact mathematical passage
U\in\operatorname{GL}_m(\OO), \qquad V\in\operatorname{GL}_n(\OO)
\operatorname{GL}_mUse in source · 1 passage
Search aliases: GL_m operatorname
Use in source · Read passage · 04_arithmetic.tex, line 705
[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].
Exact mathematical passage
U\in\operatorname{GL}_m(\OO), \qquad V\in\operatorname{GL}_n(\OO)
\operatorname{GL}_nUse in source · 1 passage
Search aliases: GL_n operatorname
Use in source · Read passage · 04_arithmetic.tex, line 705
[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].
Exact mathematical passage
U\in\operatorname{GL}_m(\OO), \qquad V\in\operatorname{GL}_n(\OO)
H_eUse in source · 4 passages
Search aliases: H_e
Use in source · Read passage · 03_weighted.tex, line 247
A packet decomposition resolves H=∑_e∈ EH_e and assigns a certified filtration gain D^-1H_e∈ F^w(e) B, w(e)∈ Z_>0. Each e is a directed transfer edge. For an ordered transfer word γ=(e_1,…,e_r), define W(γ)=∑_j=1^rw(e_j), A_γ=A_e_1⋯ A_e_r, A_e=D^-1H_e.
Exact mathematical passage
H=\sum_{e\in E}H_e
H_{e_1}Use in source · 1 passage
Search aliases: H_e_1
Use in source · Read passage · 03_weighted.tex, line 349
Here A_γ=(D^-1H_e_1)⋯(D^-1H_e_r) in the written order. Multiplicativity places it in F^W(γ)R, so the quotient kills it when W(γ)≥ L. Positive weights also imply |γ|≤ W(γ), making the retained search finite.
Exact mathematical passage
A_\gamma=(D^{-1}H_{e_1})\cdots(D^{-1}H_{e_r})
H_{e_r}Use in source · 1 passage
Search aliases: H_e_r
Use in source · Read passage · 03_weighted.tex, line 349
Here A_γ=(D^-1H_e_1)⋯(D^-1H_e_r) in the written order. Multiplicativity places it in F^W(γ)R, so the quotient kills it when W(γ)≥ L. Positive weights also imply |γ|≤ W(γ), making the retained search finite.
Exact mathematical passage
A_\gamma=(D^{-1}H_{e_1})\cdots(D^{-1}H_{e_r})
\hlineUse in source · 1 passage
Search aliases: hline
Use in source · visual_atlas.tex, line 464
The values are c|rrrrrr L · 1 · 2 · 3 · 4 · 5 · 6 W(L) · 2 · 3 · 4 · 4 · 4 · 4 # solutions · 4 · 8 · 16 · 16 · 16 · 16. Once L≥3, the nonzero weights have been fully exposed and the admissible fiber cardinality stabilizes at sixteen. The input space still grows as 2^3L. The number of attainable targets grows accordingly.
Exact mathematical passage
\begin{array}{c|rrrrrr} L&1&2&3&4&5&6\\\hline W(L)&2&3&4&4&4&4\\ \#\text{ solutions}&4&8&16&16&16&16. \end{array}
I_mUse in source · 1 passage
Search aliases: I_m
Use in source · Read passage · 02_coordinates.tex, line 69
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{m,\ell}:I_m\longrightarrow I_\ell \qquad(m\geq\ell)
I_{X^{\Pcal}}Use in source · 1 passage
Search aliases: I_X^Pcal I_X^ P Pcal mathcal
Use in source · Read passage · 02_coordinates.tex, line 104
[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
C_{\Pcal\leftarrow\Qcal}J_{\Qcal\leftarrow\Pcal}=I_{X^{\Pcal}}, \qquad J_{\Qcal\leftarrow\Pcal}C_{\Pcal\leftarrow\Qcal}=I_{X^{\Qcal}}.
I_{X^{\Qcal}}Use in source · 1 passage
Search aliases: I_X^Qcal I_X^ Q Qcal mathcal
Use in source · Read passage · 02_coordinates.tex, line 104
[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
C_{\Pcal\leftarrow\Qcal}J_{\Qcal\leftarrow\Pcal}=I_{X^{\Pcal}}, \qquad J_{\Qcal\leftarrow\Pcal}C_{\Pcal\leftarrow\Qcal}=I_{X^{\Qcal}}.
\idUse in source · 1 passage
Search aliases: id mathrm
Use in source · Read passage · 02_coordinates.tex, line 74
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{\ell,\ell}=\id_{I_\ell}, \qquad \rho_{n,\ell}=\rho_{m,\ell}\circ\rho_{n,m} \qquad(n\geq m\geq\ell).
\id_{I_\ell}Use in source · 1 passage
Search aliases: id_I_ℓ id ell mathrm ell
Use in source · Read passage · 02_coordinates.tex, line 74
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{\ell,\ell}=\id_{I_\ell}, \qquad \rho_{n,\ell}=\rho_{m,\ell}\circ\rho_{n,m} \qquad(n\geq m\geq\ell).
\iffUse in source · 1 passage
Search aliases: iff
Use in source · visual_atlas.tex, line 412
For a specific attainable target (2,4), 2x≡2pmod8iff x∈{1,5}, 4y≡4pmod8iff y∈{1,3,5,7}. The coordinate choices are independent, giving 2·4=8 solutions. The entire carrier is partitioned consistently: 8 attainable targets × 8 inputs per target=64. For an inadmissible target, such as (1,4) or (2,2), the fiber is empty. The same nonzero cardinality formula must not be applied to it.
Exact mathematical passage
2x\equiv2\pmod8\iff x\in\{1,5\},\qquad 4y\equiv4\pmod8\iff y\in\{1,3,5,7\}.
\imUse in source · 1 passage
Search aliases: im operatorname
Use in source · Read passage · 04_arithmetic.tex, line 64
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
\im E_i=\Ker f_i(A), \qquad \sum_iE_i=I, \qquad E_iE_j=0 \quad(i\ne j).
J^X_{P_b,P_a}Use in source · 1 passage
Search aliases: J^X_P_b,P_a
Use in source · Read passage · 02_coordinates.tex, line 193
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\boxed{ J^Y_{Q_b,Q_a}\,\widehat T_{Q_a,P_a} = \widehat T_{Q_b,P_b}\,J^X_{P_b,P_a}. }
J^Y_{Q_b,Q_a}Use in source · 1 passage
Search aliases: J^Y_Q_b,Q_a
Use in source · Read passage · 02_coordinates.tex, line 193
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\boxed{ J^Y_{Q_b,Q_a}\,\widehat T_{Q_a,P_a} = \widehat T_{Q_b,P_b}\,J^X_{P_b,P_a}. }
J_{4,2}Use in source · 4 passages
Search aliases: J_4,2
Use in source · visual_atlas.tex, line 113
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
\widehat T_N=R_NTR_N^{-1},\qquad J_{4,2}=R_4R_2^{-1}.
J_{8,2}Use in source · 1 passage
Search aliases: J_8,2
Use in source · visual_atlas.tex, line 117
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
J_{4,2}\widehat T_2=\widehat T_4J_{4,2},\qquad J_{8,4}J_{4,2}=J_{8,2}.
J_{8,4}Use in source · 1 passage
Search aliases: J_8,4
Use in source · visual_atlas.tex, line 117
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
J_{4,2}\widehat T_2=\widehat T_4J_{4,2},\qquad J_{8,4}J_{4,2}=J_{8,2}.
j_AUse in source · 3 passages
Search aliases: j_A
Use in source · Read passage · 02_coordinates.tex, line 500
[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.
Exact mathematical passage
j_A=j\pmod A, \qquad j_B=j\pmod B.
J_{a,a}Use in source · 1 passage
Search aliases: J_a,a
Use in source · Read passage · 02_coordinates.tex, line 177
Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children to their parents. Coordinate coherence itself does not require them.
Exact mathematical passage
J_{c,b}J_{b,a}=J_{c,a}, \qquad J_{a,a}=I, \qquad J_{b,a}^{-1}=J_{a,b}.
J_{a,b}Use in source · 1 passage
Search aliases: J_a,b
Use in source · Read passage · 02_coordinates.tex, line 177
Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children to their parents. Coordinate coherence itself does not require them.
Exact mathematical passage
J_{c,b}J_{b,a}=J_{c,a}, \qquad J_{a,a}=I, \qquad J_{b,a}^{-1}=J_{a,b}.
j_BUse in source · 3 passages
Search aliases: j_B
Use in source · Read passage · 02_coordinates.tex, line 500
[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.
Exact mathematical passage
j_A=j\pmod A, \qquad j_B=j\pmod B.
J_{b,a}Use in source · 1 passage
Search aliases: J_b,a
Use in source · Read passage · 02_coordinates.tex, line 177
Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children to their parents. Coordinate coherence itself does not require them.
Exact mathematical passage
J_{c,b}J_{b,a}=J_{c,a}, \qquad J_{a,a}=I, \qquad J_{b,a}^{-1}=J_{a,b}.
J_{b,a}^{-1}Use in source · 1 passage
Search aliases: J_b,a^-1
Use in source · Read passage · 02_coordinates.tex, line 177
Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children to their parents. Coordinate coherence itself does not require them.
Exact mathematical passage
J_{c,b}J_{b,a}=J_{c,a}, \qquad J_{a,a}=I, \qquad J_{b,a}^{-1}=J_{a,b}.
J_{c,a}Use in source · 1 passage
Search aliases: J_c,a
Use in source · Read passage · 02_coordinates.tex, line 177
Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children to their parents. Coordinate coherence itself does not require them.
Exact mathematical passage
J_{c,b}J_{b,a}=J_{c,a}, \qquad J_{a,a}=I, \qquad J_{b,a}^{-1}=J_{a,b}.
J_{c,b}Use in source · 1 passage
Search aliases: J_c,b
Use in source · Read passage · 02_coordinates.tex, line 177
Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children to their parents. Coordinate coherence itself does not require them.
Exact mathematical passage
J_{c,b}J_{b,a}=J_{c,a}, \qquad J_{a,a}=I, \qquad J_{b,a}^{-1}=J_{a,b}.
J_{\Rcal\leftarrow\Pcal}Use in source · 1 passage
Search aliases: J_Rcal←Pcal J_ R← P Rcal leftarrow Pcal mathcal leftarrow mathcal
Use in source · Read passage · 02_coordinates.tex, line 110
[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
J_{\Rcal\leftarrow\Qcal}J_{\Qcal\leftarrow\Pcal} = J_{\Rcal\leftarrow\Pcal},
J_{\Rcal\leftarrow\Qcal}Use in source · 1 passage
Search aliases: J_Rcal←Qcal J_ R← Q Rcal leftarrow Qcal mathcal leftarrow mathcal
Use in source · Read passage · 02_coordinates.tex, line 110
[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
J_{\Rcal\leftarrow\Qcal}J_{\Qcal\leftarrow\Pcal} = J_{\Rcal\leftarrow\Pcal},
kUse in source · 32 passages
Search aliases: k
Use in source · Read passage · 02_coordinates.tex, line 408
[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j, 0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.
Exact mathematical passage
\eta\sum_{k=0}^{N-1}(\omega^\vee)^{jk}\omega^{rk} = \begin{cases} 1,&r=j,\\ 0,&r\ne j, \end{cases} \qquad 0\le j,r<N.
K^\timesUse in source · 1 passage
Search aliases: K^× times
Use in source · Read passage · 04_arithmetic.tex, line 121
[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).
Exact mathematical passage
r\in K^\times
\KerUse in source · 10 passages
Search aliases: Ker operatorname
Use in source · Read passage · 04_arithmetic.tex, line 22
[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).
Exact mathematical passage
\Ker f(A)
\kerUse in source · 6 passages
Search aliases: ker
Use in source · Read passage · 04_arithmetic.tex, line 139
[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).
Exact mathematical passage
\ker F
lUse in source · 1 passage
Search aliases: l
Use in source · regrouping_not_averaging.tex, line 52
=.35pt,] (4.57000,2.26000) rectangle ++(0.57000,0.57000); node[] at (4.85500,2.54500) 2; draw[fill=black!0,draw=black!45,line width=.35pt,] (5.14000,2.26000) rectangle ++(0.57000,0.57000); node[] at (5.42500,2.54500) 0; draw[fill=black!12,draw=black!45,line width=.35pt,] (5.71000,2.26000) rectangle ++(0.57000,0.57000); node[] at (5.99500,2.54500) 4; node[] at (3.15000,2.20000) ∑_rm children; node[] at (1.14000,0.10000) 4×1=4 free coordinates; node[] at (5.14000,0.10000) 2×2=4 free coordinates; d…
Exact mathematical passage
\sum_{\rm children}
\lceilUse in source · 3 passages
Search aliases: lceil
Use in source · Read passage · 03_weighted.tex, line 338
, where A=∑_eA_e=D^-1H. With the supplied two-sided inverse D^-1 of D, this gives the explicit two-sided inverse q_L (S^pr_L,ND^-1) of q_L(D+H). If every edge weight is at least m and L≤ mN, then the nilpotence condition follows from the filtration data because m|γ|≤ W(γ). For m>0, the executable choice N=lceil L/mrceil satisfies L≤ mN. This elementary ceiling inequality is also proved in Lean. The choice m=1,N=L gives the standard cutoff. In the DVR example q_L:R→ R/π^L R, this reads q_L(T)^-1 …
Exact mathematical passage
N=\lceil L/m\rceil
\lengthUse in source · 1 passage
Search aliases: length operatorname
Use in source · Read passage · 04_arithmetic.tex, line 329
[Total gluing and worst precision] If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O, then length_ O(G_Λ, P) = ∑_j=1^ra_j, s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}). Thus the lattice index measures the total amount of gluing, while the separation weight measures the worst precision loss in one direction.
Exact mathematical passage
\length_{\OO}(G_{\Lambda,\Pcal}) = \sum_{j=1}^ra_j, \qquad s_\Lambda(\Pcal)=\max\bigl(\{0\}\cup\{a_j:1\leq j\leq r\}\bigr).
\operatorname{length}Use in source · 1 passage
Search aliases: length operatorname
Use in source · Read passage · 05_defects.tex, line 158
olomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z-T_f)^-1 dz. Consequently, Ch(T_f)-h(T_c)C · ≤ fraclength(Γ)2π sup_z∈Γ|h(z)| Δ · × sup_z∈Γ (z-T_c)^-1 sup_z∈Γ (z-T_f)^-1 .
Exact mathematical passage
\|Ch(T_f)-h(T_c)C\| &\leq \frac{\operatorname{length}(\Gamma)}{2\pi} \sup_{z\in\Gamma}|h(z)|\,\|\Delta\|\\ &\quad\times \sup_{z\in\Gamma}\|(z-T_c)^{-1}\| \sup_{z\in\Gamma}\|(z-T_f)^{-1}\|.
\length_{\OO}Use in source · 1 passage
Search aliases: length_OO length_ O length OO operatorname mathcal
Use in source · Read passage · 04_arithmetic.tex, line 329
[Total gluing and worst precision] If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O, then length_ O(G_Λ, P) = ∑_j=1^ra_j, s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}). Thus the lattice index measures the total amount of gluing, while the separation weight measures the worst precision loss in one direction.
Exact mathematical passage
\length_{\OO}(G_{\Lambda,\Pcal}) = \sum_{j=1}^ra_j, \qquad s_\Lambda(\Pcal)=\max\bigl(\{0\}\cup\{a_j:1\leq j\leq r\}\bigr).
\LongleftrightarrowUse in source · 3 passages
Search aliases: Longleftrightarrow
Use in source · Read passage · 04_arithmetic.tex, line 378
[Integral splitting criterion] Every packet projector preserves Λ if and only if Λ=⊕_α(Λ∩ V_α). Equivalently, s_Λ( P)=0 Longleftrightarrow G_Λ, P=0.
Exact mathematical passage
s_\Lambda(\Pcal)=0 \quad\Longleftrightarrow\quad G_{\Lambda,\Pcal}=0.
\lVertUse in source · 1 passage
Search aliases: lVert
Use in source · Read passage · 03_weighted.tex, line 281
lineskip [Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.
Exact mathematical passage
\lVert A\rVert<1
MUse in source · 22 passages
Search aliases: M
Use in source · Read passage · 02_coordinates.tex, line 279
[htbp] centering includegraphics[width=0.96textwidth]03_refinement_incidence.pdf captionThe incidence matrix M_r,s=1 when the fine residue s belongs to the coarse residue r. Its rows implement exact aggregation from fine packets to coarse packets. Common refinements.
Exact mathematical passage
M_{r,s}=1
M_{m,n}Use in source · 2 passages
Search aliases: M_m,n
Use in source · Read passage · 04_arithmetic.tex, line 704
[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].
Exact mathematical passage
M\in M_{m,n}(\OO)
M_{r,s}Use in source · 1 passage
Search aliases: M_r,s
Use in source · Read passage · 02_coordinates.tex, line 279
[htbp] centering includegraphics[width=0.96textwidth]03_refinement_incidence.pdf captionThe incidence matrix M_r,s=1 when the fine residue s belongs to the coarse residue r. Its rows implement exact aggregation from fine packets to coarse packets. Common refinements.
Exact mathematical passage
M_{r,s}=1
\midUse in source · 3 passages
Search aliases: mid
Use in source · Read passage · 04_arithmetic.tex, line 838
[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
p^{\min(s_i,L)}\mid (b_i)_{\mathrm{val}}
\mathbb NUse in source · 1 passage
Search aliases: N mathbb
Use in source · Read passage · 03_weighted.tex, line 445
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
w(e)=(w_1(e),\ldots,w_r(e))\in\mathbb N^r\setminus\{0\}.
N^{-1}Use in source · 1 passage
Search aliases: N^-1
Use in source · Read passage · 02_coordinates.tex, line 427
0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.
Exact mathematical passage
\eta=N^{-1}
N^2Use in source · 6 passages
Search aliases: N^2
Use in source · Read passage · 04_arithmetic.tex, line 130
[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).
Exact mathematical passage
\boxed{N^2=rN.}
\mathbb N^rUse in source · 1 passage
Search aliases: N^r mathbb
Use in source · Read passage · 03_weighted.tex, line 445
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
w(e)=(w_1(e),\ldots,w_r(e))\in\mathbb N^r\setminus\{0\}.
N_{01}Use in source · 2 passages
Search aliases: N_01
Use in source · Read passage · 01_orientation.tex, line 56
A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operator on one selected summand may also fail to intertwine with the full matrix. The later sections isolate these four effects one at a time.
Exact mathematical passage
X=X_0\oplus X_1, \qquad D=\begin{pmatrix}D_0&0\\0&D_1\end{pmatrix}, \qquad N=\begin{pmatrix}0&N_{01}\\N_{10}&0\end{pmatrix}.
N_{10}Use in source · 2 passages
Search aliases: N_10
Use in source · Read passage · 01_orientation.tex, line 56
A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operator on one selected summand may also fail to intertwine with the full matrix. The later sections isolate these four effects one at a time.
Exact mathematical passage
X=X_0\oplus X_1, \qquad D=\begin{pmatrix}D_0&0\\0&D_1\end{pmatrix}, \qquad N=\begin{pmatrix}0&N_{01}\\N_{10}&0\end{pmatrix}.
\notUse in source · 1 passage
Search aliases: not
Use in source · Read passage · 04_arithmetic.tex, line 464
be a rational projector on V, let Λ be a lattice, and let s=s_Λ(E). For every m≥0, the rule [x]_m+s⟼[Ex]_m from Λ/π^m+sΛ to V/π^mΛ is well defined. Here m is an integer. If d is an integer with 0≤ d<s, then for every integer m≥0 there are x,y∈Λ such that x≡ ypmodπ^m+dΛ but Exnot≡ Eypmodπ^mΛ. Thus s extra digits are both sufficient and necessary for a uniform precision guarantee.
Exact mathematical passage
Ex\not\equiv Ey\pmod{\pi^m\Lambda}.
o_mUse in source · 1 passage
Search aliases: o_m
Use in source · visual_atlas.tex, line 481
vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONTraditional paper: Section B.1 (p. 47) Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.
Exact mathematical passage
o_m=x_{2m+1}
\OO^2Use in source · 1 passage
Search aliases: OO^2 O^2 OO mathcal
Use in source · Read passage · 04_arithmetic.tex, line 103
[A resultant bound that is not exact] Let A=diag(0,π^s) on the standard lattice Λ= O^2. The factors x and x-π^s have resultant -π^s, up to the harmless choice of resultant order, but the two spectral projectors are the integral coordinate projectors. Their exact separation weight is zero. The resultant sees the closeness of the two eigenvalues. It does not see that the chosen lattice already splits along their eigenspaces.
Exact mathematical passage
\Lambda=\OO^2
p^LUse in source · 10 passages
Search aliases: p^L
Use in source · Read passage · 04_arithmetic.tex, line 832
[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
(\Z/(p^L)\Z)^d
p^{\min(s,L)}Use in source · 3 passages
Search aliases: p^min(s,L) min
Use in source · Read passage · 04_arithmetic.tex, line 856
Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fibre has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global count. If one coordinate fibre is empty, the global fibre is empty. Lean proves both the global solvability equivalence and the conditional count, inc…
Exact mathematical passage
\bigl(\exists x\in\mathbb Z/p^L\mathbb Z:\quad p^s x=b\pmod{p^L}\bigr) \quad\Longleftrightarrow\quad p^{\min(s,L)}\mid b_{\mathrm{val}},
p^{\min(s_i,L)}Use in source · 1 passage
Search aliases: p^min(s_i,L) min
Use in source · Read passage · 04_arithmetic.tex, line 838
[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
p^{\min(s_i,L)}\mid (b_i)_{\mathrm{val}}
p^sUse in source · 2 passages
Search aliases: p^s
Use in source · Read passage · 04_arithmetic.tex, line 856
Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fibre has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global count. If one coordinate fibre is empty, the global fibre is empty. Lean proves both the global solvability equivalence and the conditional count, inc…
Exact mathematical passage
\bigl(\exists x\in\mathbb Z/p^L\mathbb Z:\quad p^s x=b\pmod{p^L}\bigr) \quad\Longleftrightarrow\quad p^{\min(s,L)}\mid b_{\mathrm{val}},
p^{s_1}Use in source · 1 passage
Search aliases: p^s_1
Use in source · Read passage · 04_arithmetic.tex, line 833
[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
\diag(p^{s_1},\ldots,p^{s_d})x=b \pmod{p^L}.
p^{s_d}Use in source · 1 passage
Search aliases: p^s_d
Use in source · Read passage · 04_arithmetic.tex, line 833
[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
\diag(p^{s_1},\ldots,p^{s_d})x=b \pmod{p^L}.
p^{s_i}Use in source · 1 passage
Search aliases: p^s_i
Use in source · Read passage · 04_arithmetic.tex, line 886
If a Smith form is rank deficient, a literal zero diagonal factor requires the corresponding target coordinate to vanish modulo p^L and then contributes all p^L possible coordinate values. In the concrete quotient it can equivalently be represented by any weight s_i≥ L, since p^s_i=0 in Z/(p^L) Z. The theorem then gives exactly that condition and factor. The numbered theorem takes the diagonal data as its input.
Exact mathematical passage
p^{s_i}=0
p^{\sum_{i=1}^d\min(s_i,L)}Use in source · 1 passage
Search aliases: p^∑_i=1^dmin(s_i,L) sum min
Use in source · Read passage · 04_arithmetic.tex, line 844
ongruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
\boxed{ p^{\sum_{i=1}^d\min(s_i,L)}. }
P_0Use in source · 1 passage
Search aliases: P_0
Use in source · Read passage · 02_coordinates.tex, line 352
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
P_0=Q_{00}+Q_{01}, \qquad P_1=Q_{10}+Q_{11}
P_1Use in source · 1 passage
Search aliases: P_1
Use in source · Read passage · 02_coordinates.tex, line 352
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
P_0=Q_{00}+Q_{01}, \qquad P_1=Q_{10}+Q_{11}
P_{12,7}Use in source · 3 passages
Search aliases: P_12,7
Use in source · Read passage · 02_coordinates.tex, line 530
[Residues modulo twelve] The pair (jpmod4,jpmod3)=(3,1) corresponds to j=7 modulo 12. Hence P_12,7=P_4,3P_3,1. This factorization concerns addresses. It does not say that the two factors have the same integral spectral behavior.
Exact mathematical passage
P_{12,7}=P_{4,3}P_{3,1}.
P_{2,0}Use in source · 5 passages
Search aliases: P_2,0
Use in source · visual_atlas.tex, line 46
vpage3Follow one coefficient through the towerMODEL A / LOSSLESS ADDRESS REFINEMENTTraditional paper: Theorem 3.3 (p. 3), Theorem 4.6 (p. 11) Take a vector x=(x_0,…,x_15). A packet is not a single summary number: it retains every coefficient at its selected addresses. For example, P_2,0x=(x_0,0,x_2,0,…,x_14,0). The rail below lists the addresses in each packet. Its display order groups descendants under their parents. The matrix example on V5 instead uses increasing residue order.
Exact mathematical passage
P_{2,0}x=(x_0,0,x_2,0,\ldots,x_{14},0).
P_{2,1}Use in source · 2 passages
Search aliases: P_2,1
Use in source · division_free_masks.tex, line 36
75000,2.55000) rectangle ++(0.75000,0.60000); node[] at (4.12500,2.85000) 0; draw[fill=black!22,draw=black!45,line width=.35pt,] (4.50000,2.55000) rectangle ++(0.75000,0.60000); node[] at (4.87500,2.85000) 1; draw[fill=black!0,draw=black!45,line width=.35pt,] (5.25000,2.55000) rectangle ++(0.75000,0.60000); node[] at (5.62500,2.85000) 0; node[anchor=east] at (-0.20000,2.00000) P_2,1x; draw[fill=black!0,draw=black!45,line width=.35pt,] (0.00000,1.70000) rectangle ++(0.75000,0.60000); node[] at (0…
Exact mathematical passage
P_{2,1}x
P_{3,1}Use in source · 3 passages
Search aliases: P_3,1
Use in source · Read passage · 02_coordinates.tex, line 530
[Residues modulo twelve] The pair (jpmod4,jpmod3)=(3,1) corresponds to j=7 modulo 12. Hence P_12,7=P_4,3P_3,1. This factorization concerns addresses. It does not say that the two factors have the same integral spectral behavior.
Exact mathematical passage
P_{12,7}=P_{4,3}P_{3,1}.
P_{4,0}Use in source · 1 passage
Search aliases: P_4,0
Use in source · visual_atlas.tex, line 59
The address 6 is even, is 2 modulo 4, and is 6 modulo 8. Its coefficient therefore lies in the successive ranges P_8,6X⊆ P_4,2X⊆ P_2,0X. The corresponding identities hold as operators: P_2,0=P_4,0+P_4,2, P_4,2=P_8,2+P_8,6. Applying either composite parent selector to P_8,6x leaves it unchanged. Applying the odd selector kills it. This is absorption between a parent and a child, rather than an approximate resemblance between nearby scales.
Exact mathematical passage
P_{2,0}=P_{4,0}+P_{4,2},\qquad P_{4,2}=P_{8,2}+P_{8,6}.
P_{4,2}Use in source · 2 passages
Search aliases: P_4,2
Use in source · visual_atlas.tex, line 55
The address 6 is even, is 2 modulo 4, and is 6 modulo 8. Its coefficient therefore lies in the successive ranges P_8,6X⊆ P_4,2X⊆ P_2,0X. The corresponding identities hold as operators: P_2,0=P_4,0+P_4,2, P_4,2=P_8,2+P_8,6. Applying either composite parent selector to P_8,6x leaves it unchanged. Applying the odd selector kills it. This is absorption between a parent and a child, rather than an approximate resemblance between nearby scales.
Exact mathematical passage
P_{8,6}X\subseteq P_{4,2}X\subseteq P_{2,0}X.
P_{4,3}Use in source · 3 passages
Search aliases: P_4,3
Use in source · Read passage · 02_coordinates.tex, line 530
[Residues modulo twelve] The pair (jpmod4,jpmod3)=(3,1) corresponds to j=7 modulo 12. Hence P_12,7=P_4,3P_3,1. This factorization concerns addresses. It does not say that the two factors have the same integral spectral behavior.
Exact mathematical passage
P_{12,7}=P_{4,3}P_{3,1}.
P_{8,2}Use in source · 1 passage
Search aliases: P_8,2
Use in source · visual_atlas.tex, line 59
The address 6 is even, is 2 modulo 4, and is 6 modulo 8. Its coefficient therefore lies in the successive ranges P_8,6X⊆ P_4,2X⊆ P_2,0X. The corresponding identities hold as operators: P_2,0=P_4,0+P_4,2, P_4,2=P_8,2+P_8,6. Applying either composite parent selector to P_8,6x leaves it unchanged. Applying the odd selector kills it. This is absorption between a parent and a child, rather than an approximate resemblance between nearby scales.
Exact mathematical passage
P_{2,0}=P_{4,0}+P_{4,2},\qquad P_{4,2}=P_{8,2}+P_{8,6}.
P_{8,6}Use in source · 3 passages
Search aliases: P_8,6
Use in source · visual_atlas.tex, line 55
The address 6 is even, is 2 modulo 4, and is 6 modulo 8. Its coefficient therefore lies in the successive ranges P_8,6X⊆ P_4,2X⊆ P_2,0X. The corresponding identities hold as operators: P_2,0=P_4,0+P_4,2, P_4,2=P_8,2+P_8,6. Applying either composite parent selector to P_8,6x leaves it unchanged. Applying the odd selector kills it. This is absorption between a parent and a child, rather than an approximate resemblance between nearby scales.
Exact mathematical passage
P_{8,6}X\subseteq P_{4,2}X\subseteq P_{2,0}X.
P_aUse in source · 6 passages
Search aliases: P_a
Use in source · Read passage · 02_coordinates.tex, line 186
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
P_a,P_b
P_{A,j_A}Use in source · 1 passage
Search aliases: P_A,j_A
Use in source · Read passage · 02_coordinates.tex, line 506
[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.
Exact mathematical passage
P_{AB,j}=P_{A,j_A}P_{B,j_B}.
P_{AB,j}Use in source · 1 passage
Search aliases: P_AB,j
Use in source · Read passage · 02_coordinates.tex, line 506
[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.
Exact mathematical passage
P_{AB,j}=P_{A,j_A}P_{B,j_B}.
P_bUse in source · 6 passages
Search aliases: P_b
Use in source · Read passage · 02_coordinates.tex, line 186
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
P_a,P_b
P_{B,j_B}Use in source · 1 passage
Search aliases: P_B,j_B
Use in source · Read passage · 02_coordinates.tex, line 506
[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.
Exact mathematical passage
P_{AB,j}=P_{A,j_A}P_{B,j_B}.
P_jUse in source · 1 passage
Search aliases: P_j
Use in source · finite_fourier_table.tex, line 22
000,0.70000); node[] at (0.35000,1.45000) 0; draw[fill=black!0,draw=black!45,line width=.35pt,] (0.70000,1.10000) rectangle ++(0.70000,0.70000); node[] at (1.05000,1.45000) 0; draw[fill=black!12,draw=black!45,line width=.35pt,] (1.40000,1.10000) rectangle ++(0.70000,0.70000); node[] at (1.75000,1.45000) 3; node[] at (3.70000,2.05000) × 5; node[] at (6.05000,3.75000) P_j(z^r): 5=3^-1pmod7; draw[fill=black!12,draw=black!45,line width=.35pt,] (5.00000,2.50000) rectangle ++(0.70000,0.70000); node[] …
Exact mathematical passage
P_j(z^r):\ 5=3^{-1}\pmod7
P_{M,s}Use in source · 1 passage
Search aliases: P_M,s
Use in source · Read passage · 02_coordinates.tex, line 478
[Nested residue refinement] Let N,M be positive integers with N dividing M, and let 0≤ j<N. Then P_N,j = ∑_substack0≤ s<M s≡ jpmod NP_M,s. When R is nonzero, the packet system modulo M refines the packet system modulo N.
Exact mathematical passage
P_{N,j} = \sum_{\substack{0\leq s<M\\s\equiv j\pmod N}}P_{M,s}.
P_{N,i}Use in source · 3 passages
Search aliases: P_N,i
Use in source · Read passage · 02_coordinates.tex, line 383
[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.
Exact mathematical passage
P_{N,i}P_{N,j}=\delta_{ij}P_{N,j}, \qquad \sum_{j=0}^{N-1}P_{N,j}=I.
P_{N,j+1}Use in source · 1 passage
Search aliases: P_N,j+1
Use in source · Read passage · 02_coordinates.tex, line 491
[The shift graph of residue packets] Assume that R is nonzero and N>0. Multiplication by z sends the residue packet P_N,jR[[z]] into P_N,j+1R[[z]], where the index is read modulo N. Its packet graph is the directed cycle 0⟶1⟶⋯⟶ N-1⟶0. Multiplication by a polynomial supported on a set of residues gives the corresponding finite set of phase shifts. This is the simplest transfer graph behind a polyphase representation.
Exact mathematical passage
P_{N,j+1}R[[z]]
P_{N,j+r}Use in source · 1 passage
Search aliases: P_N,j+r
Use in source · Read passage · 06_models.tex, line 24
A banded operator need not remain banded after residue reordering, but its nonzero couplings become a structured packet graph. With the convention (S_rx)_n=x_n-r, the only active transfer is P_N,j+rS_rP_N,j, with indices taken modulo N. More general finite stencils produce a finite set of active phase differences.
Exact mathematical passage
P_{N,j+r}S_rP_{N,j},
P_{s,t}Use in source · 4 passages
Search aliases: P_s,t
Use in source · Read passage · 03_weighted.tex, line 507
w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, hence Q≠0 and Q is a unit in Z[[t]], and the independently enumerated walk series satisfies Z_s,t(t)=Q(t)^-1P_s,t(t), Q(t)Z_s,t(t)=P_s,t(t) in Z[[t]].
Exact mathematical passage
T(t)=I-A(t),\qquad Q(t)=\det T(t),\qquad P_{s,t}(t)=\operatorname{adj}(T(t))_{s,t}.
P_{\alpha'}Use in source · 1 passage
Search aliases: P_α' alpha
Use in source · Read passage · 02_coordinates.tex, line 303
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
P_\alpha P_{\alpha'}=0
P_\alpha^2Use in source · 1 passage
Search aliases: P_α^2 alpha
Use in source · Read passage · 02_coordinates.tex, line 300
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
R_{\alpha\beta}^2=P_\alpha^2Q_\beta^2=R_{\alpha\beta}.
P_{\rho(\beta)}Use in source · 1 passage
Search aliases: P_ρ(β) rho beta
Use in source · Read passage · 02_coordinates.tex, line 57
0 · ρ(β)≠α. Hence Q_β X⊆ P_ρ(β)X. If a third packet system R refines Q, then R refines P through the composite parent map.
Exact mathematical passage
Q_\beta X\subseteq P_{\rho(\beta)}X
\Pcal_mUse in source · 1 passage
Search aliases: Pcal_m P_m Pcal mathcal
Use in source · Read passage · 02_coordinates.tex, line 73
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\Pcal_m
\Pcal_\ellUse in source · 2 passages
Search aliases: Pcal_ℓ P_ℓ Pcal ell mathcal ell
Use in source · Read passage · 02_coordinates.tex, line 68
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
(\Pcal_\ell)_{\ell\geq0}
\mathrm{pr}Use in source · 3 passages
Search aliases: pr mathrm
Use in source · Read passage · 03_weighted.tex, line 321
et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is…
Exact mathematical passage
S^{\mathrm{pr}}_{L,N} = \sum_{n=0}^{N-1}(-1)^n \sum_{\substack{\gamma\in E^n\\W(\gamma)<L}}A_\gamma.
\QUse in source · 1 passage
Search aliases: Q mathbb
Use in source · Read passage · 04_arithmetic.tex, line 671
The two characteristic factors x-1 and x+1 have the same reduction in F_2[x]. The rational matrices E_+ and E_- themselves cannot be reduced modulo 2, because their entries contain 1/2. Consequently the integral special fibre does not inherit their separate idempotent splitting, even though the two projectors remain distinct over Q_2. This is the simplest model of delayed dyadic separation.
Exact mathematical passage
\Q_2
\mathbb QUse in source · 2 passages
Search aliases: Q mathbb
Use in source · Read passage · 04_arithmetic.tex, line 562
0 · -1 act on Z_p^2. Over Q_p, the sign projectors are E_+=fracI+A2 = 1 · 1/2
Exact mathematical passage
\mathbb Q_p
qUse in source · 21 passages
Search aliases: q
Use in source · Read passage · 03_weighted.tex, line 311
[Finite precision] Let F^bullet R be a multiplicative decreasing filtration and let q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n
Exact mathematical passage
q_L:R\to R_L
q^{(n-r)L+\sum_{i=1}^r\min(s_i,L)}Use in source · 1 passage
Search aliases: q^(n-r)L+∑_i=1^rmin(s_i,L) sum min
Use in source · Read passage · 04_arithmetic.tex, line 770
rdinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L (r<i≤ m). When it is solvable, the number of solutions in ( O/π^L O)^n is q^(n-r)L+∑_i=1^rmin(s_i,L).
Exact mathematical passage
\boxed{ q^{(n-r)L+\sum_{i=1}^r\min(s_i,L)}. }
q^LUse in source · 1 passage
Search aliases: q^L
Use in source · Read passage · 04_arithmetic.tex, line 787
ultiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L choices. Multiplying the independent counts gives the formula.
Exact mathematical passage
q^L
q^{\min(s_i,L)}Use in source · 1 passage
Search aliases: q^min(s_i,L) min
Use in source · Read passage · 04_arithmetic.tex, line 784
Set x=Vy. Multiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L choices. Multiplying the independent counts gives the formula.
Exact mathematical passage
q^{\min(s_i,L)}.
q^{\sum_i\min(s_i,L)}Use in source · 1 passage
Search aliases: q^∑_imin(s_i,L) sum min
Use in source · Read passage · 04_arithmetic.tex, line 871
For an arbitrary DVR O with residue field cardinality q, the analogous formula q^∑_imin(s_i,L) follows from the solvability and fibre count for each coordinate. The current revision adds direct proofs of this DVR coordinate arithmetic; the current correspondence records those targets and the rectangular transport separately. The inherited abstract product interface remains available when the coordinate facts are supplied as certificates. A Smith form and its changes of basis are inputs to the tr…
Exact mathematical passage
q^{\sum_i\min(s_i,L)}
Q_{00}Use in source · 1 passage
Search aliases: Q_00
Use in source · Read passage · 02_coordinates.tex, line 352
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
P_0=Q_{00}+Q_{01}, \qquad P_1=Q_{10}+Q_{11}
Q_{01}Use in source · 1 passage
Search aliases: Q_01
Use in source · Read passage · 02_coordinates.tex, line 352
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
P_0=Q_{00}+Q_{01}, \qquad P_1=Q_{10}+Q_{11}
q_1Use in source · 2 passages
Search aliases: q_1
Use in source · Read passage · 03_weighted.tex, line 422
[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.
Exact mathematical passage
Z_{u,v}(t)=\frac{P(t)}{Q(t)}, \qquad Q(t)=1+q_1t+\cdots+q_dt^d.
Q_{10}Use in source · 1 passage
Search aliases: Q_10
Use in source · Read passage · 02_coordinates.tex, line 352
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
P_0=Q_{00}+Q_{01}, \qquad P_1=Q_{10}+Q_{11}
Q_{11}Use in source · 1 passage
Search aliases: Q_11
Use in source · Read passage · 02_coordinates.tex, line 352
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
P_0=Q_{00}+Q_{01}, \qquad P_1=Q_{10}+Q_{11}
\Q_2Use in source · 1 passage
Search aliases: Q_2 Q mathbb
Use in source · Read passage · 04_arithmetic.tex, line 671
The two characteristic factors x-1 and x+1 have the same reduction in F_2[x]. The rational matrices E_+ and E_- themselves cannot be reduced modulo 2, because their entries contain 1/2. Consequently the integral special fibre does not inherit their separate idempotent splitting, even though the two projectors remain distinct over Q_2. This is the simplest model of delayed dyadic separation.
Exact mathematical passage
\Q_2
\mathbb Q_2Use in source · 1 passage
Search aliases: Q_2 mathbb
Use in source · Read passage · 04_arithmetic.tex, line 590
Modulo 2, the two eigenvalues become equal. The rational sign packets remain distinct over Q_2, but the standard lattice does not split into their intersections until one more dyadic digit is retained.
Exact mathematical passage
\mathbb Q_2
Q_aUse in source · 2 passages
Search aliases: Q_a
Use in source · Read passage · 02_coordinates.tex, line 186
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
Q_a,Q_b
Q_bUse in source · 2 passages
Search aliases: Q_b
Use in source · Read passage · 02_coordinates.tex, line 186
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
Q_a,Q_b
q_dUse in source · 2 passages
Search aliases: q_d
Use in source · Read passage · 03_weighted.tex, line 422
[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.
Exact mathematical passage
Z_{u,v}(t)=\frac{P(t)}{Q(t)}, \qquad Q(t)=1+q_1t+\cdots+q_dt^d.
q_kUse in source · 2 passages
Search aliases: q_k
Use in source · Read passage · 03_weighted.tex, line 557
[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).
Exact mathematical passage
Q(t)=\sum_kq_kt^k, \qquad Z_{s,t}(t)=\sum_{n\ge0}a_n(s,t)t^n.
q_LUse in source · 11 passages
Search aliases: q_L
Use in source · Read passage · 03_weighted.tex, line 311
[Finite precision] Let F^bullet R be a multiplicative decreasing filtration and let q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n
Exact mathematical passage
q_L:R\to R_L
\mathbb Q_pUse in source · 1 passage
Search aliases: Q_p mathbb
Use in source · Read passage · 04_arithmetic.tex, line 562
0 · -1 act on Z_p^2. Over Q_p, the sign projectors are E_+=fracI+A2 = 1 · 1/2
Exact mathematical passage
\mathbb Q_p
Q_\alphaUse in source · 3 passages
Search aliases: Q_α alpha
Use in source · Read passage · 02_coordinates.tex, line 240
Expand the two coarse projectors into their children: P_bTP_a = (∑_ρ(β)=bQ_β) T (∑_ρ(α)=aQ_α). Distributing the finite sums gives the first identity. For the coordinate identity, use [T]_ P= A_ PT S_ P and insert I_X= S_ Q A_ Q.
Exact mathematical passage
P_bTP_a = \left(\sum_{\rho(\beta)=b}Q_\beta\right) T \left(\sum_{\rho(\alpha)=a}Q_\alpha\right).
Q_{\beta'}Use in source · 1 passage
Search aliases: Q_β' beta
Use in source · Read passage · 02_coordinates.tex, line 303
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
Q_\beta Q_{\beta'}=0
Q_\beta^2Use in source · 1 passage
Search aliases: Q_β^2 beta
Use in source · Read passage · 02_coordinates.tex, line 300
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
R_{\alpha\beta}^2=P_\alpha^2Q_\beta^2=R_{\alpha\beta}.
Q_\gammaUse in source · 3 passages
Search aliases: Q_γ gamma
Use in source · Read passage · 02_coordinates.tex, line 62
Multiply the identity P_α=∑_ρ(γ)=αQ_γ by Q_β on either side. Pairwise annihilation in the fine family leaves Q_β when ρ(β)=α and leaves zero otherwise. For transitivity, substitute the child expansion of each Q_β into the expansion of P_α and group the resulting terms by the composite parent.
Exact mathematical passage
P_\alpha=\sum_{\rho(\gamma)=\alpha}Q_\gamma
\QQ_2Use in source · 1 passage
Search aliases: QQ_2 QQ
Use in source · visual_atlas.tex, line 353
vfig[.94]dyadic_eigenlattice platecaptionAn ordinary integer-lattice picture, not a drawing of the topology of QQ_2. Filled dots have even second coordinate. Unfilled dots lie in the other coset. The hollow point at (1/2,0) records the image of an integral vector under E_+.
Exact mathematical passage
\QQ_2
r^{-1}Use in source · 4 passages
Search aliases: r^-1
Use in source · Read passage · 04_arithmetic.tex, line 19
[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).
Exact mathematical passage
E_f=r^{-1}v(A)g(A)
r^{-2}Use in source · 1 passage
Search aliases: r^-2
Use in source · Read passage · 04_arithmetic.tex, line 156
The B ezout identity gives N=rI-UF. The two stated commutation relations and FG=0 give N(UF)=VGUF=VUGF=VUFG=0. Therefore N^2=N(rI-UF)=rN. Since r≠0, E^2=r^-2N^2=E. If Fx=0, the evaluated B ezout identity gives VGx=rx, hence Ex=x. If Gx=0, then Ex=0. The lattice assertion follows directly from rE=VG.
Exact mathematical passage
E^2=r^{-2}N^2=E
R_2^{-1}Use in source · 1 passage
Search aliases: R_2^-1
Use in source · visual_atlas.tex, line 113
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
\widehat T_N=R_NTR_N^{-1},\qquad J_{4,2}=R_4R_2^{-1}.
R_4Use in source · 1 passage
Search aliases: R_4
Use in source · visual_atlas.tex, line 113
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
\widehat T_N=R_NTR_N^{-1},\qquad J_{4,2}=R_4R_2^{-1}.
R_cUse in source · 5 passages
Search aliases: R_c
Use in source · Read passage · 05_defects.tex, line 119
Inverse witnesses from one side and norm control. For continuous linear maps, the same identity gives the certified norm bound C R_f-R_c C ≤ R_c Δ R_f , from a left inverse witness for the coarse scalar shift and a right inverse witness for the fine scalar shift. These witnesses suffice for the displayed identity. On their own, they need not define inverses on both sides and hence need not define resolvents. If Δ=0, Lean also proves, for every finite polynomial p, the exact compatibility C p(T_f…
Exact mathematical passage
\boxed{ \|C R_f-R_c C\| \le \|R_c\|\,\|\Delta\|\,\|R_f\|, }
R_fUse in source · 5 passages
Search aliases: R_f
Use in source · Read passage · 05_defects.tex, line 119
Inverse witnesses from one side and norm control. For continuous linear maps, the same identity gives the certified norm bound C R_f-R_c C ≤ R_c Δ R_f , from a left inverse witness for the coarse scalar shift and a right inverse witness for the fine scalar shift. These witnesses suffice for the displayed identity. On their own, they need not define inverses on both sides and hence need not define resolvents. If Δ=0, Lean also proves, for every finite polynomial p, the exact compatibility C p(T_f…
Exact mathematical passage
\boxed{ \|C R_f-R_c C\| \le \|R_c\|\,\|\Delta\|\,\|R_f\|, }
R_LUse in source · 5 passages
Search aliases: R_L
Use in source · Read passage · 03_weighted.tex, line 187
[Exact finite precision truncation] Under the hypotheses of [source reference], define R_L = ∑_W(γ)<L(-1)^|γ|A_γ D^-1. Then T^-1-R_L∈ F^L B, TR_L-I∈ F^L B, R_LT-I∈ F^L B. Thus R_L is the exact inverse in B/F^L B.
Exact mathematical passage
R_L = \sum_{W(\gamma)<L}(-1)^{|\gamma|}A_\gamma D^{-1}.
R_N^{-1}Use in source · 1 passage
Search aliases: R_N^-1
Use in source · visual_atlas.tex, line 113
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
\widehat T_N=R_NTR_N^{-1},\qquad J_{4,2}=R_4R_2^{-1}.
R_{\alpha\beta}Use in source · 4 passages
Search aliases: R_αβ alpha beta
Use in source · Read passage · 02_coordinates.tex, line 292
[Common refinement of commuting packet systems] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on X. Assume P_α Q_β=Q_β P_α for all α and β. Remove the zero operators from the family R_αβ=P_α Q_β. The remaining operators form a packet system that refines both P and Q. The same statement holds for any finite collection of pairwise commuting packet systems.
Exact mathematical passage
R_{\alpha\beta}=P_\alpha Q_\beta.
R_{\alpha\beta}^2Use in source · 1 passage
Search aliases: R_αβ^2 alpha beta
Use in source · Read passage · 02_coordinates.tex, line 300
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
R_{\alpha\beta}^2=P_\alpha^2Q_\beta^2=R_{\alpha\beta}.
R_\gammaUse in source · 6 passages
Search aliases: R_γ gamma
Use in source · Read passage · 02_coordinates.tex, line 326
Let R_γ be a packet in R. Because R refines P, there is a unique α with P_α R_γ=R_γ. Likewise there is a unique β with Q_β R_γ=R_γ. Therefore P_α Q_β R_γ=R_γ. Every packet of R lies in one product packet. Multiplying ∑_γ R_γ=I_X by P_α Q_β and using the parent identities gives P_α Q_β =∑_substackγ: P_α R_γ=R_γ
Exact mathematical passage
R_\gamma
\RcalUse in source · 10 passages
Search aliases: Rcal R mathcal
Use in source · Read passage · 02_coordinates.tex, line 57
0 · ρ(β)≠α. Hence Q_β X⊆ P_ρ(β)X. If a third packet system R refines Q, then R refines P through the composite parent map.
Exact mathematical passage
\Rcal
\rceilUse in source · 3 passages
Search aliases: rceil
Use in source · Read passage · 03_weighted.tex, line 338
, where A=∑_eA_e=D^-1H. With the supplied two-sided inverse D^-1 of D, this gives the explicit two-sided inverse q_L (S^pr_L,ND^-1) of q_L(D+H). If every edge weight is at least m and L≤ mN, then the nilpotence condition follows from the filtration data because m|γ|≤ W(γ). For m>0, the executable choice N=lceil L/mrceil satisfies L≤ mN. This elementary ceiling inequality is also proved in Lean. The choice m=1,N=L gives the standard cutoff. In the DVR example q_L:R→ R/π^L R, this reads q_L(T)^-1 …
Exact mathematical passage
N=\lceil L/m\rceil
\ResUse in source · 4 passages
Search aliases: Res operatorname
Use in source · Read passage · 04_arithmetic.tex, line 73
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
\Res(f_i,g_i)E_i\in\End_{\OO}(\Lambda)
\operatorname{Res}Use in source · 1 passage
Search aliases: Res operatorname
Use in source · visual_atlas.tex, line 345
For the original sign factors the concrete resultant is Res(x-1,x+1)=2 and the bound is sharp. That equality is special to this displayed calculation, not a universal minimality theorem for every resultant certificate.
Exact mathematical passage
\operatorname{Res}(x-1,x+1)=2
\rVertUse in source · 1 passage
Search aliases: rVert
Use in source · Read passage · 03_weighted.tex, line 281
lineskip [Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.
Exact mathematical passage
\lVert A\rVert<1
\mathcal SUse in source · 3 passages
Search aliases: S mathcal
Use in source · visual_atlas.tex, line 27
A_N, S_N · Split into range-restricted packet vectors, then sum them. · A permutation/interleaving of eight coordinates.
Exact mathematical passage
\mathcal A_N,\mathcal S_N
S^{\mathrm{pr}}_{L,N}Use in source · 3 passages
Search aliases: S^pr_L,N mathrm
Use in source · Read passage · 03_weighted.tex, line 321
et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is…
Exact mathematical passage
S^{\mathrm{pr}}_{L,N} = \sum_{n=0}^{N-1}(-1)^n \sum_{\substack{\gamma\in E^n\\W(\gamma)<L}}A_\gamma.
S^rUse in source · 1 passage
Search aliases: S^r
Use in source · Read passage · 06_models.tex, line 6
Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer blocks. Refining from N to M with Nmid M splits each coarse phase into its descendants without changing the reconstructed signal. The same formulas extend…
Exact mathematical passage
(S^r\delta_0)_n=\delta_{n,r}
s_0Use in source · 2 passages
Search aliases: s_0
Use in source · visual_atlas.tex, line 453
vpage19Precision exposes directional freedomMODEL E / A DEPTH-BY-DEPTH FIBER BUDGETTraditional paper: Theorem 10.6 (p. 36) Take p=2 and the supplied diagonal weights (s_0,s_1,s_2)=(0,1,3). At depth L, an admissible target has 2^W(L) solutions, W(L)=min(0,L)+min(1,L)+min(3,L). Each direction contributes only the amount visible at the current depth.
Exact mathematical passage
(s_0,s_1,s_2)=(0,1,3)
s_1Use in source · 8 passages
Search aliases: s_1
Use in source · Read passage · 04_arithmetic.tex, line 711
[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].
Exact mathematical passage
UMV = \diag(\pi^{s_1},\ldots,\pi^{s_r},0,\ldots,0),
\mathcal S_2Use in source · 1 passage
Search aliases: S_2 mathcal
Use in source · visual_atlas.tex, line 87
Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.
Exact mathematical passage
J=\mathcal A_4\mathcal S_2,\qquad C=\mathcal A_2\mathcal S_4, \qquad CJ=I,\quad JC=I.
s_2Use in source · 3 passages
Search aliases: s_2
Use in source · Read passage · 04_arithmetic.tex, line 580
0 · 1 . For every odd prime p, both projectors preserve Z_p^2. At p=2, s_2(E_+)=s_2(E_-)=1. The factors x-1 and x+1 have resultant 2.
Exact mathematical passage
s_2(E_+)=s_2(E_-)=1.
s_{2,\Z_2^2}Use in source · 1 passage
Search aliases: s_2,Z_2^2 s_2, Z_2^2 Z mathbb
Use in source · Read passage · 04_arithmetic.tex, line 650
[Odd-adic versus dyadic behavior] For every prime p≠2, every entry of E_+ and E_- is p-integral, and each projector has exact local denominator weight zero. At p=2, each projector has exact local denominator weight one. Multiplication by 2 makes it integral, while the unscaled matrix is not integral: s_2, Z_2^2(E_+)=s_2, Z_2^2(E_-)=1. The characteristic factors x-1 and x+1 have resultant 2. Its 2-adic valuation is one, exactly the two local projector weights above, and its p-adic valuation is ze…
Exact mathematical passage
s_{2,\Z_2^2}(E_+)=s_{2,\Z_2^2}(E_-)=1.
\mathcal S_4Use in source · 1 passage
Search aliases: S_4 mathcal
Use in source · visual_atlas.tex, line 87
Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.
Exact mathematical passage
J=\mathcal A_4\mathcal S_2,\qquad C=\mathcal A_2\mathcal S_4, \qquad CJ=I,\quad JC=I.
s_dUse in source · 1 passage
Search aliases: s_d
Use in source · Read passage · 04_arithmetic.tex, line 833
[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
\diag(p^{s_1},\ldots,p^{s_d})x=b \pmod{p^L}.
s_iUse in source · 15 passages
Search aliases: s_i
Use in source · Read passage · 04_arithmetic.tex, line 758
[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L (r<i≤ m). When it is solvable, the number of solutions in ( O/π^L O)^n is q^(n-r)L+∑_i=1^rmin(s_i,L).
Exact mathematical passage
\widetilde b_i\equiv0 \pmod{\pi^{\min(s_i,L)}} \qquad(1\leq i\leq r)
S_NUse in source · 2 passages
Search aliases: S_N
Use in source · visual_atlas.tex, line 250
vpage11The residual tells you when to stopMODEL B / A CERTIFIED CUTOFF, NOT A HEURISTICTraditional paper: Theorem 5.10 (p. 17), Corollary 5.11 (p. 18), Section B.2 (p. 47) For S_N=∑_k=0^N-1(-A)^k, telescoping gives (I+A)S_N=S_N(I+A)=I-(-A)^N. Thus the exact residual is known. To certify precision L, it is enough to show that every residual entry is divisible by 2^L.
Exact mathematical passage
S_N=\sum_{k=0}^{N-1}(-A)^k
\mathcal S_NUse in source · 2 passages
Search aliases: S_N mathcal
Use in source · visual_atlas.tex, line 27
A_N, S_N · Split into range-restricted packet vectors, then sum them. · A permutation/interleaving of eight coordinates.
Exact mathematical passage
\mathcal A_N,\mathcal S_N
S_rUse in source · 2 passages
Search aliases: S_r
Use in source · Read passage · 06_models.tex, line 23
A banded operator need not remain banded after residue reordering, but its nonzero couplings become a structured packet graph. With the convention (S_rx)_n=x_n-r, the only active transfer is P_N,j+rS_rP_N,j, with indices taken modulo N. More general finite stencils produce a finite set of active phase differences.
Exact mathematical passage
(S_rx)_n=x_{n-r}
s_rUse in source · 5 passages
Search aliases: s_r
Use in source · Read passage · 04_arithmetic.tex, line 711
[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].
Exact mathematical passage
UMV = \diag(\pi^{s_1},\ldots,\pi^{s_r},0,\ldots,0),
s_{\Lambda_s}Use in source · 1 passage
Search aliases: s_Λ_s Lambda
Use in source · Read passage · 04_arithmetic.tex, line 522
with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Only the lattice has changed.
Exact mathematical passage
s_{\Lambda_s}(E)=s.
\mathrm{Smith}Use in source · 1 passage
Search aliases: Smith mathrm
Use in source · Read passage · 06_models.tex, line 79
The symmetric weight w_Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2 indicates where small errors in reflection coordinates produce large impedance or admittance errors.
Exact mathematical passage
w_{\mathrm{Smith}}(\Gamma) = \frac1{|1-\Gamma|^2} + \frac1{|1+\Gamma|^2}
\mathrm{source}Use in source · 1 passage
Search aliases: source mathrm
Use in source · visual_atlas.tex, line 289
headThe first-edge decomposition Let a_n(s,t) count walks of total weight n from s to t. Removing the first edge gives a_n(s,t)=1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). The empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most n, so enumeration at each fixed weight is finite.
Exact mathematical passage
a_n(s,t)=\mathbf1_{s=t,\,n=0} +\sum_{\substack{e:\,\mathrm{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\mathrm{target}(e),t).
\operatorname{source}Use in source · 1 passage
Search aliases: source operatorname
Use in source · Read passage · 03_weighted.tex, line 499
[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_n(s,t) = 1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, henc…
Exact mathematical passage
a_n(s,t) = \mathbf 1_{s=t,\,n=0} +\sum_{\substack{e:\,\operatorname{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\operatorname{target}(e),t).
\supUse in source · 1 passage
Search aliases: sup
Use in source · Read passage · 05_defects.tex, line 158
olomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z-T_f)^-1 dz. Consequently, Ch(T_f)-h(T_c)C · ≤ fraclength(Γ)2π sup_z∈Γ|h(z)| Δ · × sup_z∈Γ (z-T_c)^-1 sup_z∈Γ (z-T_f)^-1 .
Exact mathematical passage
\|Ch(T_f)-h(T_c)C\| &\leq \frac{\operatorname{length}(\Gamma)}{2\pi} \sup_{z\in\Gamma}|h(z)|\,\|\Delta\|\\ &\quad\times \sup_{z\in\Gamma}\|(z-T_c)^{-1}\| \sup_{z\in\Gamma}\|(z-T_f)^{-1}\|.
\sup_{z\in\Gamma}Use in source · 1 passage
Search aliases: sup_z∈Γ sup in Gamma
Use in source · Read passage · 05_defects.tex, line 158
olomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z-T_f)^-1 dz. Consequently, Ch(T_f)-h(T_c)C · ≤ fraclength(Γ)2π sup_z∈Γ|h(z)| Δ · × sup_z∈Γ (z-T_c)^-1 sup_z∈Γ (z-T_f)^-1 .
Exact mathematical passage
\|Ch(T_f)-h(T_c)C\| &\leq \frac{\operatorname{length}(\Gamma)}{2\pi} \sup_{z\in\Gamma}|h(z)|\,\|\Delta\|\\ &\quad\times \sup_{z\in\Gamma}\|(z-T_c)^{-1}\| \sup_{z\in\Gamma}\|(z-T_f)^{-1}\|.
\operatorname{synthesize}Use in source · 1 passage
Search aliases: synthesize operatorname
Use in source · Read passage · 06_models.tex, line 214
The verified contract is synthesize(analyze(x,N))=x. The Lean counterpart uses packet coordinates restricted to projector ranges and proves both inverse laws. The finite Python routine checks the concrete array realization.
Exact mathematical passage
\operatorname{synthesize}(\operatorname{analyze}(x,N))=x
\mathsf TUse in source · 3 passages
Search aliases: T mathsf
Use in source · Read passage · 03_weighted.tex, line 395
[Rationality of the transfer series] The weighted transfer series satisfies Z_u,v(t) = u^ T(I-A(t))^-1v. It is a rational function whose denominator divides det(I-A(t)).
Exact mathematical passage
Z_{u,v}(t) = u^{\mathsf T}(I-A(t))^{-1}v.
T^{-1}Use in source · 6 passages
Search aliases: T^-1
Use in source · Read passage · 03_weighted.tex, line 58
B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
T^{-1} = \sum_\gamma(-1)^{|\gamma|}A_\gamma D^{-1}.
t^2Use in source · 7 passages
Search aliases: t^2
Use in source · Read passage · 03_weighted.tex, line 463
[A transfer series with one loop] Take vertices 0, 1, and 2. Give the edge 0→1 weight one, the loop 1→1 weight two, and the edge 1→2 weight one. Every path from 0 to 2 uses the first and last edges and may traverse the loop any number of times. Hence Z_0,2(t) = t^2(1+t^2+t^4+⋯) = fract^21-t^2. There is one path at every positive even weight and no path at an odd weight. The denominator records the increase of two units produced by the loop.
Exact mathematical passage
Z_{0,2}(t) = t^2(1+t^2+t^4+\cdots) = \frac{t^2}{1-t^2}.
t^3Use in source · 4 passages
Search aliases: t^3
Use in source · Read passage · 03_weighted.tex, line 581
[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rational expansion.
Exact mathematical passage
Z_{0,2}(t) = \frac{t^3}{1-t^2-t^3-2t^4+t^5+t^6+t^7-t^9}.
t^4Use in source · 3 passages
Search aliases: t^4
Use in source · Read passage · 03_weighted.tex, line 463
[A transfer series with one loop] Take vertices 0, 1, and 2. Give the edge 0→1 weight one, the loop 1→1 weight two, and the edge 1→2 weight one. Every path from 0 to 2 uses the first and last edges and may traverse the loop any number of times. Hence Z_0,2(t) = t^2(1+t^2+t^4+⋯) = fract^21-t^2. There is one path at every positive even weight and no path at an odd weight. The denominator records the increase of two units produced by the loop.
Exact mathematical passage
Z_{0,2}(t) = t^2(1+t^2+t^4+\cdots) = \frac{t^2}{1-t^2}.
t^5Use in source · 2 passages
Search aliases: t^5
Use in source · Read passage · 03_weighted.tex, line 581
[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rational expansion.
Exact mathematical passage
Z_{0,2}(t) = \frac{t^3}{1-t^2-t^3-2t^4+t^5+t^6+t^7-t^9}.
t^6Use in source · 2 passages
Search aliases: t^6
Use in source · Read passage · 03_weighted.tex, line 581
[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rational expansion.
Exact mathematical passage
Z_{0,2}(t) = \frac{t^3}{1-t^2-t^3-2t^4+t^5+t^6+t^7-t^9}.
t^7Use in source · 2 passages
Search aliases: t^7
Use in source · Read passage · 03_weighted.tex, line 581
[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rational expansion.
Exact mathematical passage
Z_{0,2}(t) = \frac{t^3}{1-t^2-t^3-2t^4+t^5+t^6+t^7-t^9}.
t^9Use in source · 2 passages
Search aliases: t^9
Use in source · Read passage · 03_weighted.tex, line 581
[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rational expansion.
Exact mathematical passage
Z_{0,2}(t) = \frac{t^3}{1-t^2-t^3-2t^4+t^5+t^6+t^7-t^9}.
t^dUse in source · 1 passage
Search aliases: t^d
Use in source · Read passage · 03_weighted.tex, line 422
[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.
Exact mathematical passage
Z_{u,v}(t)=\frac{P(t)}{Q(t)}, \qquad Q(t)=1+q_1t+\cdots+q_dt^d.
t^kUse in source · 1 passage
Search aliases: t^k
Use in source · Read passage · 03_weighted.tex, line 557
[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).
Exact mathematical passage
Q(t)=\sum_kq_kt^k, \qquad Z_{s,t}(t)=\sum_{n\ge0}a_n(s,t)t^n.
T^{\Pcal}_{ba}Use in source · 2 passages
Search aliases: T^Pcal_ba T^ P_ba Pcal mathcal
Use in source · Read passage · 02_coordinates.tex, line 224
[Refinement of operator blocks] Suppose that Q refines P through ρ. Then T^ P_ba = ∑_substackρ(β)=b ρ(α)=a T^ Q_βα. The matrix of T in fine coordinates is related to the matrix in coarse coordinates by [T]_ P = C_ P← Q[T]_ QJ_ Q← P.
Exact mathematical passage
T^{\Pcal}_{ba} = \sum_{\substack{\rho(\beta)=b\\\rho(\alpha)=a}} T^{\Qcal}_{\beta\alpha}.
T^{\Pcal}_{\beta\alpha}Use in source · 1 passage
Search aliases: T^Pcal_βα T^ P_βα Pcal beta alpha mathcal beta alpha
Use in source · Read passage · 02_coordinates.tex, line 218
For T∈End(X) and a packet system P, write T^ P_βα=P_β T P_α.
Exact mathematical passage
T^{\Pcal}_{\beta\alpha}=P_\beta T P_\alpha.
T^{\Qcal}_{\beta\alpha}Use in source · 2 passages
Search aliases: T^Qcal_βα T^ Q_βα Qcal beta alpha mathcal beta alpha
Use in source · Read passage · 02_coordinates.tex, line 224
[Refinement of operator blocks] Suppose that Q refines P through ρ. Then T^ P_ba = ∑_substackρ(β)=b ρ(α)=a T^ Q_βα. The matrix of T in fine coordinates is related to the matrix in coarse coordinates by [T]_ P = C_ P← Q[T]_ QJ_ Q← P.
Exact mathematical passage
T^{\Pcal}_{ba} = \sum_{\substack{\rho(\beta)=b\\\rho(\alpha)=a}} T^{\Qcal}_{\beta\alpha}.
T_0Use in source · 3 passages
Search aliases: T_0
Use in source · Read passage · 05_defects.tex, line 228
[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .
Exact mathematical passage
C_{0,L}T_L-T_0C_{0,L} = \sum_{\ell=0}^{L-1} C_{0,\ell}\Delta_\ell C_{\ell+1,L}.
t_1Use in source · 1 passage
Search aliases: t_1
Use in source · Read passage · 03_weighted.tex, line 452
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
t_1,\ldots,t_r
t_1^{w_1(e)}Use in source · 1 passage
Search aliases: t_1^w_1(e)
Use in source · Read passage · 03_weighted.tex, line 449
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
t_1^{w_1(e)}\cdots t_r^{w_r(e)}.
T_{2,1}Use in source · 1 passage
Search aliases: T_2,1
Use in source · Read passage · 06_models.tex, line 59
y any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this domain gives T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). For N=2 and k=1, T_2,1(z)=frac1z, z≠0,-1. Thus the nontrivial action at the first phase level exchanges normalized impedance and admittance. Calling these values ``packets'' is geometric terminology for the sampled orbit…
Exact mathematical passage
T_{2,1}(z)=\frac1z, \qquad z\ne0,-1. \label{eq:smith-inversion}
T_{ba}Use in source · 1 passage
Search aliases: T_ba
Use in source · visual_atlas.tex, line 29
T_ba · Input packet a, output packet b: P_bTP_a. · Blocks of a finite cyclic filter.
Exact mathematical passage
T_{ba}
T_c'Use in source · 1 passage
Search aliases: T_c'
Use in source · visual_atlas.tex, line 547
Had we chosen the transported operator T_c'=CT_fC^-1, the defect would be zero and every polynomial action would intertwine exactly. That is a different choice of coarse operator, not a stronger conclusion from the faulty one.
Exact mathematical passage
T_c'=CT_fC^{-1}
T_c^nUse in source · 7 passages
Search aliases: T_c^n
Use in source · Read passage · 05_defects.tex, line 27
[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
CT_f^n-T_c^nC = \sum_{k=0}^{n-1}T_c^{n-1-k}\Delta T_f^k.
T_c^{n+1}Use in source · 1 passage
Search aliases: T_c^n+1
Use in source · Read passage · 05_defects.tex, line 36
The statement is clear for n=1. For the induction step, write CT_f^n+1-T_c^n+1C = (CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC) and apply the induction hypothesis.
Exact mathematical passage
CT_f^{n+1}-T_c^{n+1}C = (CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC)
T_c^{n-1-k}Use in source · 3 passages
Search aliases: T_c^n-1-k
Use in source · Read passage · 05_defects.tex, line 27
[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
CT_f^n-T_c^nC = \sum_{k=0}^{n-1}T_c^{n-1-k}\Delta T_f^k.
T_f^kUse in source · 3 passages
Search aliases: T_f^k
Use in source · Read passage · 05_defects.tex, line 27
[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
CT_f^n-T_c^nC = \sum_{k=0}^{n-1}T_c^{n-1-k}\Delta T_f^k.
T_f^nUse in source · 7 passages
Search aliases: T_f^n
Use in source · Read passage · 05_defects.tex, line 27
[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
CT_f^n-T_c^nC = \sum_{k=0}^{n-1}T_c^{n-1-k}\Delta T_f^k.
T_f^{n+1}Use in source · 1 passage
Search aliases: T_f^n+1
Use in source · Read passage · 05_defects.tex, line 36
The statement is clear for n=1. For the induction step, write CT_f^n+1-T_c^n+1C = (CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC) and apply the induction hypothesis.
Exact mathematical passage
CT_f^{n+1}-T_c^{n+1}C = (CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC)
T_LUse in source · 3 passages
Search aliases: T_L
Use in source · Read passage · 05_defects.tex, line 228
[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .
Exact mathematical passage
C_{0,L}T_L-T_0C_{0,L} = \sum_{\ell=0}^{L-1} C_{0,\ell}\Delta_\ell C_{\ell+1,L}.
T_{N,k}Use in source · 1 passage
Search aliases: T_N,k
Use in source · Read passage · 06_models.tex, line 51
t z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this domain gives T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). For N=2 and k=1, T_2,1(z)=frac1z, z≠0,-1. Thus the nontrivial action at the first phase level exchanges normalized impeda…
Exact mathematical passage
T_{N,k}(z) = \frac{(1+\omega_N^k)z+(1-\omega_N^k)} {(1-\omega_N^k)z+(1+\omega_N^k)}.
t_rUse in source · 1 passage
Search aliases: t_r
Use in source · Read passage · 03_weighted.tex, line 452
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
t_1,\ldots,t_r
t_r^{w_r(e)}Use in source · 1 passage
Search aliases: t_r^w_r(e)
Use in source · Read passage · 03_weighted.tex, line 449
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
t_1^{w_1(e)}\cdots t_r^{w_r(e)}.
T_\ellUse in source · 4 passages
Search aliases: T_ℓ ell
Use in source · Read passage · 05_defects.tex, line 216
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
T_\ell\in\End(X_\ell)
T_{\ell+1}Use in source · 2 passages
Search aliases: T_ℓ+1 ell
Use in source · Read passage · 05_defects.tex, line 217
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
\Delta_\ell=C_\ell T_{\ell+1}-T_\ell C_\ell.
\mathrm{target}Use in source · 1 passage
Search aliases: target mathrm
Use in source · visual_atlas.tex, line 289
headThe first-edge decomposition Let a_n(s,t) count walks of total weight n from s to t. Removing the first edge gives a_n(s,t)=1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). The empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most n, so enumeration at each fixed weight is finite.
Exact mathematical passage
a_n(s,t)=\mathbf1_{s=t,\,n=0} +\sum_{\substack{e:\,\mathrm{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\mathrm{target}(e),t).
\operatorname{target}Use in source · 1 passage
Search aliases: target operatorname
Use in source · Read passage · 03_weighted.tex, line 499
[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_n(s,t) = 1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, henc…
Exact mathematical passage
a_n(s,t) = \mathbf 1_{s=t,\,n=0} +\sum_{\substack{e:\,\operatorname{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\operatorname{target}(e),t).
UUse in source · 29 passages
Search aliases: U
Use in source · Read passage · 02_coordinates.tex, line 398
Let U act on coefficient sequences by (UF)(z)=F(ω z). If ω^N=1, then U^N=I.
Exact mathematical passage
U
u^{\mathsf T}Use in source · 1 passage
Search aliases: u^ T mathsf
Use in source · Read passage · 03_weighted.tex, line 395
[Rationality of the transfer series] The weighted transfer series satisfies Z_u,v(t) = u^ T(I-A(t))^-1v. It is a rational function whose denominator divides det(I-A(t)).
Exact mathematical passage
Z_{u,v}(t) = u^{\mathsf T}(I-A(t))^{-1}v.
U^kUse in source · 3 passages
Search aliases: U^k
Use in source · Read passage · 02_coordinates.tex, line 418
0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.
Exact mathematical passage
\boxed{ P_{N,j} = \eta\sum_{k=0}^{N-1}(\omega^\vee)^{jk}U^k. }
U^NUse in source · 2 passages
Search aliases: U^N
Use in source · Read passage · 02_coordinates.tex, line 402
Let U act on coefficient sequences by (UF)(z)=F(ω z). If ω^N=1, then U^N=I.
Exact mathematical passage
U^N=I
u_1Use in source · 1 passage
Search aliases: u_1
Use in source · Read passage · 04_arithmetic.tex, line 819
Let M be a matrix over a DVR. Smith normal form gives invertible matrices U,V such that UMV = diag(π^s_1u_1,…,π^s_ru_r,0,…,0), where each u_i is a unit and 0≤ s_1≤⋯≤ s_r. The exponents s_i are directional lifting weights.
Exact mathematical passage
UMV = \diag(\pi^{s_1}u_1,\ldots,\pi^{s_r}u_r,0,\ldots,0),
u_rUse in source · 1 passage
Search aliases: u_r
Use in source · Read passage · 04_arithmetic.tex, line 819
Let M be a matrix over a DVR. Smith normal form gives invertible matrices U,V such that UMV = diag(π^s_1u_1,…,π^s_ru_r,0,…,0), where each u_i is a unit and 0≤ s_1≤⋯≤ s_r. The exponents s_i are directional lifting weights.
Exact mathematical passage
UMV = \diag(\pi^{s_1}u_1,\ldots,\pi^{s_r}u_r,0,\ldots,0),
V_1Use in source · 2 passages
Search aliases: V_1
Use in source · Read passage · 04_arithmetic.tex, line 345
[Two complementary packets] Suppose V=V_1⊕ V_2 with projector E onto V_1 and complementary projector I-E. Then Λ/(Λ_1⊕Λ_2) ≅ EΛ/Λ_1 ≅ (I-E)Λ/Λ_2. Consequently s_Λ(E)=s_Λ(I-E) = exp_π(Λ/(Λ_1⊕Λ_2)).
Exact mathematical passage
V=V_1\oplus V_2
V_2Use in source · 2 passages
Search aliases: V_2
Use in source · Read passage · 04_arithmetic.tex, line 345
[Two complementary packets] Suppose V=V_1⊕ V_2 with projector E onto V_1 and complementary projector I-E. Then Λ/(Λ_1⊕Λ_2) ≅ EΛ/Λ_1 ≅ (I-E)Λ/Λ_2. Consequently s_Λ(E)=s_Λ(I-E) = exp_π(Λ/(Λ_1⊕Λ_2)).
Exact mathematical passage
V=V_1\oplus V_2
v_2Use in source · 3 passages
Search aliases: v_2
Use in source · Read passage · 04_arithmetic.tex, line 664
The reduced denominators of the displayed entries are 1 or 2, which proves the local integrality and minimality statements directly. The resultant is Res(x-1,x+1)=2. Thus v_2(2)=1 and v_p(2)=0 for primes p≠2.
Exact mathematical passage
v_2(2)=1
v_3Use in source · 1 passage
Search aliases: v_3
Use in source · Read passage · 06_models.tex, line 125
Sign projector resultant weight at the odd prime 3 · v_3(2)=0
Exact mathematical passage
v_3(2)=0
v_pUse in source · 1 passage
Search aliases: v_p
Use in source · Read passage · 04_arithmetic.tex, line 664
The reduced denominators of the displayed entries are 1 or 2, which proves the local integrality and minimality statements directly. The resultant is Res(x-1,x+1)=2. Thus v_2(2)=1 and v_p(2)=0 for primes p≠2.
Exact mathematical passage
v_p(2)=0
\mathrm{val}Use in source · 4 passages
Search aliases: val mathrm
Use in source · Read passage · 04_arithmetic.tex, line 838
[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
p^{\min(s_i,L)}\mid (b_i)_{\mathrm{val}}
\veeUse in source · 5 passages
Search aliases: vee
Use in source · Read passage · 02_coordinates.tex, line 406
[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j,
Exact mathematical passage
\omega,\omega^\vee,\eta\in R
w^2Use in source · 3 passages
Search aliases: w^2
Use in source · visual_atlas.tex, line 495
Here e=1+3w+5w^2 and o=2+4w+6w^2, so direct polynomial multiplication gives y_rm even=1+8w+16w^2+17w^3, y_rm odd=4+12w+20w^2+6w^3. This is the same result as ordinary convolution followed by residue extraction. No numerical transform or limiting argument is needed because every sequence is finitely supported.
Exact mathematical passage
e=1+3w+5w^2
w^3Use in source · 1 passage
Search aliases: w^3
Use in source · visual_atlas.tex, line 496
Here e=1+3w+5w^2 and o=2+4w+6w^2, so direct polynomial multiplication gives y_rm even=1+8w+16w^2+17w^3, y_rm odd=4+12w+20w^2+6w^3. This is the same result as ordinary convolution followed by residue extraction. No numerical transform or limiting argument is needed because every sequence is finitely supported.
Exact mathematical passage
y_{\rm even}=1+8w+16w^2+17w^3, \qquad y_{\rm odd}=4+12w+20w^2+6w^3.
w_1Use in source · 2 passages
Search aliases: w_1
Use in source · Read passage · 03_weighted.tex, line 445
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
w(e)=(w_1(e),\ldots,w_r(e))\in\mathbb N^r\setminus\{0\}.
w_rUse in source · 2 passages
Search aliases: w_r
Use in source · Read passage · 03_weighted.tex, line 445
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
w(e)=(w_1(e),\ldots,w_r(e))\in\mathbb N^r\setminus\{0\}.
w_{\mathrm{Smith}}Use in source · 1 passage
Search aliases: w_Smith mathrm
Use in source · Read passage · 06_models.tex, line 79
The symmetric weight w_Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2 indicates where small errors in reflection coordinates produce large impedance or admittance errors.
Exact mathematical passage
w_{\mathrm{Smith}}(\Gamma) = \frac1{|1-\Gamma|^2} + \frac1{|1+\Gamma|^2}
\widehat TUse in source · 7 passages
Search aliases: widehat T widehat
Use in source · Read passage · 02_coordinates.tex, line 189
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\widehat T_{Q,P}=\cA_QT\cS_P
\widehat T_2Use in source · 3 passages
Search aliases: widehat T_2 widehat
Use in source · visual_atlas.tex, line 117
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
J_{4,2}\widehat T_2=\widehat T_4J_{4,2},\qquad J_{8,4}J_{4,2}=J_{8,2}.
\widehat T_4Use in source · 3 passages
Search aliases: widehat T_4 widehat
Use in source · visual_atlas.tex, line 117
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
J_{4,2}\widehat T_2=\widehat T_4J_{4,2},\qquad J_{8,4}J_{4,2}=J_{8,2}.
\widehat T_NUse in source · 1 passage
Search aliases: widehat T_N widehat
Use in source · visual_atlas.tex, line 113
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
\widehat T_N=R_NTR_N^{-1},\qquad J_{4,2}=R_4R_2^{-1}.
\widehat T_{Q,P}Use in source · 1 passage
Search aliases: widehat T_Q,P widehat
Use in source · Read passage · 02_coordinates.tex, line 189
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\widehat T_{Q,P}=\cA_QT\cS_P
\widehat T_{Q_a,P_a}Use in source · 1 passage
Search aliases: widehat T_Q_a,P_a widehat
Use in source · Read passage · 02_coordinates.tex, line 193
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\boxed{ J^Y_{Q_b,Q_a}\,\widehat T_{Q_a,P_a} = \widehat T_{Q_b,P_b}\,J^X_{P_b,P_a}. }
\widehat T_{Q_b,P_b}Use in source · 1 passage
Search aliases: widehat T_Q_b,P_b widehat
Use in source · Read passage · 02_coordinates.tex, line 193
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\boxed{ J^Y_{Q_b,Q_a}\,\widehat T_{Q_a,P_a} = \widehat T_{Q_b,P_b}\,J^X_{P_b,P_a}. }
\widetilde bUse in source · 5 passages
Search aliases: widetilde b widetilde
Use in source · Read passage · 04_arithmetic.tex, line 754
[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L (r<i≤ m). When it is solvable, the number of solutions in ( O/π^L O)^n is q^(n-r)L+∑_i=1^rmin(s_i,L).
Exact mathematical passage
\widetilde b=Ub
\widetilde b_iUse in source · 3 passages
Search aliases: widetilde b_i widetilde
Use in source · Read passage · 04_arithmetic.tex, line 758
[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L (r<i≤ m). When it is solvable, the number of solutions in ( O/π^L O)^n is q^(n-r)L+∑_i=1^rmin(s_i,L).
Exact mathematical passage
\widetilde b_i\equiv0 \pmod{\pi^{\min(s_i,L)}} \qquad(1\leq i\leq r)
\widetilde TUse in source · 4 passages
Search aliases: widetilde T widetilde
Use in source · Read passage · 03_weighted.tex, line 165
[Stability of inversion at finite precision] Let T and widetilde T be invertible elements of F^0 B. If T-widetilde T∈ F^L B, then T^-1-widetilde T^-1∈ F^L B.
Exact mathematical passage
\widetilde T
\widetilde T^{-1}Use in source · 2 passages
Search aliases: widetilde T^-1 widetilde
Use in source · Read passage · 03_weighted.tex, line 170
[Stability of inversion at finite precision] Let T and widetilde T be invertible elements of F^0 B. If T-widetilde T∈ F^L B, then T^-1-widetilde T^-1∈ F^L B.
Exact mathematical passage
T^{-1}-\widetilde T^{-1}\in F^L\Bcal.
X_0Use in source · 2 passages
Search aliases: X_0
Use in source · Read passage · 01_orientation.tex, line 56
A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operator on one selected summand may also fail to intertwine with the full matrix. The later sections isolate these four effects one at a time.
Exact mathematical passage
X=X_0\oplus X_1, \qquad D=\begin{pmatrix}D_0&0\\0&D_1\end{pmatrix}, \qquad N=\begin{pmatrix}0&N_{01}\\N_{10}&0\end{pmatrix}.
x_0Use in source · 4 passages
Search aliases: x_0
Use in source · visual_atlas.tex, line 45
vpage3Follow one coefficient through the towerMODEL A / LOSSLESS ADDRESS REFINEMENTTraditional paper: Theorem 3.3 (p. 3), Theorem 4.6 (p. 11) Take a vector x=(x_0,…,x_15). A packet is not a single summary number: it retains every coefficient at its selected addresses. For example, P_2,0x=(x_0,0,x_2,0,…,x_14,0). The rail below lists the addresses in each packet. Its display order groups descendants under their parents. The matrix example on V5 instead uses increasing residue order.
Exact mathematical passage
x=(x_0,\ldots,x_{15})
X_{00}Use in source · 1 passage
Search aliases: X_00
Use in source · Read passage · 02_coordinates.tex, line 348
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
X=X_{00}\oplus X_{01}\oplus X_{10}\oplus X_{11}.
x_{00}Use in source · 2 passages
Search aliases: x_00
Use in source · Read passage · 02_coordinates.tex, line 357
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
x=(x_{00},x_{01},x_{10},x_{11})
X_{01}Use in source · 1 passage
Search aliases: X_01
Use in source · Read passage · 02_coordinates.tex, line 348
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
X=X_{00}\oplus X_{01}\oplus X_{10}\oplus X_{11}.
x_{01}Use in source · 2 passages
Search aliases: x_01
Use in source · Read passage · 02_coordinates.tex, line 357
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
x=(x_{00},x_{01},x_{10},x_{11})
X_1Use in source · 2 passages
Search aliases: X_1
Use in source · Read passage · 01_orientation.tex, line 56
A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operator on one selected summand may also fail to intertwine with the full matrix. The later sections isolate these four effects one at a time.
Exact mathematical passage
X=X_0\oplus X_1, \qquad D=\begin{pmatrix}D_0&0\\0&D_1\end{pmatrix}, \qquad N=\begin{pmatrix}0&N_{01}\\N_{10}&0\end{pmatrix}.
x_1Use in source · 3 passages
Search aliases: x_1
Use in source · Read passage · 05_defects.tex, line 263
[A visible one step defect] Let X_f=bbk^2, let X_c=bbk, and let C(x_1,x_2)=x_1. Take T_f= a · b
Exact mathematical passage
C(x_1,x_2)=x_1.
X_{10}Use in source · 1 passage
Search aliases: X_10
Use in source · Read passage · 02_coordinates.tex, line 348
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
X=X_{00}\oplus X_{01}\oplus X_{10}\oplus X_{11}.
x_{10}Use in source · 2 passages
Search aliases: x_10
Use in source · Read passage · 02_coordinates.tex, line 357
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
x=(x_{00},x_{01},x_{10},x_{11})
X_{11}Use in source · 1 passage
Search aliases: X_11
Use in source · Read passage · 02_coordinates.tex, line 348
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
X=X_{00}\oplus X_{01}\oplus X_{10}\oplus X_{11}.
x_{11}Use in source · 2 passages
Search aliases: x_11
Use in source · Read passage · 02_coordinates.tex, line 357
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
x=(x_{00},x_{01},x_{10},x_{11})
x_{14}Use in source · 1 passage
Search aliases: x_14
Use in source · visual_atlas.tex, line 46
vpage3Follow one coefficient through the towerMODEL A / LOSSLESS ADDRESS REFINEMENTTraditional paper: Theorem 3.3 (p. 3), Theorem 4.6 (p. 11) Take a vector x=(x_0,…,x_15). A packet is not a single summary number: it retains every coefficient at its selected addresses. For example, P_2,0x=(x_0,0,x_2,0,…,x_14,0). The rail below lists the addresses in each packet. Its display order groups descendants under their parents. The matrix example on V5 instead uses increasing residue order.
Exact mathematical passage
P_{2,0}x=(x_0,0,x_2,0,\ldots,x_{14},0).
x_{15}Use in source · 1 passage
Search aliases: x_15
Use in source · visual_atlas.tex, line 45
vpage3Follow one coefficient through the towerMODEL A / LOSSLESS ADDRESS REFINEMENTTraditional paper: Theorem 3.3 (p. 3), Theorem 4.6 (p. 11) Take a vector x=(x_0,…,x_15). A packet is not a single summary number: it retains every coefficient at its selected addresses. For example, P_2,0x=(x_0,0,x_2,0,…,x_14,0). The rail below lists the addresses in each packet. Its display order groups descendants under their parents. The matrix example on V5 instead uses increasing residue order.
Exact mathematical passage
x=(x_0,\ldots,x_{15})
x_2Use in source · 3 passages
Search aliases: x_2
Use in source · Read passage · 05_defects.tex, line 263
[A visible one step defect] Let X_f=bbk^2, let X_c=bbk, and let C(x_1,x_2)=x_1. Take T_f= a · b
Exact mathematical passage
C(x_1,x_2)=x_1.
x_{2m}Use in source · 1 passage
Search aliases: x_2m
Use in source · visual_atlas.tex, line 481
vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONTraditional paper: Section B.1 (p. 47) Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.
Exact mathematical passage
e_m=x_{2m}
x_{2m+1}Use in source · 1 passage
Search aliases: x_2m+1
Use in source · visual_atlas.tex, line 481
vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONTraditional paper: Section B.1 (p. 47) Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.
Exact mathematical passage
o_m=x_{2m+1}
x_3Use in source · 1 passage
Search aliases: x_3
Use in source · regrouping_not_averaging.tex, line 73
5pt,] (9.17000,2.26000) rectangle ++(0.57000,0.57000); node[] at (9.45500,2.54500) 1; draw[fill=black!0,draw=black!45,line width=.35pt,] (9.74000,2.26000) rectangle ++(0.57000,0.57000); node[] at (10.02500,2.54500) 0; draw[fill=black!12,draw=black!45,line width=.35pt,] (10.31000,2.26000) rectangle ++(0.57000,0.57000); node[] at (10.59500,2.54500) 1; node[] at (9.74000,1.55000) (x_0+x_2, x_1+x_3); node[] at (9.74000,0.10000) 4⟶2 scalars: loss;
Exact mathematical passage
(x_0+x_2,\ x_1+x_3)
x_6Use in source · 1 passage
Search aliases: x_6
Use in source · visual_atlas.tex, line 10
headAddress resolution A coefficient such as x_6 can sit in the even packet, then the residue-2 packet modulo 4, then the residue-6 packet modulo 8. Its value does not become more accurate. Its address becomes more specific, while the full object remains exactly reconstructible.
Exact mathematical passage
x_6
X_cUse in source · 7 passages
Search aliases: X_c
Use in source · Read passage · 05_defects.tex, line 5
Let X_f and X_c be vector spaces or Banach spaces over a common field, and let C:X_f⟶ X_c be a linear map, and let T_f∈End(X_f), T_c∈End(X_c). Define Δ=CT_f-T_cC.
Exact mathematical passage
X_c
X_fUse in source · 7 passages
Search aliases: X_f
Use in source · Read passage · 05_defects.tex, line 5
Let X_f and X_c be vector spaces or Banach spaces over a common field, and let C:X_f⟶ X_c be a linear map, and let T_f∈End(X_f), T_c∈End(X_c). Define Δ=CT_f-T_cC.
Exact mathematical passage
X_f
x_jUse in source · 2 passages
Search aliases: x_j
Use in source · Read passage · 06_models.tex, line 8
Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer blocks. Refining from N to M with Nmid M splits each coarse phase into its descendants without changing the reconstructed signal. The same formulas extend…
Exact mathematical passage
x_j[m]=x[Nm+j].
x_nUse in source · 2 passages
Search aliases: x_n
Use in source · Read passage · 06_models.tex, line 5
Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer blocks. Refining from N to M with Nmid M splits each coarse phase into its descendants without changing the reconstructed signal. The same formulas extend…
Exact mathematical passage
x=(x_n)_{n\in\Z}\in c_{00}(\Z;R)
x_{n-r}Use in source · 1 passage
Search aliases: x_n-r
Use in source · Read passage · 06_models.tex, line 23
A banded operator need not remain banded after residue reordering, but its nonzero couplings become a structured packet graph. With the convention (S_rx)_n=x_n-r, the only active transfer is P_N,j+rS_rP_N,j, with indices taken modulo N. More general finite stencils produce a finite set of active phase differences.
Exact mathematical passage
(S_rx)_n=x_{n-r}
X_N^{\rm pkt}Use in source · 1 passage
Search aliases: X_N^rm pkt rm
Use in source · visual_atlas.tex, line 79
Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.
Exact mathematical passage
X_N^{\rm pkt}=\bigoplus_j P_{N,j}X, \qquad \mathcal A_Nx=(P_{N,j}x)_j, \qquad \mathcal S_N(x_j)_j=\sum_jx_j.
x_\betaUse in source · 1 passage
Search aliases: x_β beta
Use in source · Read passage · 02_coordinates.tex, line 172
Every coarse packet is also uniquely reconstructed from its fine children. If vectors x_β∈ Q_β X sum to a coarse packet x, applying Q_γ recovers x_γ=Q_γ x. This is the uniqueness clause of the canonical nested-refinement statement.
Exact mathematical passage
x_\beta\in Q_\beta X
x_\gammaUse in source · 1 passage
Search aliases: x_γ gamma
Use in source · Read passage · 02_coordinates.tex, line 172
Every coarse packet is also uniquely reconstructed from its fine children. If vectors x_β∈ Q_β X sum to a coarse packet x, applying Q_γ recovers x_γ=Q_γ x. This is the uniqueness clause of the canonical nested-refinement statement.
Exact mathematical passage
x_\gamma=Q_\gamma x
x_{\rho(\beta)}Use in source · 2 passages
Search aliases: x_ρ(β) rho beta
Use in source · Read passage · 02_coordinates.tex, line 121
w QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
\bigl(J_{\Qcal\leftarrow\Pcal}x\bigr)_\beta = Q_\beta x_{\rho(\beta)}, \qquad \bigl(C_{\Pcal\leftarrow\Qcal}y\bigr)_\alpha = \sum_{\rho(\beta)=\alpha}y_\beta.
X_\ellUse in source · 2 passages
Search aliases: X_ℓ ell
Use in source · Read passage · 05_defects.tex, line 213
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_\ell:X_{\ell+1}\longrightarrow X_\ell
X_{\ell+1}Use in source · 1 passage
Search aliases: X_ℓ+1 ell
Use in source · Read passage · 05_defects.tex, line 213
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_\ell:X_{\ell+1}\longrightarrow X_\ell
YUse in source · 4 passages
Search aliases: Y
Use in source · Read passage · 02_coordinates.tex, line 187
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
Y
y_0Use in source · 1 passage
Search aliases: y_0
Use in source · smith_coordinate_transport.tex, line 165
(6.70000,0.88000) 2; node[font=sffamilyscriptsize] at (6.70000,1.32000) 3; node[font=sffamilyscriptsize] at (6.70000,1.76000) 4; node[font=sffamilyscriptsize] at (6.70000,2.20000) 5; node[font=sffamilyscriptsize] at (6.70000,2.64000) 6; node[font=sffamilyscriptsize] at (6.70000,3.08000) 7; node[] at (5.10000,2.30000) x=Vy; node[] at (5.10000,1.30000) x_0=y_0-y_1; node[] at (5.10000,0.50000) x_1=y_1; node[] at (5.40000,-1.00000) The eight solutions are permuted, not created or discarded.;
Exact mathematical passage
x_0=y_0-y_1
y_1Use in source · 2 passages
Search aliases: y_1
Use in source · smith_coordinate_transport.tex, line 165
(6.70000,0.88000) 2; node[font=sffamilyscriptsize] at (6.70000,1.32000) 3; node[font=sffamilyscriptsize] at (6.70000,1.76000) 4; node[font=sffamilyscriptsize] at (6.70000,2.20000) 5; node[font=sffamilyscriptsize] at (6.70000,2.64000) 6; node[font=sffamilyscriptsize] at (6.70000,3.08000) 7; node[] at (5.10000,2.30000) x=Vy; node[] at (5.10000,1.30000) x_0=y_0-y_1; node[] at (5.10000,0.50000) x_1=y_1; node[] at (5.40000,-1.00000) The eight solutions are permuted, not created or discarded.;
Exact mathematical passage
x_0=y_0-y_1
y_iUse in source · 1 passage
Search aliases: y_i
Use in source · Read passage · 04_arithmetic.tex, line 779
Set x=Vy. Multiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L choices. Multiplying the independent counts gives the formula.
Exact mathematical passage
\pi^{s_i}y_i\equiv\widetilde b_i\pmod{\pi^L} \qquad(1\leq i\leq r)
y_\alphaUse in source · 1 passage
Search aliases: y_α alpha
Use in source · Read passage · 02_coordinates.tex, line 161
[Orthogonal Hilbert packets] If X is a real or complex Hilbert space and every packet system in the tower consists of orthogonal projections, equip each finite packet direct sum with the Hilbert inner product ⟨(x_α),(y_α)⟩=∑_α⟨ x_α,y_α⟩. Then analysis and synthesis are unitary. Every coordinate change in [source reference] is unitary.
Exact mathematical passage
\langle(x_\alpha),(y_\alpha)\rangle=\sum_\alpha\langle x_\alpha,y_\alpha\rangle
y_\betaUse in source · 1 passage
Search aliases: y_β beta
Use in source · Read passage · 02_coordinates.tex, line 121
w QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
\bigl(J_{\Qcal\leftarrow\Pcal}x\bigr)_\beta = Q_\beta x_{\rho(\beta)}, \qquad \bigl(C_{\Pcal\leftarrow\Qcal}y\bigr)_\alpha = \sum_{\rho(\beta)=\alpha}y_\beta.
\ZUse in source · 9 passages
Search aliases: Z mathbb
Use in source · Read passage · 03_weighted.tex, line 249
A packet decomposition resolves H=∑_e∈ EH_e and assigns a certified filtration gain D^-1H_e∈ F^w(e) B, w(e)∈ Z_>0. Each e is a directed transfer edge. For an ordered transfer word γ=(e_1,…,e_r), define W(γ)=∑_j=1^rw(e_j), A_γ=A_e_1⋯ A_e_r, A_e=D^-1H_e.
Exact mathematical passage
D^{-1}H_e\in F^{w(e)}\cB, \qquad w(e)\in\Z_{>0}.
\mathbb Z^2Use in source · 1 passage
Search aliases: Z^2 mathbb
Use in source · dyadic_eigenlattice.tex, line 53
0 · 2, det B=2; node[] at (6.45000,0.60000) Filled: B Z^2={(x,y):y is even}; node[] at (6.45000,-0.35000) Unfilled: the other lattice coset.; node[] at (6.45000,-1.25000) B^-1 is integral at odd primes, not at 2.;
Exact mathematical passage
B\mathbb Z^2=\{(x,y):y\text{ is even}\}
z^rUse in source · 1 passage
Search aliases: z^r
Use in source · finite_fourier_table.tex, line 22
000,0.70000); node[] at (0.35000,1.45000) 0; draw[fill=black!0,draw=black!45,line width=.35pt,] (0.70000,1.10000) rectangle ++(0.70000,0.70000); node[] at (1.05000,1.45000) 0; draw[fill=black!12,draw=black!45,line width=.35pt,] (1.40000,1.10000) rectangle ++(0.70000,0.70000); node[] at (1.75000,1.45000) 3; node[] at (3.70000,2.05000) × 5; node[] at (6.05000,3.75000) P_j(z^r): 5=3^-1pmod7; draw[fill=black!12,draw=black!45,line width=.35pt,] (5.00000,2.50000) rectangle ++(0.70000,0.70000); node[] …
Exact mathematical passage
P_j(z^r):\ 5=3^{-1}\pmod7
Z_0Use in source · 2 passages
Search aliases: Z_0
Use in source · Read passage · 06_models.tex, line 42
Let Z and Z_0≠0 be complex impedances, put z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this domain gives T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). For N=2 and k=1, T_2,1(z)=frac1z, z≠0,-1. Thus the nontrivial action at the firs…
Exact mathematical passage
Z_0\ne0
Z_{0,1}Use in source · 2 passages
Search aliases: Z_0,1
Use in source · visual_atlas.tex, line 310
From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.
Exact mathematical passage
QZ_{0,1}=2t^2
Z_{0,2}Use in source · 2 passages
Search aliases: Z_0,2
Use in source · Read passage · 03_weighted.tex, line 463
[A transfer series with one loop] Take vertices 0, 1, and 2. Give the edge 0→1 weight one, the loop 1→1 weight two, and the edge 1→2 weight one. Every path from 0 to 2 uses the first and last edges and may traverse the loop any number of times. Hence Z_0,2(t) = t^2(1+t^2+t^4+⋯) = fract^21-t^2. There is one path at every positive even weight and no path at an odd weight. The denominator records the increase of two units produced by the loop.
Exact mathematical passage
Z_{0,2}(t) = t^2(1+t^2+t^4+\cdots) = \frac{t^2}{1-t^2}.
\Z_2Use in source · 1 passage
Search aliases: Z_2 Z mathbb
Use in source · Read passage · 04_arithmetic.tex, line 696
The failure of the character formula frac12(I± A) to be integral does not invalidate coefficient selectors such as even and odd residue extraction. It shows that two distinct constructions must be distinguished: integral address selector ≠ character-average spectral projector when the group order is not a unit. The first can remain exact over Z_2. The second can acquire a denominator weight.
Exact mathematical passage
\Z_2
\mathbb Z_2Use in source · 1 passage
Search aliases: Z_2 mathbb
Use in source · Read passage · 04_arithmetic.tex, line 587
The displayed matrices are complementary idempotents. The scalar 2 is a unit in Z_p for odd p, so both matrices are integral there. Over Z_2, multiplying by 2 clears the denominators, while neither projector preserves the standard lattice. Thus both weights are one. The resultant of the two linear factors is their root difference, which is 2.
Exact mathematical passage
\mathbb Z_2
\Z_2^2Use in source · 1 passage
Search aliases: Z_2^2 Z mathbb
Use in source · Read passage · 04_arithmetic.tex, line 650
[Odd-adic versus dyadic behavior] For every prime p≠2, every entry of E_+ and E_- is p-integral, and each projector has exact local denominator weight zero. At p=2, each projector has exact local denominator weight one. Multiplication by 2 makes it integral, while the unscaled matrix is not integral: s_2, Z_2^2(E_+)=s_2, Z_2^2(E_-)=1. The characteristic factors x-1 and x+1 have resultant 2. Its 2-adic valuation is one, exactly the two local projector weights above, and its p-adic valuation is ze…
Exact mathematical passage
s_{2,\Z_2^2}(E_+)=s_{2,\Z_2^2}(E_-)=1.
\mathbb Z_2^2Use in source · 1 passage
Search aliases: Z_2^2 mathbb
Use in source · Read passage · 04_arithmetic.tex, line 436
[Strict growth under refinement] Take the standard lattice Z_2^2 and the involution from [source reference]. The one packet system (I) has weight zero. Refining it into the two rational sign packets produces weight one. The coarse lattice is already split because there is only one summand. The fine eigenspaces are glued together by one dyadic digit.
Exact mathematical passage
\mathbb Z_2^2
\Z_{>0}Use in source · 1 passage
Search aliases: Z_>0 Z mathbb
Use in source · Read passage · 03_weighted.tex, line 249
A packet decomposition resolves H=∑_e∈ EH_e and assigns a certified filtration gain D^-1H_e∈ F^w(e) B, w(e)∈ Z_>0. Each e is a directed transfer edge. For an ordered transfer word γ=(e_1,…,e_r), define W(γ)=∑_j=1^rw(e_j), A_γ=A_e_1⋯ A_e_r, A_e=D^-1H_e.
Exact mathematical passage
D^{-1}H_e\in F^{w(e)}\cB, \qquad w(e)\in\Z_{>0}.
z_kUse in source · 1 passage
Search aliases: z_k
Use in source · smith_chart_orbit.tex, line 25
) circle (.075); node[] at (-1.38027,-1.38027) 5; fill (0.00000,-1.60000) circle (.075); node[] at (0.00000,-1.95200) 6; fill (1.13137,-1.13137) circle (.075); node[] at (1.38027,-1.38027) 7; draw[dashed] (11.28333,0) circle (2.46667); draw[black!45] (7.90000,0.00000) -- (14.21250,0.00000); draw[black!45] (8.20000,-2.85000) -- (8.20000,2.85000); node[] at (11.28333,3.90000) z_k=(1+Γ_k)/(1-Γ_k); fill (13.75000,0.00000) circle (.075); node[] at (13.98433,0.00000) 0; fill (10.75575,2.40959) circle …
Exact mathematical passage
z_k=(1+\Gamma_k)/(1-\Gamma_k)
\mathbb Z_pUse in source · 1 passage
Search aliases: Z_p mathbb
Use in source · Read passage · 04_arithmetic.tex, line 587
The displayed matrices are complementary idempotents. The scalar 2 is a unit in Z_p for odd p, so both matrices are integral there. Over Z_2, multiplying by 2 clears the denominators, while neither projector preserves the standard lattice. Thus both weights are one. The resultant of the two linear factors is their root difference, which is 2.
Exact mathematical passage
\mathbb Z_p
\mathbb Z_p^2Use in source · 2 passages
Search aliases: Z_p^2 mathbb
Use in source · Read passage · 04_arithmetic.tex, line 562
0 · -1 act on Z_p^2. Over Q_p, the sign projectors are E_+=fracI+A2 = 1 · 1/2
Exact mathematical passage
\mathbb Z_p^2
\ZZUse in source · 4 passages
Search aliases: ZZ
Use in source · visual_atlas.tex, line 32
L · Precision depth in a quotient such as ZZ/2^LZZ. · L=5 gives arithmetic modulo 32.
Exact mathematical passage
\ZZ/2^L\ZZ
\ZZ^2Use in source · 1 passage
Search aliases: ZZ^2 ZZ
Use in source · visual_atlas.tex, line 350
vpage15Where the dyadic digit is lostMODEL D / THE EIGENLATTICE HAS INDEX TWOTraditional paper: Proposition 9.3 (p. 32) The eigenspaces are spanned by (1,0) and (-1,2). Their integral eigenvectors do not form an integral basis of ZZ^2. The change-of-basis determinant is two.
Exact mathematical passage
\ZZ^2
\ZZ_{(p)}^2Use in source · 1 passage
Search aliases: ZZ_(p)^2 ZZ
Use in source · visual_atlas.tex, line 361
0 · 2, AB=Bdiag(1,-1), E_+=Bdiag(1,0)B^-1. The index-two eigenlattice explains the denominator in B^-1 and in the projector. Over the localized lattice Λ_p=ZZ_(p)^2, the number 2 is a unit for every odd prime p. Hence the local denominator weight is zero there and one at p=2.
Exact mathematical passage
\Lambda_p=\ZZ_{(p)}^2
|^2Use in source · 1 passage
Search aliases: |^2
Use in source · Read passage · 06_models.tex, line 79
The symmetric weight w_Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2 indicates where small errors in reflection coordinates produce large impedance or admittance errors.
Exact mathematical passage
w_{\mathrm{Smith}}(\Gamma) = \frac1{|1-\Gamma|^2} + \frac1{|1+\Gamma|^2}
\pmUse in source · 3 passages
Search aliases: ± pm
Use in source · Read passage · 04_arithmetic.tex, line 685
The failure of the character formula frac12(I± A) to be integral does not invalidate coefficient selectors such as even and odd residue extraction. It shows that two distinct constructions must be distinguished: integral address selector ≠ character-average spectral projector when the group order is not a unit. The first can remain exact over Z_2. The second can acquire a denominator weight.
Exact mathematical passage
\frac12(I\pm A)
\cdotUse in source · 139 passages
Search aliases: · cdot
Use in source · visual_atlas.tex, line 416
For a specific attainable target (2,4), 2x≡2pmod8iff x∈{1,5}, 4y≡4pmod8iff y∈{1,3,5,7}. The coordinate choices are independent, giving 2·4=8 solutions. The entire carrier is partitioned consistently: 8 attainable targets × 8 inputs per target=64. For an inadmissible target, such as (1,4) or (2,2), the fiber is empty. The same nonzero cardinality formula must not be applied to it.
Exact mathematical passage
2\cdot4=8
\timesUse in source · 23 passages
Search aliases: × times
Use in source · Read passage · 02_coordinates.tex, line 407
[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j,
Exact mathematical passage
N\times N
\alpha'Use in source · 2 passages
Search aliases: α' alpha
Use in source · Read passage · 02_coordinates.tex, line 303
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
(\alpha,\beta)\ne(\alpha',\beta')
\beta'Use in source · 2 passages
Search aliases: β' beta
Use in source · Read passage · 02_coordinates.tex, line 303
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
(\alpha,\beta)\ne(\alpha',\beta')
\GammaUse in source · 17 passages
Search aliases: Γ Gamma
Use in source · Read passage · 05_defects.tex, line 149
Analytic functional calculus. [Holomorphic functional calculus defect] Let X_f and X_c be complex Banach spaces, and let C:X_f→ X_c, T_f, and T_c be bounded complex-linear maps. Let h be holomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z…
Exact mathematical passage
\Gamma
\Gamma_kUse in source · 3 passages
Search aliases: Γ_k Gamma
Use in source · visual_atlas.tex, line 510
AR PHASE ORBITTraditional paper: Section 12.3 (p. 42), Section 12.3, Smith-phase identity (p. 43) Smith normal form concerns linear congruences. The Smith impedance chart uses the nonlinear coordinate change Γ=fracz-1z+1, z=frac1+Γ1-Γ. They share a name, not a mathematical definition. Sample the reflection circle at the exact algebraic points Γ_k=tfrac12 e^ikπ/4.
Exact mathematical passage
\Gamma_k=\tfrac12 e^{ik\pi/4}
\deltaUse in source · 4 passages
Search aliases: δ delta
Use in source · Read passage · 02_coordinates.tex, line 383
[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.
Exact mathematical passage
P_{N,i}P_{N,j}=\delta_{ij}P_{N,j}, \qquad \sum_{j=0}^{N-1}P_{N,j}=I.
\delta_0Use in source · 2 passages
Search aliases: δ_0 delta
Use in source · Read passage · 06_models.tex, line 5
Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer blocks. Refining from N to M with Nmid M splits each coarse phase into its descendants without changing the reconstructed signal. The same formulas extend…
Exact mathematical passage
\delta_0
\delta_{ij}Use in source · 2 passages
Search aliases: δ_ij delta
Use in source · Read passage · 02_coordinates.tex, line 383
[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.
Exact mathematical passage
P_{N,i}P_{N,j}=\delta_{ij}P_{N,j}, \qquad \sum_{j=0}^{N-1}P_{N,j}=I.
\delta_{n,r}Use in source · 1 passage
Search aliases: δ_n,r delta
Use in source · Read passage · 06_models.tex, line 6
Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer blocks. Refining from N to M with Nmid M splits each coarse phase into its descendants without changing the reconstructed signal. The same formulas extend…
Exact mathematical passage
(S^r\delta_0)_n=\delta_{n,r}
\Delta_\ellUse in source · 4 passages
Search aliases: Δ_ℓ Delta ell
Use in source · Read passage · 05_defects.tex, line 217
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
\Delta_\ell=C_\ell T_{\ell+1}-T_\ell C_\ell.
\varepsilonUse in source · 9 passages
Search aliases: ε varepsilon
Use in source · visual_atlas.tex, line 527
vpage22A small defect can grow under powersMODEL H / EXACT TELESCOPING, EXPLICIT GROWTHTraditional paper: Theorem 11.1 (p. 37), Corollary 11.3 (p. 38) Choose an invertible coordinate shear but independently keep the same numerical operator at both levels: C=1 · ε 0 · 1, T_f=T_c=diag(1,2), ε=frac116. Then Δ=CT_f-T_cC=ε E_01. Exact coordinate invertibility has not forced these independently selected operators to intertwine.
Exact mathematical passage
C=\begin{pmatrix}1&\varepsilon\\0&1\end{pmatrix},\quad T_f=T_c=\diag(1,2),\quad \varepsilon=\frac1{16}.
\etaUse in source · 8 passages
Search aliases: η eta
Use in source · Read passage · 02_coordinates.tex, line 406
[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j,
Exact mathematical passage
\omega,\omega^\vee,\eta\in R
\Lambda_0Use in source · 1 passage
Search aliases: Λ_0 Lambda
Use in source · Read passage · 04_arithmetic.tex, line 505
[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Only the lattice has changed.
Exact mathematical passage
\Lambda_0=\OO e_1\oplus\OO e_2,
\Lambda_1Use in source · 5 passages
Search aliases: Λ_1 Lambda
Use in source · Read passage · 04_arithmetic.tex, line 349
[Two complementary packets] Suppose V=V_1⊕ V_2 with projector E onto V_1 and complementary projector I-E. Then Λ/(Λ_1⊕Λ_2) ≅ EΛ/Λ_1 ≅ (I-E)Λ/Λ_2. Consequently s_Λ(E)=s_Λ(I-E) = exp_π(Λ/(Λ_1⊕Λ_2)).
Exact mathematical passage
\Lambda/(\Lambda_1\oplus\Lambda_2) \cong E\Lambda/\Lambda_1 \cong (I-E)\Lambda/\Lambda_2.
\Lambda_2Use in source · 4 passages
Search aliases: Λ_2 Lambda
Use in source · Read passage · 04_arithmetic.tex, line 349
[Two complementary packets] Suppose V=V_1⊕ V_2 with projector E onto V_1 and complementary projector I-E. Then Λ/(Λ_1⊕Λ_2) ≅ EΛ/Λ_1 ≅ (I-E)Λ/Λ_2. Consequently s_Λ(E)=s_Λ(I-E) = exp_π(Λ/(Λ_1⊕Λ_2)).
Exact mathematical passage
\Lambda/(\Lambda_1\oplus\Lambda_2) \cong E\Lambda/\Lambda_1 \cong (I-E)\Lambda/\Lambda_2.
\Lambda_pUse in source · 1 passage
Search aliases: Λ_p Lambda
Use in source · visual_atlas.tex, line 361
0 · 2, AB=Bdiag(1,-1), E_+=Bdiag(1,0)B^-1. The index-two eigenlattice explains the denominator in B^-1 and in the projector. Over the localized lattice Λ_p=ZZ_(p)^2, the number 2 is a unit for every odd prime p. Hence the local denominator weight is zero there and one at p=2.
Exact mathematical passage
\Lambda_p=\ZZ_{(p)}^2
\Lambda_{\Pcal}Use in source · 4 passages
Search aliases: Λ_Pcal Λ_ P Lambda Pcal Lambda mathcal
Use in source · Read passage · 04_arithmetic.tex, line 399
[Gluing exact sequence under refinement] Let Q=(Q_β) refine P=(P_α) on V, and let Λ be a lattice. Define Λ_ P = ⊕_α(Λ∩ P_α V), Λ_ Q = ⊕_β(Λ∩ Q_β V). Then Λ_ Q⊆Λ_ P⊆Λ and there is a short exact sequence 0 ⟶ Λ_ P/Λ_ Q ⟶ Λ/Λ_ Q ⟶ Λ/Λ_ P ⟶0. In particular, s_Λ( P)≤ s_Λ( Q).
Exact mathematical passage
\Lambda_{\Pcal} = \bigoplus_\alpha(\Lambda\cap P_\alpha V), \qquad \Lambda_{\Qcal} = \bigoplus_\beta(\Lambda\cap Q_\beta V).
\Lambda_{\Qcal}Use in source · 4 passages
Search aliases: Λ_Qcal Λ_ Q Lambda Qcal Lambda mathcal
Use in source · Read passage · 04_arithmetic.tex, line 399
[Gluing exact sequence under refinement] Let Q=(Q_β) refine P=(P_α) on V, and let Λ be a lattice. Define Λ_ P = ⊕_α(Λ∩ P_α V), Λ_ Q = ⊕_β(Λ∩ Q_β V). Then Λ_ Q⊆Λ_ P⊆Λ and there is a short exact sequence 0 ⟶ Λ_ P/Λ_ Q ⟶ Λ/Λ_ Q ⟶ Λ/Λ_ P ⟶0. In particular, s_Λ( P)≤ s_Λ( Q).
Exact mathematical passage
\Lambda_{\Pcal} = \bigoplus_\alpha(\Lambda\cap P_\alpha V), \qquad \Lambda_{\Qcal} = \bigoplus_\beta(\Lambda\cap Q_\beta V).
\Lambda_sUse in source · 4 passages
Search aliases: Λ_s Lambda
Use in source · Read passage · 04_arithmetic.tex, line 509
[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Only the lattice has changed.
Exact mathematical passage
\Lambda_s = \OO e_1+ \OO(\pi^{-s}e_1+e_2).
\pi^{-s}Use in source · 2 passages
Search aliases: π^-s pi
Use in source · Read passage · 04_arithmetic.tex, line 509
[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Only the lattice has changed.
Exact mathematical passage
\Lambda_s = \OO e_1+ \OO(\pi^{-s}e_1+e_2).
\pi^aUse in source · 2 passages
Search aliases: π^a pi
Use in source · Read passage · 03_weighted.tex, line 211
[A triangular transfer] Let O be complete with uniformizer π, and consider T= 1 · π^a 0 · 1 with a≥1. The only nontrivial transfer has weight a, and its square is zero. Hence T^-1 = 1 · -π^a
Exact mathematical passage
T= \begin{pmatrix} 1&\pi^a\\ 0&1 \end{pmatrix}
\pi^{a_1}Use in source · 1 passage
Search aliases: π^a_1 pi
Use in source · Read passage · 04_arithmetic.tex, line 729
[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).
Exact mathematical passage
\diag(\pi^{a_1},\ldots,\pi^{a_n}), \qquad 0\leq a_1\leq\cdots\leq a_n,
\pi^{a_j}Use in source · 3 passages
Search aliases: π^a_j pi
Use in source · Read passage · 04_arithmetic.tex, line 284
ght] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).
Exact mathematical passage
G_{\Lambda,\Pcal} \cong \bigoplus_{j=1}^r\OO/\pi^{a_j}\OO
\pi^{a_n}Use in source · 1 passage
Search aliases: π^a_n pi
Use in source · Read passage · 04_arithmetic.tex, line 729
[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).
Exact mathematical passage
\diag(\pi^{a_1},\ldots,\pi^{a_n}), \qquad 0\leq a_1\leq\cdots\leq a_n,
\pi^dUse in source · 5 passages
Search aliases: π^d pi
Use in source · Read passage · 04_arithmetic.tex, line 315
Let s=s_Λ( P). For every x∈Λ and every α, π^sE_α x∈Λ_α. Hence π^sx=∑_απ^sE_α x∈Λ_split, so π^sG_Λ, P=0. This proves exp_π(G_Λ, P)≤ s. If d<s, then for some α there is an x∈Λ with π^dE_α x∉Λ_α. The class π^d[x] is therefore nonzero in the gluing module. Thus π^d does not annihilate the gluing module, and equality follows. The final formula is the usual description of the exponent through Smith normal form.
Exact mathematical passage
\pi^dE_\alpha x\notin\Lambda_\alpha.
\pi^LUse in source · 8 passages
Search aliases: π^L pi
Use in source · Read passage · 03_weighted.tex, line 29
If O is a complete discrete valuation ring with uniformizer π and B is a separated and complete O algebra for the filtration F^L B=π^L B, then this is a standard example. Finite matrix algebras over O satisfy these conditions. Completeness of O alone does not imply completeness of every O algebra. Basic facts about completeness in this setting are reviewed in [source reference].
Exact mathematical passage
F^L\Bcal=\pi^L\Bcal,
\pi^mUse in source · 8 passages
Search aliases: π^m pi
Use in source · Read passage · 03_weighted.tex, line 237
Let O be a complete discrete valuation ring with uniformizer π, fraction field K, and filtration F^m B=π^m B on an O-algebra B. The finite quotient statements below do not require B itself to be complete. Suppose T=D+H, where D is invertible and the perturbation H raises filtration. The symbol K is reserved for the fraction field.
Exact mathematical passage
F^m\cB=\pi^m\cB
\pi^{m+d}Use in source · 3 passages
Search aliases: π^m+d pi
Use in source · Read passage · 04_arithmetic.tex, line 460
n delay] Let E be a rational projector on V, let Λ be a lattice, and let s=s_Λ(E). For every m≥0, the rule [x]_m+s⟼[Ex]_m from Λ/π^m+sΛ to V/π^mΛ is well defined. Here m is an integer. If d is an integer with 0≤ d<s, then for every integer m≥0 there are x,y∈Λ such that x≡ ypmodπ^m+dΛ but Exnot≡ Eypmodπ^mΛ. Thus s extra digits are both sufficient and necessary for a uniform precision guarantee.
Exact mathematical passage
x\equiv y\pmod{\pi^{m+d}\Lambda}
\pi^{m+s}Use in source · 5 passages
Search aliases: π^m+s pi
Use in source · Read passage · 04_arithmetic.tex, line 209
[Projector precision delay] Let E be a rational projector with π^sE(Λ)⊆Λ. If x≡ ypmodπ^m+sΛ, then Ex≡ Eypmodπ^mΛ. Thus m+s input digits suffice to determine m output digits of the projected packet.
Exact mathematical passage
x\equiv y\pmod{\pi^{m+s}\Lambda},
\pi^{\min(s_i,L)}Use in source · 2 passages
Search aliases: π^min(s_i,L) pi min
Use in source · Read passage · 04_arithmetic.tex, line 758
[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L (r<i≤ m). When it is solvable, the number of solutions in ( O/π^L O)^n is q^(n-r)L+∑_i=1^rmin(s_i,L).
Exact mathematical passage
\widetilde b_i\equiv0 \pmod{\pi^{\min(s_i,L)}} \qquad(1\leq i\leq r)
\pi^{s_1}Use in source · 2 passages
Search aliases: π^s_1 pi
Use in source · Read passage · 04_arithmetic.tex, line 711
[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].
Exact mathematical passage
UMV = \diag(\pi^{s_1},\ldots,\pi^{s_r},0,\ldots,0),
\pi^{s_i}Use in source · 2 passages
Search aliases: π^s_i pi
Use in source · Read passage · 04_arithmetic.tex, line 779
Set x=Vy. Multiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L choices. Multiplying the independent counts gives the formula.
Exact mathematical passage
\pi^{s_i}y_i\equiv\widetilde b_i\pmod{\pi^L} \qquad(1\leq i\leq r)
\pi^{s_r}Use in source · 2 passages
Search aliases: π^s_r pi
Use in source · Read passage · 04_arithmetic.tex, line 711
[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].
Exact mathematical passage
UMV = \diag(\pi^{s_1},\ldots,\pi^{s_r},0,\ldots,0),
\pi^{\val_\pi(\Res(f,g))}Use in source · 1 passage
Search aliases: π^val_π(Res(f,g)) pi val pi Res pi operatorname pi operatorname
Use in source · Read passage · 04_arithmetic.tex, line 89
[Resultant denominator bound] Under the hypotheses of [source reference], one may take r=Res(f,g) up to multiplication by a unit. Therefore π^val_π(Res(f,g))E_fΛ⊆Λ.
Exact mathematical passage
\pi^{\val_\pi(\Res(f,g))}E_f\Lambda\subseteq\Lambda.
\rho^{n-1}Use in source · 1 passage
Search aliases: ρ^n-1 rho
Use in source · Read passage · 05_defects.tex, line 70
[Norm bound] For a submultiplicative operator norm, CT_f^n-T_c^nC ≤ Δ ∑_k=0^n-1 T_c ^n-1-k T_f ^k. If T_f , T_c ≤ρ, then CT_f^n-T_c^nC ≤ nρ^n-1 Δ .
Exact mathematical passage
\boxed{ \|CT_f^n-T_c^nC\| \le n\rho^{n-1}\|\Delta\|. }
\rho_{m,\ell}Use in source · 3 passages
Search aliases: ρ_m,ℓ rho ell
Use in source · Read passage · 02_coordinates.tex, line 69
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{m,\ell}:I_m\longrightarrow I_\ell \qquad(m\geq\ell)
\rho_{n,m}Use in source · 1 passage
Search aliases: ρ_n,m rho
Use in source · Read passage · 02_coordinates.tex, line 74
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{\ell,\ell}=\id_{I_\ell}, \qquad \rho_{n,\ell}=\rho_{m,\ell}\circ\rho_{n,m} \qquad(n\geq m\geq\ell).
\rho_{n,\ell}Use in source · 1 passage
Search aliases: ρ_n,ℓ rho ell
Use in source · Read passage · 02_coordinates.tex, line 74
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{\ell,\ell}=\id_{I_\ell}, \qquad \rho_{n,\ell}=\rho_{m,\ell}\circ\rho_{n,m} \qquad(n\geq m\geq\ell).
\rho_{\ell,\ell}Use in source · 1 passage
Search aliases: ρ_ℓ,ℓ rho ell ell
Use in source · Read passage · 02_coordinates.tex, line 74
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{\ell,\ell}=\id_{I_\ell}, \qquad \rho_{n,\ell}=\rho_{m,\ell}\circ\rho_{n,m} \qquad(n\geq m\geq\ell).
\sigmaUse in source · 1 passage
Search aliases: σ sigma
Use in source · Read passage · 05_defects.tex, line 170
The assertion is immediate if either space is zero. Otherwise, use the rectifiable-cycle form of the holomorphic functional calculus in [source reference]. For each operator T, compare Γ with a positive simple system around its spectrum. Their difference has zero winding outside U∖σ(T), so Theorem~4.1 applied to the resolvent-valued integrand identifies the two integrals. Consequently, h(T) = frac12π i ∫_Γ h(z)(z-T)^-1 dz. Subtract the two formulas after composing with C, then use [source refere…
Exact mathematical passage
U\setminus\sigma(T)
\PhiUse in source · 2 passages
Search aliases: Φ Phi
Use in source · Read passage · 04_arithmetic.tex, line 268
[Gluing module and exact separation weight] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).
Exact mathematical passage
\Phi:G_{\Lambda,\Pcal} \longrightarrow \bigoplus_{\alpha\in I}E_\alpha\Lambda/\Lambda_\alpha, \qquad [x]\longmapsto(E_\alpha x+\Lambda_\alpha)_\alpha
\chiUse in source · 6 passages
Search aliases: χ chi
Use in source · Read passage · 04_arithmetic.tex, line 11
[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).
Exact mathematical passage
\chi_A(x)=f(x)g(x)
\chi_AUse in source · 6 passages
Search aliases: χ_A chi
Use in source · Read passage · 04_arithmetic.tex, line 11
[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).
Exact mathematical passage
\chi_A(x)=f(x)g(x)
\omegaUse in source · 25 passages
Search aliases: ω omega
Use in source · Read passage · 02_coordinates.tex, line 399
Let U act on coefficient sequences by (UF)(z)=F(ω z). If ω^N=1, then U^N=I.
Exact mathematical passage
(UF)(z)=F(\omega z).
\omega^{-1}Use in source · 3 passages
Search aliases: ω^-1 omega
Use in source · Read passage · 02_coordinates.tex, line 427
0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.
Exact mathematical passage
\omega^\vee=\omega^{-1}
\omega^{-jk}Use in source · 2 passages
Search aliases: ω^-jk omega
Use in source · Read passage · 02_coordinates.tex, line 457
The field specialization. [Character realization] Let bbk be a field, assume that N is nonzero in bbk, and suppose that bbk contains a primitive Nth root of unity ω. Define (UF)(z)=F(ω z). Then P_N,j = frac1N∑_k=0^N-1ω^-jkU^k.
Exact mathematical passage
P_{N,j} = \frac1N\sum_{k=0}^{N-1}\omega^{-jk}U^k.
\omega^3Use in source · 1 passage
Search aliases: ω^3 omega
Use in source · finite_fourier_table.tex, line 41
.35pt,] (5.00000,1.10000) rectangle ++(0.70000,0.70000); node[] at (5.35000,1.45000) 0; draw[fill=black!0,draw=black!45,line width=.35pt,] (5.70000,1.10000) rectangle ++(0.70000,0.70000); node[] at (6.05000,1.45000) 0; draw[fill=black!12,draw=black!45,line width=.35pt,] (6.40000,1.10000) rectangle ++(0.70000,0.70000); node[] at (6.75000,1.45000) 1; node[] at (10.15000,3.05000) ω=2, ω^3=1; node[] at (10.15000,2.15000) ω^-1=4, η=5; node[] at (10.15000,1.25000) j,r∈{0,1,2}; draw[black!35] (0.00000,…
Exact mathematical passage
\omega=2,\quad\omega^3=1
\omega^{k(n-j)}Use in source · 1 passage
Search aliases: ω^k(n-j) omega
Use in source · Read passage · 02_coordinates.tex, line 466
On z^n, the right side acts by the scalar frac1N∑_k=0^N-1ω^k(n-j). This is one when n≡ jpmod N and zero otherwise by the geometric sum identity in the field bbk.
Exact mathematical passage
\frac1N\sum_{k=0}^{N-1}\omega^{k(n-j)}.
\omega^NUse in source · 3 passages
Search aliases: ω^N omega
Use in source · Read passage · 02_coordinates.tex, line 402
Let U act on coefficient sequences by (UF)(z)=F(ω z). If ω^N=1, then U^N=I.
Exact mathematical passage
\omega^N=1
\omega^{nk}Use in source · 1 passage
Search aliases: ω^nk omega
Use in source · Read passage · 02_coordinates.tex, line 432
On the monomial z^n, the right-hand side acts by η∑_k=0^N-1(ω^vee)^jkω^nk. The identity ω^N=1 reduces this expression to the table entry indexed by r=nbmod N. The finite certificate therefore gives exactly the coefficient mask defining P_N,j. Over a field, the primitive-root geometric sum supplies the certificate.
Exact mathematical passage
\eta\sum_{k=0}^{N-1}(\omega^\vee)^{jk}\omega^{nk}.
\omega^{rk}Use in source · 1 passage
Search aliases: ω^rk omega
Use in source · Read passage · 02_coordinates.tex, line 408
[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j, 0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.
Exact mathematical passage
\eta\sum_{k=0}^{N-1}(\omega^\vee)^{jk}\omega^{rk} = \begin{cases} 1,&r=j,\\ 0,&r\ne j, \end{cases} \qquad 0\le j,r<N.
\omega^\veeUse in source · 5 passages
Search aliases: ω^vee omega vee
Use in source · Read passage · 02_coordinates.tex, line 406
[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j,
Exact mathematical passage
\omega,\omega^\vee,\eta\in R
\omega_N^kUse in source · 4 passages
Search aliases: ω_N^k omega
Use in source · Read passage · 06_models.tex, line 46
Let Z and Z_0≠0 be complex impedances, put z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this domain gives T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). For N=2 and k=1, T_2,1(z)=frac1z, z≠0,-1. Thus the nontrivial action at the firs…
Exact mathematical passage
\omega_N^k
\ell^2Use in source · 1 passage
Search aliases: ℓ^2 ell
Use in source · visual_atlas.tex, line 317
headThree boundaries of this calculation This is a formal series identity. No analytic convergence at a chosen real or complex value of t is needed. The canonical determinant need not be the minimal denominator after cancellation. Finally, positive weights prevent infinitely many walks at a fixed weight: a single weight-zero loop would already give the infinitely many distinct walks 1,ℓ,ℓ^2,… at weight zero.
Exact mathematical passage
1,\ell,\ell^2,\ldots
\mapstoUse in source · 3 passages
Search aliases: ↦ mapsto
Use in source · Read passage · 06_models.tex, line 47
Let Z and Z_0≠0 be complex impedances, put z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this domain gives T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). For N=2 and k=1, T_2,1(z)=frac1z, z≠0,-1. Thus the nontrivial action at the firs…
Exact mathematical passage
\Gamma\mapsto\omega_N^k\Gamma.
\hookrightarrowUse in source · 1 passage
Search aliases: ↪ hookrightarrow
Use in source · Read passage · 04_arithmetic.tex, line 725
[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).
Exact mathematical passage
\Lambda_{\mathrm{split}}\hookrightarrow\Lambda.
\RightarrowUse in source · 1 passage
Search aliases: ⇒ Rightarrow
Use in source · Read passage · 06_models.tex, line 285
The proof order is: · complete packet axioms ⇒finite-packet uniqueness and coordinate coherence · ⇒coordinate naturality and refined transfer blocks, · residue selectors and a finite Fourier table ⇒Fourier formulas, · coprime moduli⇒dyadic--odd formulas, · quotient, filtration, and edge-weight certificates ⇒explicit pruned inverse, · finite positive-weight graph ⇒semantic enumeration and first-edge recurrence · ⇒det/adj rationality and recurrence, · minimal evaluated B ezout data and inverse-uni…
Exact mathematical passage
\begin{aligned} &\text{complete packet axioms} \Rightarrow\text{finite-packet uniqueness and coordinate coherence}\\ &\qquad\Rightarrow\text{coordinate naturality and refined transfer blocks},\\ &\text{residue selectors and a finite Fourier table} \Rightarrow\text{Fourier formulas},\\ &\text{coprime moduli}\Rightarrow\text{dyadic--odd formulas},\\ &\text{quotient, filtration, and edge-weight certificates} \Rightarrow\text{explicit pruned inverse},\\ &\text{finite positive-weight graph} \Rightarrow\text{semantic enumeration and first-edge recurrence}\\ &\qquad\Rightarrow\det/\operatorname{adj}\text{ rationality and recurrence},\\ &\text{minimal evaluated B\'ezout data and inverse-unit lattice action}\\ &\qquad\Rightarrow\text{separation bound, upward closure, and precision delay},\\ &\text{concrete }\mathbb Z/p^L\mathbb Z\text{ coordinate fibres} \Rightarrow\text{finite Smith product count},\\ &\text{one-step typed defect} \Rightarrow\text{powers, polynomial actions, and witnessed resolvents}. \end{aligned}
\existsUse in source · 1 passage
Search aliases: ∃ exists
Use in source · Read passage · 04_arithmetic.tex, line 856
Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fibre has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global count. If one coordinate fibre is empty, the global fibre is empty. Lean proves both the global solvability equivalence and the conditional count, inc…
Exact mathematical passage
\bigl(\exists x\in\mathbb Z/p^L\mathbb Z:\quad p^s x=b\pmod{p^L}\bigr) \quad\Longleftrightarrow\quad p^{\min(s,L)}\mid b_{\mathrm{val}},
\notinUse in source · 5 passages
Search aliases: ∉ notin
Use in source · Read passage · 04_arithmetic.tex, line 315
Let s=s_Λ( P). For every x∈Λ and every α, π^sE_α x∈Λ_α. Hence π^sx=∑_απ^sE_α x∈Λ_split, so π^sG_Λ, P=0. This proves exp_π(G_Λ, P)≤ s. If d<s, then for some α there is an x∈Λ with π^dE_α x∉Λ_α. The class π^d[x] is therefore nonzero in the gluing module. Thus π^d does not annihilate the gluing module, and equality follows. The final formula is the usual description of the exponent through Smith normal form.
Exact mathematical passage
\pi^dE_\alpha x\notin\Lambda_\alpha.
\prod_{i=1}^rUse in source · 1 passage
Search aliases: ∏_i=1^r prod
Use in source · Read passage · 04_arithmetic.tex, line 56
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
\chi_A(x)=\prod_{i=1}^rf_i(x),
\prod_{j\ne i}Use in source · 1 passage
Search aliases: ∏_j≠ i prod ne
Use in source · Read passage · 04_arithmetic.tex, line 60
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
g_i=\prod_{j\ne i}f_j.
\sum_eUse in source · 1 passage
Search aliases: ∑_e sum
Use in source · Read passage · 03_weighted.tex, line 329
W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is a two-sided inverse of q_L(I+A), where A=∑_eA_e=D^-1H. With the supplied two-sided inverse D^-1 of D, this gives the explicit two-sided inverse q_L (S^pr_L,ND^-1) of q_L(D+H). If every edge weight is at least m and L≤ mN, then the nilpotence condition follows from the filtration data because m|γ|≤ W(γ). For m>0, the executable choice N=lceil L/mrceil satisfies L≤ mN. This elem…
Exact mathematical passage
A=\sum_eA_e=D^{-1}H
\sum_{e:\alpha\to\beta}Use in source · 1 passage
Search aliases: ∑_e:α→β sum alpha to beta
Use in source · Read passage · 03_weighted.tex, line 99
ra of operators. Suppose D=∑_α P_α D P_α belongs to B and is invertible on every packet. Assume that the assembled inverse D^-1=∑_α D_α^-1P_α∈ B=F^0 B, D_α=P_α DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be finite, assume w(e)≥1 for every edge, and write N=∑_e:α→βN_e, N_e=P_β N_eP_α, N_e∈ F^w(e) B. For an edge e:α→β, put A_e=(P_β DP_β)^-1N_e. Then the block from P_α X to P_β X of (D+N)^-1 is P_β(D+N)^-1P_α = ∑_γ:αleadstoβ (-1)^|γ|A_γ D_α^-1, where D…
Exact mathematical passage
N=\sum_{e:\alpha\to\beta}N_e, \qquad N_e=P_\beta N_eP_\alpha, \qquad N_e\in F^{w(e)}\Bcal.
\sum_{e\in E}Use in source · 4 passages
Search aliases: ∑_e∈ E sum in
Use in source · Read passage · 03_weighted.tex, line 40
[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
T=D+\sum_{e\in E}N_e, \qquad N_e\in F^{w(e)}\Bcal, \qquad w(e)\geq1.
\sum_iUse in source · 2 passages
Search aliases: ∑_i sum
Use in source · Read passage · 04_arithmetic.tex, line 64
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
\im E_i=\Ker f_i(A), \qquad \sum_iE_i=I, \qquad E_iE_j=0 \quad(i\ne j).
\sum_{i=1}^dUse in source · 1 passage
Search aliases: ∑_i=1^d sum
Use in source · Read passage · 04_arithmetic.tex, line 844
ongruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
\boxed{ p^{\sum_{i=1}^d\min(s_i,L)}. }
\sum_{i=1}^rUse in source · 1 passage
Search aliases: ∑_i=1^r sum
Use in source · Read passage · 04_arithmetic.tex, line 770
rdinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L (r<i≤ m). When it is solvable, the number of solutions in ( O/π^L O)^n is q^(n-r)L+∑_i=1^rmin(s_i,L).
Exact mathematical passage
\boxed{ q^{(n-r)L+\sum_{i=1}^r\min(s_i,L)}. }
\sum_jUse in source · 2 passages
Search aliases: ∑_j sum
Use in source · visual_atlas.tex, line 79
Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.
Exact mathematical passage
X_N^{\rm pkt}=\bigoplus_j P_{N,j}X, \qquad \mathcal A_Nx=(P_{N,j}x)_j, \qquad \mathcal S_N(x_j)_j=\sum_jx_j.
\sum_{j=0}^{N-1}Use in source · 1 passage
Search aliases: ∑_j=0^N-1 sum
Use in source · Read passage · 02_coordinates.tex, line 383
[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.
Exact mathematical passage
P_{N,i}P_{N,j}=\delta_{ij}P_{N,j}, \qquad \sum_{j=0}^{N-1}P_{N,j}=I.
\sum_{j=1}^rUse in source · 3 passages
Search aliases: ∑_j=1^r sum
Use in source · Read passage · 03_weighted.tex, line 52
[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
A_\gamma=A_{e_r}\cdots A_{e_1}, \qquad W(\gamma)=\sum_{j=1}^rw(e_j).
\sum_kUse in source · 2 passages
Search aliases: ∑_k sum
Use in source · Read passage · 03_weighted.tex, line 557
[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).
Exact mathematical passage
Q(t)=\sum_kq_kt^k, \qquad Z_{s,t}(t)=\sum_{n\ge0}a_n(s,t)t^n.
\sum_{k=0}^2Use in source · 1 passage
Search aliases: ∑_k=0^2 sum
Use in source · visual_atlas.tex, line 166
On the monomial z^n, define UF(z)=F(2z). The proposed projector acts by 5∑_k=0^2 4^jk2^nk. The identity 2^3=1 makes this depend only on nbmod3. The nine-entry table therefore establishes the formula for every exponent once periodicity is used, rather than requiring a separate test at every coefficient.
Exact mathematical passage
5\sum_{k=0}^2 4^{jk}2^{nk}.
\sum_{k=0}^{n}Use in source · 1 passage
Search aliases: ∑_k=0^n sum
Use in source · Read passage · 03_weighted.tex, line 564
[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).
Exact mathematical passage
\boxed{ \sum_{k=0}^{n}q_k a_{n-k}(s,t)=0 \qquad\bigl(n>\deg P_{s,t}\bigr). }
\sum_{k=0}^{N-1}Use in source · 7 passages
Search aliases: ∑_k=0^N-1 sum
Use in source · Read passage · 02_coordinates.tex, line 408
[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j, 0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.
Exact mathematical passage
\eta\sum_{k=0}^{N-1}(\omega^\vee)^{jk}\omega^{rk} = \begin{cases} 1,&r=j,\\ 0,&r\ne j, \end{cases} \qquad 0\le j,r<N.
\sum_{k=0}^{n-1}Use in source · 5 passages
Search aliases: ∑_k=0^n-1 sum
Use in source · Read passage · 05_defects.tex, line 27
[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
CT_f^n-T_c^nC = \sum_{k=0}^{n-1}T_c^{n-1-k}\Delta T_f^k.
\sum_{k=0}^{r-1}Use in source · 2 passages
Search aliases: ∑_k=0^r-1 sum
Use in source · Read passage · 03_weighted.tex, line 278
[Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.
Exact mathematical passage
\boxed{(D+H)^{-1}=\left(\sum_{k=0}^{r-1}(-A)^k\right)D^{-1}.}
\sum_{k\ge0}Use in source · 2 passages
Search aliases: ∑_k≥0 sum ge
Use in source · Read passage · 03_weighted.tex, line 282
[Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.
Exact mathematical passage
\sum_{k\ge0}(-A)^k
\sum_{n=0}^NUse in source · 1 passage
Search aliases: ∑_n=0^N sum
Use in source · Read passage · 05_defects.tex, line 80
[Polynomial defect] For p(z)=∑_n=0^Na_nz^n, we have Cp(T_f)-p(T_c)C = ∑_n=1^Na_n ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
p(z)=\sum_{n=0}^Na_nz^n,
\sum_{n=0}^{N-1}Use in source · 1 passage
Search aliases: ∑_n=0^N-1 sum
Use in source · Read passage · 03_weighted.tex, line 321
et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is…
Exact mathematical passage
S^{\mathrm{pr}}_{L,N} = \sum_{n=0}^{N-1}(-1)^n \sum_{\substack{\gamma\in E^n\\W(\gamma)<L}}A_\gamma.
\sum_{n=1}^NUse in source · 1 passage
Search aliases: ∑_n=1^N sum
Use in source · Read passage · 05_defects.tex, line 84
[Polynomial defect] For p(z)=∑_n=0^Na_nz^n, we have Cp(T_f)-p(T_c)C = ∑_n=1^Na_n ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
Cp(T_f)-p(T_c)C = \sum_{n=1}^Na_n \sum_{k=0}^{n-1}T_c^{n-1-k}\Delta T_f^k.
\sum_{n\ge0}Use in source · 2 passages
Search aliases: ∑_n≥0 sum ge
Use in source · Read passage · 03_weighted.tex, line 485
For fixed start and target vertices s,t, define Z_s,t(t) = ∑_n≥0a_n(s,t)t^n, where a_n(s,t) counts directed walks from s to t with total weight n. Vertices and edges may repeat. Thus ``path zeta'' is retained as the name of the series and Lean API, but the walks are not required to visit each vertex only once.
Exact mathematical passage
Z_{s,t}(t) = \sum_{n\ge0}a_n(s,t)t^n,
\sum_{\rm children}Use in source · 1 passage
Search aliases: ∑_rm children sum rm
Use in source · regrouping_not_averaging.tex, line 52
=.35pt,] (4.57000,2.26000) rectangle ++(0.57000,0.57000); node[] at (4.85500,2.54500) 2; draw[fill=black!0,draw=black!45,line width=.35pt,] (5.14000,2.26000) rectangle ++(0.57000,0.57000); node[] at (5.42500,2.54500) 0; draw[fill=black!12,draw=black!45,line width=.35pt,] (5.71000,2.26000) rectangle ++(0.57000,0.57000); node[] at (5.99500,2.54500) 4; node[] at (3.15000,2.20000) ∑_rm children; node[] at (1.14000,0.10000) 4×1=4 free coordinates; node[] at (5.14000,0.10000) 2×2=4 free coordinates; d…
Exact mathematical passage
\sum_{\rm children}
\sum_{r\geq0}Use in source · 2 passages
Search aliases: ∑_r≥0 sum geq
Use in source · Read passage · 03_weighted.tex, line 72
Put A=∑_e∈ EA_e. Since left multiplication by D^-1 preserves the filtration, A∈ F^1 B. Completeness gives the convergent Neumann series (I+A)^-1=∑_r≥0(-A)^r. Because T=D(I+A), T^-1=(I+A)^-1D^-1. Expanding each power of A gives the sum over words, with the rightmost edge acting first. Multiplicativity of the filtration gives A_γ∈ F^W(γ) B. For each L, only finitely many words have weight below L because E is finite and every edge has positive weight. Thus the word expansion converges to the same …
Exact mathematical passage
(I+A)^{-1}=\sum_{r\geq0}(-A)^r.
\sum_{\substack{0\leq s<M\\s\equiv j\pmod N}}Use in source · 1 passage
Search aliases: ∑_substack0≤ s<M s≡ jpmod N sum substack leq s equiv pmod
Use in source · Read passage · 02_coordinates.tex, line 478
[Nested residue refinement] Let N,M be positive integers with N dividing M, and let 0≤ j<N. Then P_N,j = ∑_substack0≤ s<M s≡ jpmod NP_M,s. When R is nonzero, the packet system modulo M refines the packet system modulo N.
Exact mathematical passage
P_{N,j} = \sum_{\substack{0\leq s<M\\s\equiv j\pmod N}}P_{M,s}.
\sum_{\substack{e:\,\mathrm{source}(e)=s\\w(e)\le n}}Use in source · 1 passage
Search aliases: ∑_substacke: source(e)=s w(e)≤ n sum substack mathrm w le
Use in source · visual_atlas.tex, line 289
headThe first-edge decomposition Let a_n(s,t) count walks of total weight n from s to t. Removing the first edge gives a_n(s,t)=1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). The empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most n, so enumeration at each fixed weight is finite.
Exact mathematical passage
a_n(s,t)=\mathbf1_{s=t,\,n=0} +\sum_{\substack{e:\,\mathrm{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\mathrm{target}(e),t).
\sum_{\substack{e:\,\operatorname{source}(e)=s\\w(e)\le n}}Use in source · 1 passage
Search aliases: ∑_substacke: source(e)=s w(e)≤ n sum substack operatorname w le
Use in source · Read passage · 03_weighted.tex, line 499
[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_n(s,t) = 1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, henc…
Exact mathematical passage
a_n(s,t) = \mathbf 1_{s=t,\,n=0} +\sum_{\substack{e:\,\operatorname{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\operatorname{target}(e),t).
\sum_{\substack{\gamma:\ P_\alpha R_\gamma=R_\gamma\\Q_\beta R_\gamma=R_\gamma}}Use in source · 1 passage
Search aliases: ∑_substackγ: P_α R_γ=R_γ Q_β R_γ=R_γ ∑_substackγ: P_α R_γ=R_γ{ Q_β R_γ=R_γ sum substack gamma alpha gamma gamma Q beta gamma gamma sum substack gamma alpha gamma gamma mathbb beta gamma gamma
Use in source · Read passage · 02_coordinates.tex, line 339
a packet in R. Because R refines P, there is a unique α with P_α R_γ=R_γ. Likewise there is a unique β with Q_β R_γ=R_γ. Therefore P_α Q_β R_γ=R_γ. Every packet of R lies in one product packet. Multiplying ∑_γ R_γ=I_X by P_α Q_β and using the parent identities gives P_α Q_β =∑_substackγ: P_α R_γ=R_γ{ Q_β R_γ=R_γR_γ. Thus the packets with the same pair of parents reconstruct the product packet. If that product packet is nonzero, the sum has a child, so the parent map onto the supported product fa…
Exact mathematical passage
P_\alpha Q_\beta =\sum_{\substack{\gamma:\ P_\alpha R_\gamma=R_\gamma\\Q_\beta R_\gamma=R_\gamma}}R_\gamma.
\sum_{\substack{\gamma\in E^n\\W(\gamma)<L}}Use in source · 1 passage
Search aliases: ∑_substackγ∈ E^n W(γ)<L sum substack gamma in W gamma
Use in source · Read passage · 03_weighted.tex, line 321
et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is…
Exact mathematical passage
S^{\mathrm{pr}}_{L,N} = \sum_{n=0}^{N-1}(-1)^n \sum_{\substack{\gamma\in E^n\\W(\gamma)<L}}A_\gamma.
\sum_{\substack{\rho(\beta)=b\\\rho(\alpha)=a}}Use in source · 1 passage
Search aliases: ∑_substackρ(β)=b ρ(α)=a sum substack rho beta rho alpha
Use in source · Read passage · 02_coordinates.tex, line 224
[Refinement of operator blocks] Suppose that Q refines P through ρ. Then T^ P_ba = ∑_substackρ(β)=b ρ(α)=a T^ Q_βα. The matrix of T in fine coordinates is related to the matrix in coarse coordinates by [T]_ P = C_ P← Q[T]_ QJ_ Q← P.
Exact mathematical passage
T^{\Pcal}_{ba} = \sum_{\substack{\rho(\beta)=b\\\rho(\alpha)=a}} T^{\Qcal}_{\beta\alpha}.
\sum_{W(\gamma)<L}Use in source · 2 passages
Search aliases: ∑_W(γ)<L sum gamma
Use in source · Read passage · 03_weighted.tex, line 187
[Exact finite precision truncation] Under the hypotheses of [source reference], define R_L = ∑_W(γ)<L(-1)^|γ|A_γ D^-1. Then T^-1-R_L∈ F^L B, TR_L-I∈ F^L B, R_LT-I∈ F^L B. Thus R_L is the exact inverse in B/F^L B.
Exact mathematical passage
R_L = \sum_{W(\gamma)<L}(-1)^{|\gamma|}A_\gamma D^{-1}.
\sum_{\alpha,\beta}Use in source · 1 passage
Search aliases: ∑_α,β sum alpha beta
Use in source · Read passage · 02_coordinates.tex, line 304
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
\sum_{\alpha,\beta}R_{\alpha\beta} = \left(\sum_\alpha P_\alpha\right) \left(\sum_\beta Q_\beta\right) =I_X.
\sum_\betaUse in source · 2 passages
Search aliases: ∑_β sum beta
Use in source · Read passage · 02_coordinates.tex, line 304
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
\sum_{\alpha,\beta}R_{\alpha\beta} = \left(\sum_\alpha P_\alpha\right) \left(\sum_\beta Q_\beta\right) =I_X.
\sum_{\beta\succ b,\,\alpha\succ a}Use in source · 1 passage
Search aliases: ∑_βsucc b, αsucc a sum beta succ alpha succ
Use in source · visual_atlas.tex, line 125
headRefined blocks still reconstruct the same map A coarse block expands as P_bTP_a=∑_βsucc b, αsucc aP_β TP_α. The terms are maps embedded in the original source and target spaces. Adding their norms, or adding their matrix entries after forgetting the embeddings, is not this operator identity.
Exact mathematical passage
P_bTP_a=\sum_{\beta\succ b,\,\alpha\succ a}P_\beta TP_\alpha.
\sum_\gammaUse in source · 2 passages
Search aliases: ∑_γ sum gamma
Use in source · Read passage · 02_coordinates.tex, line 338
Let R_γ be a packet in R. Because R refines P, there is a unique α with P_α R_γ=R_γ. Likewise there is a unique β with Q_β R_γ=R_γ. Therefore P_α Q_β R_γ=R_γ. Every packet of R lies in one product packet. Multiplying ∑_γ R_γ=I_X by P_α Q_β and using the parent identities gives P_α Q_β =∑_substackγ: P_α R_γ=R_γ
Exact mathematical passage
\sum_\gamma R_\gamma=I_X
\sum_{\gamma:\alpha\leadsto\beta}Use in source · 1 passage
Search aliases: ∑_γ:αleadstoβ sum gamma alpha leadsto beta
Use in source · Read passage · 03_weighted.tex, line 111
pha DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be finite, assume w(e)≥1 for every edge, and write N=∑_e:α→βN_e, N_e=P_β N_eP_α, N_e∈ F^w(e) B. For an edge e:α→β, put A_e=(P_β DP_β)^-1N_e. Then the block from P_α X to P_β X of (D+N)^-1 is P_β(D+N)^-1P_α = ∑_γ:αleadstoβ (-1)^|γ|A_γ D_α^-1, where D_α=P_α DP_α and the sum runs over directed paths from α to β, including the empty path when α=β.
Exact mathematical passage
P_\beta(D+N)^{-1}P_\alpha = \sum_{\gamma:\alpha\leadsto\beta} (-1)^{|\gamma|}A_\gamma D_\alpha^{-1},
\sum_{\rho(\alpha)=a}Use in source · 1 passage
Search aliases: ∑_ρ(α)=a sum rho alpha
Use in source · Read passage · 02_coordinates.tex, line 240
Expand the two coarse projectors into their children: P_bTP_a = (∑_ρ(β)=bQ_β) T (∑_ρ(α)=aQ_α). Distributing the finite sums gives the first identity. For the coordinate identity, use [T]_ P= A_ PT S_ P and insert I_X= S_ Q A_ Q.
Exact mathematical passage
P_bTP_a = \left(\sum_{\rho(\beta)=b}Q_\beta\right) T \left(\sum_{\rho(\alpha)=a}Q_\alpha\right).
\sum_{\rho(\beta)=b}Use in source · 1 passage
Search aliases: ∑_ρ(β)=b sum rho beta
Use in source · Read passage · 02_coordinates.tex, line 240
Expand the two coarse projectors into their children: P_bTP_a = (∑_ρ(β)=bQ_β) T (∑_ρ(α)=aQ_α). Distributing the finite sums gives the first identity. For the coordinate identity, use [T]_ P= A_ PT S_ P and insert I_X= S_ Q A_ Q.
Exact mathematical passage
P_bTP_a = \left(\sum_{\rho(\beta)=b}Q_\beta\right) T \left(\sum_{\rho(\alpha)=a}Q_\alpha\right).
\sum_{\rho(\gamma)=\alpha}Use in source · 1 passage
Search aliases: ∑_ρ(γ)=α sum rho gamma alpha
Use in source · Read passage · 02_coordinates.tex, line 62
Multiply the identity P_α=∑_ρ(γ)=αQ_γ by Q_β on either side. Pairwise annihilation in the fine family leaves Q_β when ρ(β)=α and leaves zero otherwise. For transitivity, substitute the child expansion of each Q_β into the expansion of P_α and group the resulting terms by the composite parent.
Exact mathematical passage
P_\alpha=\sum_{\rho(\gamma)=\alpha}Q_\gamma
\sum_{\ell=0}^{L-1}Use in source · 2 passages
Search aliases: ∑_ℓ=0^L-1 sum ell
Use in source · Read passage · 05_defects.tex, line 228
[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .
Exact mathematical passage
C_{0,L}T_L-T_0C_{0,L} = \sum_{\ell=0}^{L-1} C_{0,\ell}\Delta_\ell C_{\ell+1,L}.
\setminusUse in source · 2 passages
Search aliases: ∖ setminus
Use in source · Read passage · 03_weighted.tex, line 445
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
w(e)=(w_1(e),\ldots,w_r(e))\in\mathbb N^r\setminus\{0\}.
\circUse in source · 1 passage
Search aliases: ∘ circ
Use in source · Read passage · 02_coordinates.tex, line 74
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{\ell,\ell}=\id_{I_\ell}, \qquad \rho_{n,\ell}=\rho_{m,\ell}\circ\rho_{n,m} \qquad(n\geq m\geq\ell).
\inftyUse in source · 6 passages
Search aliases: ∞ infty
Use in source · Read passage · 03_weighted.tex, line 129
[Minimum path weight] For packets α and β, let d(α,β) be the least total weight of a directed path from α to β, with d=∞ when no path exists. Use the convention F^∞ B=⋂_L≥0F^L B=0. If α≠β, then P_β(D+N)^-1P_α ∈ F^d(α,β) B. If there is no path from α to β, the block is zero.
Exact mathematical passage
d=\infty
\intUse in source · 4 passages
Search aliases: ∫ int
Use in source · Read passage · 05_defects.tex, line 150
X_c be complex Banach spaces, and let C:X_f→ X_c, T_f, and T_c be bounded complex-linear maps. Let h be holomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z-T_f)^-1 dz. Consequently, Ch(T_f)-h(T_c)C · ≤ fraclength(Γ)2π sup_z∈Γ|h(z)| Δ
Exact mathematical passage
Ch(T_f)-h(T_c)C = \frac{1}{2\pi i} \int_\Gamma h(z)(z-T_c)^{-1}\Delta(z-T_f)^{-1}\,dz.
\int_0^tUse in source · 2 passages
Search aliases: ∫_0^t int
Use in source · Read passage · 05_defects.tex, line 182
[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t is complex. For real t≥0, Ce^tT_f-e^tT_cC ≤ Δ ∫_0^t e^(t-s)T_c e^sT_f ds.
Exact mathematical passage
Ce^{tT_f}-e^{tT_c}C = \int_0^t e^{(t-s)T_c}\Delta e^{sT_f}\,ds,
\int_\GammaUse in source · 2 passages
Search aliases: ∫_Γ int Gamma
Use in source · Read passage · 05_defects.tex, line 150
X_c be complex Banach spaces, and let C:X_f→ X_c, T_f, and T_c be bounded complex-linear maps. Let h be holomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z-T_f)^-1 dz. Consequently, Ch(T_f)-h(T_c)C · ≤ fraclength(Γ)2π sup_z∈Γ|h(z)| Δ
Exact mathematical passage
Ch(T_f)-h(T_c)C = \frac{1}{2\pi i} \int_\Gamma h(z)(z-T_c)^{-1}\Delta(z-T_f)^{-1}\,dz.
\congUse in source · 4 passages
Search aliases: ≅ cong
Use in source · Read passage · 04_arithmetic.tex, line 284
ght] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).
Exact mathematical passage
G_{\Lambda,\Pcal} \cong \bigoplus_{j=1}^r\OO/\pi^{a_j}\OO
\neqUse in source · 1 passage
Search aliases: ≠ neq
Use in source · Read passage · 04_arithmetic.tex, line 689
The failure of the character formula frac12(I± A) to be integral does not invalidate coefficient selectors such as even and odd residue extraction. It shows that two distinct constructions must be distinguished: integral address selector ≠ character-average spectral projector when the group order is not a unit. The first can remain exact over Z_2. The second can acquire a denominator weight.
Exact mathematical passage
\boxed{ \text{integral address selector} \quad\neq\quad \text{character-average spectral projector} }
\oplusUse in source · 9 passages
Search aliases: ⊕ oplus
Use in source · Read passage · 01_orientation.tex, line 56
A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operator on one selected summand may also fail to intertwine with the full matrix. The later sections isolate these four effects one at a time.
Exact mathematical passage
X=X_0\oplus X_1, \qquad D=\begin{pmatrix}D_0&0\\0&D_1\end{pmatrix}, \qquad N=\begin{pmatrix}0&N_{01}\\N_{10}&0\end{pmatrix}.
\bigoplus_jUse in source · 1 passage
Search aliases: ⊕_j bigoplus
Use in source · visual_atlas.tex, line 79
Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.
Exact mathematical passage
X_N^{\rm pkt}=\bigoplus_j P_{N,j}X, \qquad \mathcal A_Nx=(P_{N,j}x)_j, \qquad \mathcal S_N(x_j)_j=\sum_jx_j.
\bigoplus_{j=1}^nUse in source · 1 passage
Search aliases: ⊕_j=1^n bigoplus
Use in source · Read passage · 04_arithmetic.tex, line 735
[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).
Exact mathematical passage
G_{\Lambda,\Pcal} \cong \bigoplus_{j=1}^n\OO/\pi^{a_j}\OO, \qquad s_\Lambda(\Pcal)=\max\bigl(\{0\}\cup\{a_j:1\leq j\leq n\}\bigr).
\bigoplus_{j=1}^rUse in source · 2 passages
Search aliases: ⊕_j=1^r bigoplus
Use in source · Read passage · 04_arithmetic.tex, line 284
ght] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).
Exact mathematical passage
G_{\Lambda,\Pcal} \cong \bigoplus_{j=1}^r\OO/\pi^{a_j}\OO
\bigoplus_{\alpha}Use in source · 1 passage
Search aliases: ⊕_α bigoplus alpha
Use in source · Read passage · 01_orientation.tex, line 18
Λ be an O lattice in V. A rational projector E need not preserve Λ. Its separation weight is the least s such that π^sEΛ⊆Λ. A Bézout construction gives an upper bound through a polynomial resultant. That bound need not be exact because it does not see the position of the lattice. The exact answer is controlled by the finite length module G_Λ, P = Λ/⊕_α(Λ∩ V_α), where V=⊕_α V_α is the rational packet decomposition. The largest Smith exponent of this gluing module is exactly the common denominator…
Exact mathematical passage
G_{\Lambda,\Pcal} = \Lambda\Big/\bigoplus_{\alpha}(\Lambda\cap V_\alpha),
\bigoplus_\betaUse in source · 1 passage
Search aliases: ⊕_β bigoplus beta
Use in source · Read passage · 04_arithmetic.tex, line 399
[Gluing exact sequence under refinement] Let Q=(Q_β) refine P=(P_α) on V, and let Λ be a lattice. Define Λ_ P = ⊕_α(Λ∩ P_α V), Λ_ Q = ⊕_β(Λ∩ Q_β V). Then Λ_ Q⊆Λ_ P⊆Λ and there is a short exact sequence 0 ⟶ Λ_ P/Λ_ Q ⟶ Λ/Λ_ Q ⟶ Λ/Λ_ P ⟶0. In particular, s_Λ( P)≤ s_Λ( Q).
Exact mathematical passage
\Lambda_{\Pcal} = \bigoplus_\alpha(\Lambda\cap P_\alpha V), \qquad \Lambda_{\Qcal} = \bigoplus_\beta(\Lambda\cap Q_\beta V).
\otimesUse in source · 2 passages
Search aliases: ⊗ otimes
Use in source · Read passage · 04_arithmetic.tex, line 4
Let O be a discrete valuation ring with fraction field K and uniformizer π. Let Λ be a finite free O module and put V=K⊗_ OΛ. Suppose A∈End_ O(Λ).
Exact mathematical passage
V=K\otimes_{\OO}\Lambda.
\otimes_\OOUse in source · 1 passage
Search aliases: ⊗_OO ⊗_ O otimes OO otimes mathcal
Use in source · Read passage · 04_arithmetic.tex, line 121
[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).
Exact mathematical passage
F,G,U,V\in\End_K(\Lambda\otimes_\OO K)
\otimes_{\OO}Use in source · 1 passage
Search aliases: ⊗_OO ⊗_ O otimes OO otimes mathcal
Use in source · Read passage · 04_arithmetic.tex, line 4
Let O be a discrete valuation ring with fraction field K and uniformizer π. Let Λ be a finite free O module and put V=K⊗_ OΛ. Suppose A∈End_ O(Λ).
Exact mathematical passage
V=K\otimes_{\OO}\Lambda.
\langleUse in source · 1 passage
Search aliases: ⟨ langle
Use in source · Read passage · 02_coordinates.tex, line 161
[Orthogonal Hilbert packets] If X is a real or complex Hilbert space and every packet system in the tower consists of orthogonal projections, equip each finite packet direct sum with the Hilbert inner product ⟨(x_α),(y_α)⟩=∑_α⟨ x_α,y_α⟩. Then analysis and synthesis are unitary. Every coordinate change in [source reference] is unitary.
Exact mathematical passage
\langle(x_\alpha),(y_\alpha)\rangle=\sum_\alpha\langle x_\alpha,y_\alpha\rangle
\rangleUse in source · 1 passage
Search aliases: ⟩ rangle
Use in source · Read passage · 02_coordinates.tex, line 161
[Orthogonal Hilbert packets] If X is a real or complex Hilbert space and every packet system in the tower consists of orthogonal projections, equip each finite packet direct sum with the Hilbert inner product ⟨(x_α),(y_α)⟩=∑_α⟨ x_α,y_α⟩. Then analysis and synthesis are unitary. Every coordinate change in [source reference] is unitary.
Exact mathematical passage
\langle(x_\alpha),(y_\alpha)\rangle=\sum_\alpha\langle x_\alpha,y_\alpha\rangle
\longmapstoUse in source · 4 passages
Search aliases: ⟼ longmapsto
Use in source · Read passage · 04_arithmetic.tex, line 268
[Gluing module and exact separation weight] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).
Exact mathematical passage
\Phi:G_{\Lambda,\Pcal} \longrightarrow \bigoplus_{\alpha\in I}E_\alpha\Lambda/\Lambda_\alpha, \qquad [x]\longmapsto(E_\alpha x+\Lambda_\alpha)_\alpha
Source coverage
Automatic extraction covers TeX and Markdown math delimiters and common equation environments in these selected files. It does not interpret diagrams, images, PDF text, program identifiers, or historical archives. Occurrences are navigation aids, not a proof review.
Download all entries, passages and source checksums (JSON) · Collection coverage
- papers/paper-ii/current/README.md · 0 mathematical passages
- papers/paper-ii/current/editable/traditional/Paper_II_Traditional_Final.tex · 0 mathematical passages
- papers/paper-ii/current/editable/traditional/atlas_macros.tex · 0 mathematical passages
- papers/paper-ii/current/editable/traditional/chapters/01_orientation.tex · 26 mathematical passages
- papers/paper-ii/current/editable/traditional/chapters/02_coordinates.tex · 251 mathematical passages
- papers/paper-ii/current/editable/traditional/chapters/03_weighted.tex · 261 mathematical passages
- papers/paper-ii/current/editable/traditional/chapters/04_arithmetic.tex · 411 mathematical passages
- papers/paper-ii/current/editable/traditional/chapters/05_defects.tex · 83 mathematical passages
- papers/paper-ii/current/editable/traditional/chapters/06_models.tex · 102 mathematical passages
- papers/paper-ii/current/editable/traditional/chapters/07_scope.tex · 0 mathematical passages
- papers/paper-ii/current/editable/traditional/generated/current_edition.tex · 0 mathematical passages
- papers/paper-ii/current/editable/traditional/generated/current_scope.tex · 0 mathematical passages
- papers/paper-ii/current/editable/traditional/generated/current_visual_scope.tex · 0 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_atlas.tex · 321 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/bezout_normalization.tex · 18 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/coefficient_tower.tex · 65 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/crt_address_grid.tex · 24 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/division_free_masks.tex · 47 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/dyadic_eigenlattice.tex · 7 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/finite_fourier_table.tex · 26 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/finite_polyphase_filter.tex · 27 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/inverse_residual_depth.tex · 17 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/modular_inverse.tex · 21 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/operator_naturality.tex · 196 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/power_defect_growth.tex · 20 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/precision_delay.tex · 8 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/regrouping_not_averaging.tex · 37 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/resolvent_amplification.tex · 13 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/smith_chart_orbit.tex · 18 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/smith_coordinate_transport.tex · 37 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/smith_depth_budget.tex · 18 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/smith_fibre_grid.tex · 37 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/three_resolutions.tex · 22 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/weighted_multigraph.tex · 12 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/weighted_word_cutoff.tex · 22 mathematical passages
- papers/paper-ii/current/editable/traditional/visual_graphics/zeta_recurrence.tex · 30 mathematical passages
PDF reading editions (3)
The text index does not extract PDF glyphs. Check the reading edition for notation in figures, tables, and PDF-only material.