diff 56bc728..f6bf760 simplicial_volume_scalar_bound_v2.txt
diff --git a/simplicial_volume_scalar_bound_v2.txt b/simplicial_volume_scalar_bound_v2.txt
index 2e57d12..0bd5a75 100644
--- a/simplicial_volume_scalar_bound_v2.txt
+++ b/simplicial_volume_scalar_bound_v2.txt
@@ -53,7 +53,7 @@ and every smooth Riemannian metric g on M satisfy
Here Scal_g^-=max{0,-Scal_g} and |M| is the real simplicial
volume of M. The left side is a local analytic quantity: it is
dimensionless, so it does not change under a constant rescaling
-gmapstolambda^2g, and it measures both how negative the scalar
+g|->lambda^2g, and it measures both how negative the scalar
curvature becomes and over how much volume. The right side is a global
topological invariant, independent of the metric. The theorem says that
negative scalar curvature is a resource: without enough of it, measured
@@ -67,7 +67,7 @@ bounded by a weighted integral that, after the deformations, is
comparable to Vol_g(M). All constants depend only on n.
-parnoindent{footnotesizettfamily src-ma2026.pdf}
+parnoindent{footnotesizettfamily ./pdf/src-ma2026.pdf}
The real simplicial volume of a closed connected oriented
n-manifold M is the ell^1-seminorm of its real fundamental class:
@@ -88,7 +88,7 @@ coefficient depending only on n. Writing
[
int_M(Scal_g^-)^{n/2},dV_g
]
-is invariant under a constant rescaling gmapstolambda^2g, so it is
+is invariant under a constant rescaling g|->lambda^2g, so it is
dimensionless; and it records both how negative the scalar curvature
becomes and over how much volume it does so.
@@ -130,7 +130,7 @@ such manifolds are equivalent by a Yamabe rescaling; see
cite[Lemma~2.1]{MinZhengZhu2026}.
-parnoindent{footnotesizettfamily src-min2026.pdf}
+parnoindent{footnotesizettfamily ./pdf/src-min2026.pdf}
Several related statements were known before the general case.
@@ -147,7 +147,7 @@ Several related statements were known before the general case.
-parnoindent{footnotesizettfamily src-braunsauer2021.pdf}
+parnoindent{footnotesizettfamily ./pdf/src-braunsauer2021.pdf}
The one external input is the qualitative vanishing
theorem of
@@ -172,7 +172,7 @@ and the last stage composes them.
by int_M(Scal_g^-)^{n/2}dV_g of the original metric. So
Theorem~thm:main follows from the normalized estimate
- |M|le C_nVol_g(M)qquadwhenever Scal_gequiv-1,
+ |M|le C_nVol_g(M) whenever Scal_gequiv-1,
which is Theorem~thm:scalar-volume.
@@ -360,9 +360,9 @@ Consequently (Vol_h(M)leint_M(Scal_g^-)^{n/2},dV_g), and
parnoindent{footnotesize J. Math. Pures Appl. (9) 55 (1976), 269--296}
parnoindent{footnotesizehttps://doi.org/10.4310/jdg/1214439291}
-parnoindent{footnotesizettfamily src-trudinger1968.pdf}
-parnoindent{footnotesizettfamily src-yamabe1960.pdf}
-parnoindent{footnotesizettfamily src-leeparker1987.pdf}
+parnoindent{footnotesizettfamily ./pdf/src-trudinger1968.pdf}
+parnoindent{footnotesizettfamily ./pdf/src-yamabe1960.pdf}
+parnoindent{footnotesizettfamily ./pdf/src-leeparker1987.pdf}
For the rest of the proof one fixes (M^n) and (g) with
(Scal_g=-1), and prove (eq:scalar-minus-one-target). Continue to assume (|M|>0).
@@ -388,8 +388,8 @@ cite[Section 3.1]{Gromov1982} and its canonical group formulation in
cite[Theorems 8.3 and 8.6]{IvanovNotes}.
-parnoindent{footnotesizettfamily src-gromov1982.pdf}
-parnoindent{footnotesizettfamily src-ivanov2017.pdf}
+parnoindent{footnotesizettfamily ./pdf/src-gromov1982.pdf}
+parnoindent{footnotesizettfamily ./pdf/src-ivanov2017.pdf}
Use the homogeneous model for group cochains. Thus an (n)-cochain
is a bounded function on (Gamma^{n+1}), invariant under simultaneous
@@ -416,20 +416,20 @@ and divide by (1+ε), with (0<εle1/2). Replacing
the cocycle by its negative first, if necessary, fixes the sign.
-parnoindent{footnotesizettfamily src-frigerio2016.pdf}
+parnoindent{footnotesizettfamily ./pdf/src-frigerio2016.pdf}
Allow a finite number of copies of every group label. More
-precisely, let (mathcal L=calL=Gammatimes{1,...,s}), with the free
+precisely, let (L={1,...,s}), with the free
left (Gamma)-action on the first factor, and evaluate (c) on
-(mathcal L^calL^{n+1}) by projection to (Gamma^{n+1}). The same
-notation will be used when further finite copies of (mathcal LcalL) are
+(L^{n+1}) by projection to (Gamma^{n+1}). The same
+notation will be used when further finite copies of (L are
introduced. Extend (c) multilinearly to finitely supported vectors
-on (mathcal LcalL).
+on (L
Let
[
-Delta_{mathcal L}calL=
-{p=(p_ell)_{ellinmathcal L}calL:
+Delta_{L}
+{p=(p_ell)_{ellinL}:
p_ellge0, sum_ell p_ell=1, p has finite support}.
]
On each finite-dimensional face define the (n)-form
@@ -443,14 +443,14 @@ Call a map (F=(F_ell)) with nonnegative coordinates and
(sum_ell F_ell^2=1) a root probability map. For the maps
used here, whose coordinate supports are locally finite, the squared
coordinates (p_ell=F_ell^2) give the associated map to
-(Delta_{mathcal L}calL).
+(Delta_{L}
The form (eta) is closed on every finite-dimensional face of
-(Delta_{mathcal L}calL). There is a smooth (Gamma)-equivariant map
+(Delta_{L} There is a smooth (Gamma)-equivariant map
[
- F:widetilde M→ell^2(mathcal L),calL),
+ F:widetilde M→ell^2(L),
F_ellge0, sum_ell F_ell^2=1,
]
with finitely many label orbits and locally finite compact supports for
@@ -522,7 +522,7 @@ over (M) consists of (Gamma)-equivariant Borel data on
(widetilde MtimesΩ), smooth in the (widetilde M) variable.
Geometrically one may regard the diagonal quotient
[
- mathcal M=(calM=(widetilde MtimesΩ)/Gamma
+ M=( MtimesΩ)/Gamma
]
as a space whose leaves are ordinary smooth manifolds. If
(Gamma_omega) is the stabilizer of (omega), the corresponding
@@ -539,7 +539,7 @@ atlas (U_a) of (M), chosen lifts (widetilde U_a), and a smooth
partition of unity (chi_a) subordinate to the atlas. For an
integrable equivariant leafwise (n)-form (alpha), set
-int_{mathcal M}calMalpha
+int_{M}
=
sum_aint_Ωint_{widetilde U_a}
(chi_acircpi),alpha_omega,dmu(omega).
@@ -567,9 +567,9 @@ uniform leafwise derivative bounds. Their affine homotopy has the
same properties. Stokes' theorem on the measured product with
([0,1]) gives
[
-int_{mathcal M}calM(p^0)^*eta
+int_{M}*eta
=
-int_{mathcal M}calM(p^1)^*eta.
+int_{M}*eta.
]
Thus every probability map constructed below continues to detect the
same number (I). Enlarging (Ω) by an independent probability
@@ -604,7 +604,7 @@ the graph and vertical Jacobi identities, see
calculation with the conventions needed here.
-parnoindent{footnotesizettfamily src-perelman2002.pdf}
+parnoindent{footnotesizettfamily ./pdf/src-perelman2002.pdf}
[Weighted graph identity]
@@ -770,7 +770,7 @@ energy. The (A^2/(2f)) term completes the longitudinal square:
&ε N^2|dF|_{h_{in}}^2
-4ε a(F/f)NAe(F)
&=ε|N,dF
- -2a(F/f)A,h_{in}(e,cdot)|_{h_{in}}^2
+ -2a(F/f)A,h_{in}(e,·)|_{h_{in}}^2
-4ε a^2(F/f)^2A^2(1-B)
&quadge-4ε a^2(F/f)^2A^2(1-B).
@@ -886,7 +886,7 @@ positive-order derivatives. In particular,
e^{phi_z}dV_{h_z}=e^phi dV_h,
|dz|_{h_z}^2=1-v_z^2<1.
]
-For the family assertion, let (mathcal TcalT) be a standard Borel
+For the family assertion, let (T be a standard Borel
parameter space indexing complete problems of a fixed dimension.
Assume a common chart radius and buffers, common bounds for the
metric, its inverse, and all their chart derivatives, and common
@@ -1638,7 +1638,7 @@ Finally, define two nonnegative quantities:
T_0={}&sum_i A_i^2[
C_i+4a^2sigma_iB_i+(L_i-1){2a(sigma_i+r)+u_i}],
- mathcal X=calX={}&sum_i[
+ X={}&sum_i[
(X_i-t_i)^2+N^2|II_i^{rem}|^2
+N^2|dlog L_i|_{h_i}^2+beta_i^2f_i^2].
@@ -1646,7 +1646,7 @@ Finally, define two nonnegative quantities:
The quantity (T_0) records the gains from the metric comparisons
and from each graph's stretching. Its full coefficient will yield
the sharp quadratic term (m^2/n) in the entropy iteration.
-The quantity (mathcal XcalX) retains the derivative squares and the
+The quantity (X retains the derivative squares and the
applied coefficients. Together with the mean and variance squares
below, it controls graph curvature; those moment squares also control
the normalization.
@@ -1666,7 +1666,7 @@ charts for the instance. Suppose (h_0) is complete, its
geometry and the positive-order derivatives of (phi_0) have
the uniform bounds above, and
[
- F=(F_i)_{iinmathcal L}calL, F_ige0,
+ F=(F_i)_{iinL}, F_ige0,
sum_iF_i^2=1
]
has finitely many free label orbits and uniformly locally finite
@@ -1677,11 +1677,11 @@ be a complete equivariant Borel leafwise metric whose coefficients,
inverse, and all coordinate derivatives have uniform bounds in the
same lifted buffered charts as (h_0). Suppose a Borel
equivariant coloring
-(chi:mathcal LcalLtimesΩto{1,...,k}), (k<∞),
+(chi:LΩto{1,...,k}), (k<∞),
has been chosen such that, for each retained (omega) and distinct
labels (i,j) with (chi(i,omega)=chi(j,omega)), the
(R_{loc})-neighborhoods in (h_{ref}(omega))
-of (supp F_i(cdot,omega)) and (supp F_j(cdot,omega))
+of (supp F_i(·,omega)) and (supp F_j(·,omega))
are disjoint. Fix the pass parameters in
(eq:sweep-parameters), with (εgeε_*) in
the boosted case.
@@ -1724,7 +1724,7 @@ Define the nonnegative reserve
Pi_p:={}&
T_0+cε(M_0^2+E_nu V_i^2+thetaVar_nu u)
- &+cfrac rεmathcal XcalX
+ &+cfrac rεX
+c_*ε r^6indic_{{rge C_*}},
@@ -1800,7 +1800,7 @@ adds exactly
Since (L_ige1), this is at least
- mathcal B_i:=calB_i:=
+ B_i:=calB_i:=
beta_i^2f_i^2+2ε abeta_i f_i^2+beta_i A_i^2
-2ε abeta_iF_i^2
-2NA_if_i|dbeta_i|_{h_{i-1}}.
@@ -1830,7 +1830,7 @@ sum is at most (delta_0) at every point. Telescoping
Gge sum_i[&
Z_i^2-2NU_i(Z_i)+N^2|II_i|^2
+N^2|dlog L_i|_{h_i}^2
- +ε N^2|dF_i|_{h_0}^2+mathcal B_i]calB_i]
+ +ε N^2|dF_i|_{h_0}^2+B_i]calB_i]
notag
&-ε N^2e_{h_+}(Y)-delta_0 .
@@ -1876,7 +1876,7 @@ After multiplication by (r/ε), the result is
Qge{}&T_0+frac rεsum_iK_i
+εE_nu g_i^2
+sum_i A_i^2[(2a-4a^2)sigma_i+2ar-1]
- +frac rε(sum_imathcal B_i-calB_i-delta_0).
+ +frac rε(sum_iB_i-calB_i-delta_0).
Indeed (rF_i^2/f_i^2=sigma_i). The terms with (B_i) and
@@ -1915,7 +1915,7 @@ Here
(tfrac14-lambda^2=2theta-4theta^2ge7theta/4).
This supplies a reserve for all three squares in the
proposition. Finish by paying the terms in
-(mathcal B_icalB_i) from fixed fractions of this reserve and of
+(B_i from fixed fractions of this reserve and of
the nonnegative (beta_i) terms.
Write (s_i=beta_{s,i}) and (b_i=εpsi(R_i)), so
@@ -2035,7 +2035,7 @@ available for this purpose. If
It is therefore covered in both ranges. Fixed fractions of
the mean square and of (eq:sweep-boost-square) remain.
One can simultaneously retain a fixed fraction of
-(sum_ibeta_i^2f_i^2) for (mathcal XcalX): reserve, for
+(sum_ibeta_i^2f_i^2) for (X: reserve, for
example, one half of this sum at the outset and use
(beta_i^2ge s_i^2+b_i^2) on the other half for the
spatial and boost estimates. Take (c_0,zeta,kappa) small
@@ -2043,7 +2043,7 @@ enough for these fixed allocations.
Insert the absorptions into (eq:sweep-expanded) and
(eq:sweep-mean-identity). A fixed fraction of
-(sum_iK_i+sum_ibeta_i^2f_i^2=mathcal XcalX) and of the
+(sum_iK_i+sum_ibeta_i^2f_i^2=X and of the
three mean and variance squares survives. The remaining
part of (eq:sweep-boost-square), after multiplication
by (r/ε), is
@@ -2051,7 +2051,7 @@ by (r/ε), is
(c_*) a positive multiple of (kappa^2).
The terms (T_0) and (-4theta^2m) have not been spent.
Choose the constant (c) in (eq:sweep-positive-reserve)
-below the fixed numerical fractions that survive for (mathcal XcalX)
+below the fixed numerical fractions that survive for (X
and the mean and variance squares. These fractions are independent
of (ε_*), because (kappa) was chosen to fit the fixed
allocations.
@@ -2081,10 +2081,10 @@ to be chosen larger than the combinatorial quantities used later.
Write (exp^{[0]}(t)=t) and
(exp^{[j+1]}(t)=exp(exp^{[j]}(t))). Call a function
-(mathcal B:[1,∞)calB:[1,∞)to[0,∞)) elementary if, for some fixed
+(B:[1,∞)calB:[1,∞)to[0,∞)) elementary if, for some fixed
(HinNcup{0}) and (A,B>0),
[
- mathcal B(q)calB(q)leq exp^{[H]}(B(1+q)^A)(qgeq1).
+ B(q)<= exp^{[H]}(B(1+q)^A)(qgeq1).
]
When the function is prescribed in a fixed dimension, its parameters
(H,A,B) are allowed to depend on that dimension.
@@ -2094,13 +2094,13 @@ Let ((M^n,g)) be a closed connected Riemannian manifold, where (ngeq3)
and (Scal_g=-1), and put (Gamma=pi_1(M)). Let
((Ω_0,mu_0)) be a standard Borel probability space with a
measure-preserving (Gamma)-action; the one-point space is allowed.
-Suppose (mathcal L_0calL_0) is a countable set with finitely many free
+Suppose (L_0 is a countable set with finitely many free
(Gamma)-orbits and
[
F^{in}:widetilde MtimesΩ_0→
- ell^2(mathcal L_0),calL_0),
+ ell^2(L_0),
F^{in}_ellgeq0,
- sum_{ellinmathcal L_0}calL_0(F^{in}_ell)^2=1
+ sum_{ellinL_0}{in}_ell)^2=1
]
is equivariant, Borel in the transverse variable, and smooth along
(widetilde M). Assume the following uniformity for this input. In
@@ -2109,28 +2109,28 @@ the supports are locally finite with a uniform bound on their number,
and all leafwise derivatives of the coordinate functions have uniform
bounds. For each orbit representative (ell) there is a compact set
(K_ellsubsetwidetilde M) such that
-(supp F^{in}_{gammaell}(,cdot,,omega)
+(supp F^{in}_{gammaell}(,·,,omega)
subsetgamma K_ell) for all (gammainGamma) and almost every
(omega). The bounds and compact sets in these assumptions may depend
on the input.
-For every prescribed elementary function (mathcal BcalB) and every finite
+For every prescribed elementary function (B and every finite
number (E_{min}), there are a standard Borel probability
(Gamma)-space ((Ω,mu)) extending ((Ω_0,mu_0)), a
-countable set (mathcal LcalL) with finitely many free (Gamma)-orbits, and
+countable set (L with finitely many free (Gamma)-orbits, and
equivariant leafwise data on (widetilde MtimesΩ): a metric (h),
a function (phi), a positive density (rho), and a root probability
-map (F) with labels (mathcal LcalL). There are also (E>E_{min}), an
-integer (qgeq n+1), and a constant (C=C(n,mathcal B)<∞calB)<∞) such
+map (F) with labels (L There are also (E>E_{min}), an
+integer (qgeq n+1), and a constant (C=C(n,B)<∞calB)<∞) such
that, on an invariant conull set of transverse parameters,
&rholeq dV_g,
W:=log(dV_h)/(rho)geq0,
- &wscal(h,phi)geq
+ &wscal(h,phi)>=
-C+2(W-frac n2log E)+E e_h(F),
- e_h(F)^{1/2}leq E^{-1/100},
- &E>mathcal B(q).calB(q).
+ e_h(F)^{1/2}<= E^{-1/100},
+ &E>B(q).
Here the extension means that there is an equivariant measure-preserving
@@ -2145,18 +2145,18 @@ compact sets containing their supports, uniformly in the transverse
parameter. All these output bounds may depend on the input and on the
chosen finite construction.
-There is, in addition, a fixed (Gamma)-invariant graph (mathcal GcalG)
-on (mathcal LcalL), independent of the transverse parameter, of finite
+There is, in addition, a fixed (Gamma)-invariant graph (G
+on (L, independent of the transverse parameter, of finite
maximum degree (Delta). It contains an edge between every pair of
distinct labels that are simultaneously nonzero at any point of any leaf.
Take
[
- qgeq max{n+1,#(Gammabackslashmathcal L),calL),Delta+1}
+ qgeq max{n+1,#(GammabackslashL),Delta+1}
]
and a Borel equivariant coloring
-(chi:mathcal LcalLtimesΩto{1,...,q}) such that labels of
-graph distance one or two have different colors. The set (mathcal LcalL)
-can be taken to consist of finitely many copies of (mathcal L_0calL_0).
+(chi:LΩto{1,...,q}) such that labels of
+graph distance one or two have different colors. The set (L
+can be taken to consist of finitely many copies of (L_0
The constant (C) is independent of the manifold, metric, input family,
all the uniform bounds just specified, and (E_{min}).
@@ -2180,24 +2180,24 @@ also occur in [cite];
the proof below establishes the forcing and entropy estimates needed here.
-First supply the colors required for the sweeps. Let (mathcal HcalH)
+First supply the colors required for the sweeps. Let (H
be any fixed invariant graph of maximum degree
-(Delta_{mathcal H}calH<∞) on a countable label set (mathcal LcalL).
+(Delta_{H}<∞) on a countable label set (L
Adjoin to the transverse space independent uniform priorities in
-([0,1]^{mathcal L}calL), with the (Gamma)-action permuting coordinates.
+([0,1]^{L}, with the (Gamma)-action permuting coordinates.
This is a standard Borel probability extension. Direct every edge
from the larger priority to the smaller, after discarding the invariant
null set on which two priorities coincide. This set is null because the
label set is countable and the priorities have continuous laws. For a
fixed vertex the probability of a strictly descending path of length
-(k) is at most (Delta_{mathcal H}calH^k/(k+1)!): there are at most
-(Delta_{mathcal H}calH^k) paths, and the priorities of the (k+1)
+(k) is at most (Delta_{H}/(k+1)!): there are at most
+(Delta_{H} paths, and the priorities of the (k+1)
distinct vertices in a given descending path have the required order
with probability (1/(k+1)!). The probability of arbitrarily long
such paths is therefore zero. Since the label set is countable, on
an invariant conull set every vertex has finite descending depth.
Greedily assign to a vertex the least color in
-({1,...,Delta_{mathcal H}calH+1}) unused by its lower-priority
+({1,...,Delta_{H}+1}) unused by its lower-priority
neighbors, recursively in that finite depth. This defines a proper
coloring. It is Borel, since each vertex's color is determined in
some finite descending neighborhood, and it is equivariant, since
@@ -2254,14 +2254,14 @@ The graph operations give
frac{dV_{h_+}}{dV_{h_0}}=e^{W_p},
phi_+=phi_0-W_p,
- N^2 e_{h_+}(Y)leq frac mr.
+ N^2 e_{h_+}(Y)<= frac mr.
The added graph tensors are nonnegative. Thus, as quadratic forms,
-(h_0leq h_{i-1}leq h_ileq h_+); the inverse forms reverse these
+(h_0leq h_{i-1}<= h_ileq h_+); the inverse forms reverse these
inequalities. In particular, for every covector (alpha),
[
- |alpha|_{h_+}leq|alpha|_{h_i}
- leq|alpha|_{h_{i-1}}leq|alpha|_{h_0}.
+ |alpha|_{h_+}<=|alpha|_{h_i}
+ <=|alpha|_{h_{i-1}}<=|alpha|_{h_0}.
]
The positive reserve (Pi_p) is defined in
(eq:sweep-positive-reserve), with coefficient exactly one on
@@ -2273,7 +2273,7 @@ threshold for (ε). The sweep inequality reads
Q&:=(rN^2)/(ε)
[wscal(h_+,phi_+)-wscal(h_0,phi_0)
+ε e_{h_0}(F)-ε e_{h_+}(Y)]
- &geq Pi_p-4theta^2m-frac rεdelta_0 .
+ &>= Pi_p-4theta^2m-frac rεdelta_0 .
The positive errors (delta_0,eta_0) can be prescribed independently
@@ -2289,8 +2289,8 @@ boost, followed individually by conformal changes, produce a metric
(h_*), a root probability map (F_*), and
(ε_0in[ε_*,2ε_*]) such that
- rho:=dV_{h_*}leq dV_g,
- Scal_{h_*}geq-C_{boot}+ε_0e_{h_*}(F_*).
+ rho:=dV_{h_*}<= dV_g,
+ Scal_{h_*}>=-C_{boot}+ε_0e_{h_*}(F_*).
The metric and map have the uniformity and support properties in
Theorem~thm:entropy. The number of sweeps may depend on the input.
@@ -2311,11 +2311,11 @@ The scalar conformal formula and the definition of (wscal) give
The same reset contracts the volume density:
dV_k=e^{-nW_p/(n-1)}dV_{h_+}
- =e^{-W_p/(n-1)}dV_{h_0}leq dV_{h_0}.
+ =e^{-W_p/(n-1)}dV_{h_0}<= dV_{h_0}.
Claim that there are dimensional (C_b) and (ε_b>0) such
that, whenever (2leq C_0leq4), (0<εleqε_b), and
-(Scal_{h_0}geq-C_0+ε e_{h_0}(F)), one can choose the errors
+(Scal_{h_0}>=-C_0+ε e_{h_0}(F)), one can choose the errors
in the sweep so that
Scal_kgeq-(C_0+C_b lε)+(1+l)ε e_k(Y).
@@ -2327,12 +2327,12 @@ First derive the budget needed for this step. The reset gives
(eq:entropy-bootstrap-reset), and dropping its nonnegative
gradient term, shows that it suffices to have
[
- (ε Q)/(rN^2)geq
+ (ε Q)/(rN^2)>=
lε e_{h_+}(Y)+C_0(1-e^{-2W_p/(n-1)})
-C_b lε e^{-2W_p/(n-1)}.
]
Multiply by (rN^2), and use
-(N^2 e_{h_+}(Y)leq m/r) and (N^2=lε). A sufficient budget
+(N^2 e_{h_+}(Y)<= m/r) and (N^2=lε). A sufficient budget
for (eq:entropy-bootstrap-step) is therefore
ε Qgeq
@@ -2381,16 +2381,16 @@ longitudinal component of (II_i) is (-X_i/N), so
+N(trII_i^{rem}-U_i(phi_{i-1})).
]
The terms ((X_i-t_i)^2), (N^2|II_i^{rem}|^2), and
-(beta_i^2f_i^2) occur in (mathcal XcalX).
+(beta_i^2f_i^2) occur in (X
For the remaining term, choose (m_b<1/4). Then
-(L_ileq C) and (|dlog L_i|_{h_i}leq C A_i|II_i|).
+(L_ileq C) and (|dlog L_i|_{h_i}<= C A_i|II_i|).
Because the metrics increase during the sweep and
(phi_{i-1}=-sum_{j<i}log L_j), Cauchy--Schwarz gives
[
sum_iN^2U_i(phi_{i-1})^2
- &leq C m^2sum_jN^2|II_j|^2
- &leq C' m^2(mathcal X+calX+sum_jt_j^2).
+ &<= C m^2sum_jN^2|II_j|^2
+ &<= C' m^2(X+calX+sum_jt_j^2).
]
For example, the first line follows by bounding
@@ -2404,7 +2404,7 @@ Choosing (m_b) small absorbs the last sum of (t_j^2) in
ε^2(1-r)^2leq C(ε^2M_0^2+m^2),
]
-fractions of the (M_0^2,E V_i^2,mathcal XcalX) terms in
+fractions of the (M_0^2,E V_i^2,X terms in
(εPi_p) control both required squares. The arbitrarily
small errors in the prescription contribute (rdelta'_0).
The term (-4theta^2m) in (eq:entropy-sweep-form) is
@@ -2489,7 +2489,7 @@ The errors can be chosen so that
wscal(h_+,phi_+)-(1+l)ε e_{h_+}(Y)
- geq{}&wscal(h_0,phi_0)-ε e_{h_0}(F)
+ >={}&wscal(h_0,phi_0)-ε e_{h_0}(F)
&+2W_p-nlog(1+l)-C_nl^2+P_p ,
@@ -2523,7 +2523,7 @@ As matrices,
(sum_iN^2df_i ⊗ df_i=h_+-h_0). Hence
m_f=n-tr(h_+^{-1}h_0)
- leq n(1-e^{-2W_p/n}).
+ <= n(1-e^{-2W_p/n}).
The inequality is arithmetic--geometric mean applied to the positive
eigenvalues of (h_+^{-1}h_0), whose product is (e^{-2W_p}).
@@ -2544,11 +2544,11 @@ For (mgeq2n), the (C_i) terms of (T_0) give
For (m_0leq mleq2n), first suppose that every (A_i^2) is at most
a small number (eta=eta(n,m_0)>0). Then
[
- 2W_p=sum_i-log(1-A_i^2)leq(1+Ceta)m.
+ 2W_p=sum_i-log(1-A_i^2)<=(1+Ceta)m.
]
By (eq:entropy-trace),
[
- m-m_fgeq m-n(1-e^{-(1+Ceta)m/n})geq c_n m.
+ m-m_fgeq m-n(1-e^{-(1+Ceta)m/n})>= c_n m.
]
For the last inequality, the same expression with (eta=0) divided
by (m) has a positive minimum on ([m_0,2n]), and then (eta)
@@ -2559,26 +2559,26 @@ small, (M_0=a(1-r)-barbar u/2u/2) is bounded away from zero unless (barbar uu)
lies in a fixed compact subinterval of ((0,∞)). In the former
case (ε M_0^2geq c_nε_*). In the latter,
(ε=m/barbar uuleq C_n) and
-(u_i=A_i^2/(εnu_i)geq c_n), since (nu_ileq1).
+(u_i=A_i^2/(εnu_i)>= c_n), since (nu_ileq1).
Each bound controls (mleq2n). This proves the (m) part for
(mgeq m_0).
For the volume part, indices with (A_igeq1/4) satisfy
[
- A_i^2(L_i-1)2argeq c,r(L_i-1)geq c,rlog L_i.
+ A_i^2(L_i-1)2argeq c,r(L_i-1)>= c,rlog L_i.
]
For all other indices, (log L_ileq C A_i^2), so their contribution
to (rW_p) is at most (Crm). If (rleq2), the bound for (m)
already covers this. If (r>2), then
-(|M_0|geq c(r+barbar u)u)), and
+(|M_0|>= c(r+barbar u)u)), and
(ε M_0^2geq cε rbarbar u=cu=c,rm).
This proves (eq:entropy-coarse-coercivity) when (mgeq m_0).
When (m<m_0), one has (barbar uuleq m_0/ε_*) small and
[
- W_p=frac m2+O(sum_iA_i^4)leq C m.
+ W_p=frac m2+O(sum_iA_i^4)<= C m.
]
If (|1-r|geq1/4), choosing (m_0) still smaller gives
-(|M_0|geq c|1-r|). The quantity
+(|M_0|>= c|1-r|). The quantity
(ε(1-r)^2) controls (m+rm): for bounded (r) use
(εgeqε_*) and (mleq m_0), and for (rgeq2)
use ((1-r)^2geq r^2/4). This proves the claim in the remaining
@@ -2602,7 +2602,7 @@ Here (T_0geq c_nm^2). Indeed, expanding the exponential in
(W_p=m/2+O(sum_iA_i^4)) gives
[
sum_i A_i^2 C_i=m-m_f
- geq c_n m^2-C_nsum_iA_i^4
+ >= c_n m^2-C_nsum_iA_i^4
]
after decreasing (m_0). The self terms
(sum_i A_i^2(L_i-1)2ar) are at least
@@ -2637,11 +2637,11 @@ Every eigenvalue of (h_0^{-1}h_+) is at least one, and their product
is (e^{2W_p}). Each eigenvalue is therefore at most that product,
which proves the comparison
[
- h_0leq h_jleq h_+leq e^{2W_p}h_0=(1+O(m))h_0.
+ h_0leq h_jleq h_+<= e^{2W_p}h_0=(1+O(m))h_0.
]
Normalize the directions (e_i) in the (h_0) inner product and write
the resulting unit vectors as (barbar e_ie_i). Put
-(xi_{ij}=(barbar e_ie_icdotbarbar e_j)^2e_j)^2). Before approximating the metric
+(xi_{ij}=(barbar e_ie_i·barbar e_j)^2e_j)^2). Before approximating the metric
losses, their exact earlier- and later-color telescopes are
[
B_i=sum_{j<i}(Ndf_j(e_i))^2,
@@ -2653,7 +2653,7 @@ The first identity follows from
(|e_i|_{h_{i-1}}=1). The second follows by telescoping the rank-one
inverse formula
[
- h_j^{-1}=h_{j-1}^{-1}-A_j^2 e_j e_j^{mathsf T}sfT
+ h_j^{-1}=h_{j-1}^{-1}-A_j^2 e_j e_j^{T}
]
from (h_i) to (h_+), and using
(N^2|df_i|_{h_i}^2=A_i^2). Now
@@ -2669,7 +2669,7 @@ in the exact formulas differs from its (h_0) expression by
Inserting (eq:entropy-BC-expansion) into (T_0) yields
T_0geq
- tr[(sum_i A_i^2barbar e_ie_ibarbar e_i^e_i^{mathsf T}sfT)^2]
+ tr[(sum_i A_i^2barbar e_ie_ibarbar e_i^e_i^{T}]
-C_n(m^3+m^2|r-1|
+msum_iA_i^2(1-sigma_i)_+).
@@ -2692,8 +2692,8 @@ inequality give, for every fixed (c'>0),
msum_iA_i^2(1-sigma_i)_+
- &leq C mε,E[u_i(|V_i|+|u_i-barbar u|)]u|)]
- &leq c'ε(E V_i^2+thetaVar u)
+ &<= C mε,E[u_i(|V_i|+|u_i-barbar u|)]u|)]
+ &<= c'ε(E V_i^2+thetaVar u)
+C_{c'}(m^2)/(theta)ε(barbar u^2+u^2+Var u).
@@ -2710,7 +2710,7 @@ is
Here (εgeqε_*), and (K,D_*,ε_*) are already
fixed dimensionally. Also
[
- |r-1|leq C(|M_0|+barbar u).u).
+ |r-1|<= C(|M_0|+barbar u).u).
]
Consequently a term (O_n(l^2)|r-1|) is bounded by a small fraction
of (ε M_0^2) plus (O_n(l^3)): use Young's inequality on
@@ -2741,10 +2741,10 @@ There is also a dimensional (ε_{exit}) such that,
if (εgeqε_{exit}) and (rleq1/4), then
- P_p&geq c_n(ε)/(lr)
+ P_p&>= c_n(ε)/(lr)
(1+(m^2)/(ε^2)),
sum_i(D_i^2+|II_i|^2+|dlog L_i|_{h_i}^2)
- &leq C_n(P_p)/(theta).
+ &<= C_n(P_p)/(theta).
The errors in the prescription may simultaneously be chosen to have
@@ -2754,15 +2754,15 @@ The errors in the prescription may simultaneously be chosen to have
Proof.
If (barbar u=m/εu=m/ε) is larger than a fixed dimensional number, then
(mgeq m_0), so the surplus retains the mean square and boost.
-Since (|M_0|geq c(r+barbar u)u)) in this range and (lleq1),
+Since (|M_0|>= c(r+barbar u)u)) in this range and (lleq1),
[
P_pgeq c_nε,
frac{(r+barbar u)^2+c_*r^6u)^2+c_*r^6indic_{{rgeq C_*}}}{r}
- geq c'_nε barbar u^u^{5/3}.
+ >= c'_nε barbar u^u^{5/3}.
]
For the last inequality, if (rleq barbar u^u^{1/3}) use (barbar u^2/ru^2/r);
if (rgeq barbar u^u^{1/3}) use (r^5), taking the threshold for (barbar uu)
-large enough that (barbar u^u^{1/3}geq C_*). It follows in all cases that
+large enough that (barbar u^u^{1/3}>= C_*). It follows in all cases that
mleq C_nε(1+P_p/ε)^{3/5}.
@@ -2785,13 +2785,13 @@ when (m>0); when (m=0), (W_p=0).
To justify the bound when the number of those indices is smaller
than (4m), use that (slog(1+b/s)) is increasing in (s>0).
Put (x=m/ε), (y=P_p/ε). The elementary bound
-(1+log(1+t)leq C(1+t)^{1/4}) for (tgeq0), applied with
+(1+log(1+t)<= C(1+t)^{1/4}) for (tgeq0), applied with
(t=y/x), makes the right side of
(eq:entropy-log-surplus), divided by (ε), at most
[
C_n x^{3/4}(x+y)^{1/4}
- leq C_n(1+y)^{7/10}
- leq C_n(1+y)^{3/4},
+ <= C_n(1+y)^{7/10}
+ <= C_n(1+y)^{3/4},
]
where (eq:entropy-m-surplus) was used in the middle inequality.
This proves (eq:entropy-pass-volume).
@@ -2803,7 +2803,7 @@ because (|1-r|geq3/4). Outside a fixed compact subinterval of
sufficiently large,
[
T_0geq m-m_fgeqε barbar u-nu-n
- geq c_nε(1+barbar u^2)u^2)
+ >= c_nε(1+barbar u^2)u^2)
]
by (eq:entropy-trace). Multiplying the appropriate retained
term by (gamma_n/(lr)) proves the first inequality in
@@ -2811,9 +2811,9 @@ term by (gamma_n/(lr)) proves the first inequality in
For the second, the retained derivative terms imply
- mathcal XcalXleq C_n N^2P_p,
+ X C_n N^2P_p,
(ε^2)/(rN^2)
- (E V_i^2+thetaVar u)leq C_nP_p.
+ (E V_i^2+thetaVar u)<= C_nP_p.
Also
[
@@ -2823,7 +2823,7 @@ The first inequality of (eq:entropy-last-surplus) controls
the (barbar u^2u^2) part of this sum divided by (N^2), and the variance
term in (eq:entropy-X-retained) controls the other part with
a factor (1/theta). Thus
-(sum_i t_i^2/N^2leq C_nP_p/theta). Since (mathcal XcalX) contains
+(sum_i t_i^2/N^2leq C_nP_p/theta). Since (X contains
((X_i-t_i)^2+N^2|II_i^{rem}|^2+N^2|dlog L_i|^2), this proves the
claimed bounds for (II_i) and (dlog L_i).
@@ -2839,10 +2839,10 @@ for (sigma_i-1). Since (rleq1/4),
sum_i(D_i^0)^2leq
(Cε^2)/(rN^2)
(1+barbar u^2+u^2+E V_i^2+Var u)
- leq C_nP_p/theta.
+ <= C_nP_p/theta.
]
The relation (ND_i=ND_i^0-beta_i f_i+err_i) finishes the
-estimate: (sum_ibeta_i^2f_i^2leqmathcal XcalX), and one chooses (sum_i|err_i|^2leq1). Its contribution (1/N^2) is
+estimate: (sum_ibeta_i^2f_i^2leqX, and one chooses (sum_i|err_i|^2leq1). Its contribution (1/N^2) is
bounded by (C_nP_p) using the first inequality of
(eq:entropy-last-surplus), (εgeq1), and (rleq1/4).
@@ -2872,7 +2872,7 @@ The estimates below show how each margin is used.
[Accumulation of entropy passes]
Suppose at parameter (ε_{in}geqε_*) that
[
- wscal(h,phi)geq-C+2W-nlogε_{in}
+ wscal(h,phi)>=-C+2W-nlogε_{in}
+ε_{in}e_h(F)+P_{in},
W=log(dV_h/rho)geq0, P_{in}geq0.
]
@@ -2885,12 +2885,12 @@ constant is (C_nsum_p l(ε_p)^2). Along the entire increasing
sequence of parameters,
sum_p l(ε_p)^2leq C_n,
- #{p:ε_p<E}leq C_n(1+log^3(1+E)).
+ #{p:ε_p<E}<= C_n(1+log^3(1+E)).
If the initial volume logarithm is zero, then, for all sufficiently
large dimensional (E),
- W_{new}leq E^{0.26}(E+P)^{0.75}.
+ W_{new}<= E^{0.26}(E+P)^{0.75}.
The same assertion holds if the initial data have parameter (E_0),
surplus (P_0), and
@@ -2921,7 +2921,7 @@ Comparison with the integrals of ((1+t)^2) and ((1+t)^{-2})
respectively proves the number bound and
[
sum_p l(ε_p)^2
- leq C_n l_*int_{t_0}^{∞}(dt)/((1+t)^2)leq C_n.
+ <= C_n l_*int_{t_0}^{∞}(dt)/((1+t)^2)<= C_n.
]
The comparison is valid on each interval ([t_p,t_{p+1}]), since
its endpoints differ by a uniformly bounded small number.
@@ -2948,12 +2948,12 @@ tensor in (eq:sweep-output); hence, if (h^0) is its starting
metric and (h_+) its final metric, the successive metrics and their
inverses satisfy
[
- h^0leq h^{(1)}leq...leq h_+,
- (h_+)^{-1}leq...leq(h^{(1)})^{-1}leq(h^0)^{-1}.
+ h^0leq h^{(1)}<=...<= h_+,
+ (h_+)^{-1}<=...<=(h^{(1)})^{-1}<=(h^0)^{-1}.
]
In particular, the starting metric is an admissible reference metric
for every sweep in the phase, and
-(e_{h_+}(Y^0)leq e_{h^0}(Y^0)) for its starting map (Y^0).
+(e_{h_+}(Y^0)<= e_{h^0}(Y^0)) for its starting map (Y^0).
Now finish a phase with an absolute gradient bound. Normalizing
the last bumps alone divides by (sqrt r), which can make the
@@ -2972,7 +2972,7 @@ parameter (E=(1+l(ε))ε), and output
surplus from that last sweep, that
- wscal(h_+,phi_+)&geq-C+2W-nlog E+E e_{h_+}(Y)+P,
+ wscal(h_+,phi_+)&>=-C+2W-nlog E+E e_{h_+}(Y)+P,
W&=log(dV_{h_+}/rho)geq0,
Wleq E^{0.26}(E+P)^{0.75}.
@@ -2983,11 +2983,11 @@ on two copies of the labels such that
wscal(widehatwidehat h,h,widehatphi)
- &geq-(C+C_n)+2widehatwidehat W-nW-nlog E
+ &>=-(C+C_n)+2widehatwidehat W-nW-nlog E
+E e_{widehatwidehat h}h(Z)+P',
widehatwidehat W&=W&=log(dV_{widehatwidehat h}h/rho)geq0,
widehatwidehat WWleq E^{0.32}(E+P')^{0.75},
- e_{widehatwidehat h}h(Z)^{1/2}&leq E^{-1/100}.
+ e_{widehatwidehat h}h(Z)^{1/2}&<= E^{-1/100}.
Here (P'geq0), and the added scalar cost (C_n) is independent
@@ -3015,7 +3015,7 @@ between phases. By (eq:entropy-trace),
N^2sum_i|df_i|_{h_+}^2=m_fleq n,
|ds|_{h_+}^2leq n/N^2.
]
-Monotonicity of the phase metrics also gives (e_{h_+}(Y^0)leq K_0^2).
+Monotonicity of the phase metrics also gives (e_{h_+}(Y^0)<= K_0^2).
Since (Y) and (Y^0) are unit vectors in orthogonal spaces, their
derivatives are perpendicular to their respective vectors. Differentiating
@@ -3026,21 +3026,21 @@ derivatives are perpendicular to their respective vectors. Differentiating
]
For (rgeq1/4), it follows that
[
- E(e(Z)-e(Y))leq
+ E(e(Z)-e(Y))<=
(Ec_E^2K_0^2)/(r+c_E^2)
+(Ec_E^2n)/(N^2(r+c_E^2)^2)
- leq 4+(16n)/(N^2)leq C_n
+ <= 4+(16n)/(N^2)<= C_n
]
when (E) is large, since (N^2=lε) tends to infinity under
(eq:entropy-step-schedule). For (rleq1/4), the norm of the
derivative of normalization is at most (1/√(r+c_E^2)), so
- e(Z)leq(n/N^2+c_E^2K_0^2)/(r+c_E^2).
+ e(Z)<=(n/N^2+c_E^2K_0^2)/(r+c_E^2).
The first inequality of (eq:entropy-last-surplus) now implies
[
- E e(Z)leq (C_n)/(l(r+c_E^2))
- leq (C_n)/(ε)P_p.
+ E e(Z)<= (C_n)/(l(r+c_E^2))
+ <= (C_n)/(ε)P_p.
]
Thus replacing (Y) by (Z) in
(eq:entropy-before-exit) costs a dimensional constant on
@@ -3072,7 +3072,7 @@ claimed volume logarithm. Differentiating (r) gives
[
|dr|leq2sqrt r(sum_i|df_i|^2)^{1/2}
leqfrac{2√(nr)}N,
- |du|leq(C_n)/(Nsqrt r).
+ |du|<=(C_n)/(Nsqrt r).
]
The coefficient (n-1) in the new weight cancels the Laplacian of
(u) in the weighted conformal formula. With
@@ -3113,7 +3113,7 @@ coefficient is
(< b_{i-1},mathbf n_i>=-U_i(bfn_i>=-U_i(phi_{i-1})), and
its magnitude satisfies
[
- |U_i(phi_{i-1})|leq |D_i|+sqrt n,|II_i|,
+ |U_i(phi_{i-1})|<= |D_i|+sqrt n,|II_i|,
]
because (D_i=trII_i-U_i(phi_{i-1})).
By orthonormality and (eq:entropy-last-surplus), the norm of
@@ -3122,7 +3122,7 @@ the sum of these normal corrections is at most
For the subtracted tangent gradients, split the indices according as
(A_ileq1/2) or (A_i>1/2). In the first set
-(|dlog L_i|leq C A_i|II_i|); in the second there are at most
+(|dlog L_i|<= C A_i|II_i|); in the second there are at most
(4m) indices. Cauchy--Schwarz and
(eq:entropy-last-surplus) consequently bound the sum of their
norms by (C_n√(mP_p/theta)). The final tangent embedding sends
@@ -3137,19 +3137,19 @@ The normal and tangent bounds therefore prove
It follows that
[
|b(u)|leqfrac C r|b(r)|
- leq C_n(((1+m)P_p)/(theta N^2r))^{1/2}
- leq frac{C_nP_p}{√(theta E)}.
+ <= C_n(((1+m)P_p)/(theta N^2r))^{1/2}
+ <= frac{C_nP_p}{√(theta E)}.
]
For the last inequality use (Easympε), (N^2=lε),
and
[
P_pgeq(c_n)/(lr)
(ε+(m^2)/(ε))
- geq (c'_n(1+m))/(lr)
+ >= (c'_n(1+m))/(lr)
]
for (εgeq1). The same first surplus bound gives
- |du|^2leq(C_n)/(N^2r)leq(C_nP_p)/(E^2).
+ |du|^2leq(C_n)/(N^2r)<=(C_nP_p)/(E^2).
All these estimates are only needed where (une0), a subset of
({rleq1/4}) for large (E). Since
@@ -3163,13 +3163,13 @@ entropy term. From (rleq E^{-0.32}) and
[
Pgeq P_pgeq c_n E^{1.32}.
]
-Using (eq:entropy-before-exit) and (e^{2u}leq C E^{0.04}),
+Using (eq:entropy-before-exit) and (e^{2u}<= C E^{0.04}),
one obtains there
- frac{e^{2u}W}{P}leq
- C_nE^{0.04+0.26}P^{-0.25}leq C_nE^{-0.03},
+ frac{e^{2u}W}{P}<=
+ C_nE^{0.04+0.26}P^{-0.25}<= C_nE^{-0.03},
- frac{e^{2u}nu}{P}leq C_nE^{-1.28}log E.
+ frac{e^{2u}nu}{P}<= C_nE^{-1.28}log E.
In the first bound use was made of (E+Pleq C_nP), valid for large (E)
on this region. Both quantities tend to zero with dimensional
@@ -3188,7 +3188,7 @@ to the left gives
-2(e^{2u}-1)W-2nu e^{2u}]
+(1-e^{-2u})nlog E .
-Use was made of (e^{-2u}(C+C_n)leq C+C_n). The last term is nonnegative
+Use was made of (e^{-2u}(C+C_n)<= C+C_n). The last term is nonnegative
for (Egeq1). Equations (eq:entropy-exit-derivatives) and
(eq:entropy-exit-volume-cost) leave at least a fixed fraction
of (P) in the bracket on the support of (u); off that support the
@@ -3201,11 +3201,11 @@ for a fixed (alpha'_n>0), proving the scalar inequality in
the dimensional replacement cost.
The volume estimate at exit follows as well. Since
-(Pleq C_ne^{2u}P'leq C_nE^{0.04}P') and (nuleq C_nlog E),
+(Pleq C_ne^{2u}P'<= C_nE^{0.04}P') and (nuleq C_nlog E),
[
widehatwidehat WWleq
C_n E^{0.26+0.04(0.75)}(E+P')^{0.75}+C_nlog E
- leq E^{0.32}(E+P')^{0.75}
+ <= E^{0.32}(E+P')^{0.75}
]
for all sufficiently large dimensional (E).
@@ -3213,18 +3213,18 @@ Finally, (eq:entropy-Z-energy) and the conformal scaling of
energy give
[
e_{widehatwidehat h}h(Z)^{1/2}
- leq e^{-u}frac{sqrt n/N+c_EK_0}{√(r+c_E^2)}.
+ <= e^{-u}frac{sqrt n/N+c_EK_0}{√(r+c_E^2)}.
]
The choice (eq:entropy-u-choice) implies
[
frac{e^{-u}}{√(r+c_E^2)}
- leq C(frac{E^{-0.02}}{c_E}+E^{0.16}).
+ <= C(frac{E^{-0.02}}{c_E}+E^{0.16}).
]
Using (sqrt E/Nleq C_n(1+log E)), (c_EK_0leq E^{-1/2}),
and (Nleqsqrt E), one concludes that
e_{widehatwidehat h}h(Z)^{1/2}
- leq C_n(
+ <= C_n(
frac{(1+K_0)(1+log E)}{E^{0.02}}+frac{E^{0.16}}{N}).
The first summand is (o(E^{-0.01})) for fixed (K_0); the second is
@@ -3249,7 +3249,7 @@ Apply Lemma~lem:entropy-bootstrap. At its output
(ε_0in[ε_*,2ε_*]). Increasing a dimensional
constant once, (eq:entropy-bootstrap-output) has the form
[
- wscal(h_*,0)geq-C_n+2W-nlogε_0
+ wscal(h_*,0)>=-C_n+2W-nlogε_0
+ε_0 e_{h_*}(F_*),
]
because (ε_0) ranges in a fixed dimensional interval.
@@ -3298,16 +3298,16 @@ is the finite input-dependent gradient bound of (F_*).
Now there holds the scalar inequality, the volume bound with exponent
(0.32), and the absolute gradient bound at (E_1).
Denote the exit metric, map, and fixed cooccurrence graph by
-(h_1,F_1,mathcal G_1calG_1).
+(h_1,F_1,G_1
Next isolate how a small gradient controls the graph in a later
phase. Suppose a phase starts with metric (h_j), root map (F_j)
-with (e_{h_j}(F_j)^{1/2}leq K), and a fixed graph (mathcal G_jcalG_j)
+with (e_{h_j}(F_j)^{1/2}<= K), and a fixed graph (G_j
containing all its possible cooccurrences. Let (R) bound the
following pairwise distance: whenever two possible enlarged supports
meet or come close enough to require separation by a buffer, there
are points (x,y) in the old nonzero supports of their respective
-parent labels with (dist_{h_j}(x,y)leq R). If a closure is
+parent labels with (dist_{h_j}(x,y)<= R). If a closure is
involved, allow an arbitrarily small additional distance. Suppose
[
KR<sqrt2.
@@ -3315,15 +3315,15 @@ involved, allow an arbitrarily small additional distance. Suppose
Choosing the additional distance small enough, the gradient bound gives
[
|F_j(x)-F_j(y)|_{ell^2}
- leq Kdist_{h_j}(x,y)<sqrt2.
+ <= Kdist_{h_j}(x,y)<sqrt2.
]
Two nonnegative unit vectors with disjoint supports have distance
exactly (sqrt2). Thus (F_j(x)) and (F_j(y)) share a nonzero
label, say (k). The first parent and (k) cooccur at (x), and
the second parent and (k) cooccur at (y). The two parents are
-therefore at graph distance at most two in (mathcal G_jcalG_j). This
+therefore at graph distance at most two in (G_j This
argument applies separately at each transverse parameter; because
-(mathcal G_jcalG_j) already contains all possible cooccurrences, the
+(G_j already contains all possible cooccurrences, the
conclusion holds in that one fixed graph.
After an exit at (E_j), one chooses a target that both meets this
@@ -3333,7 +3333,7 @@ overlap condition and grows by one exponential level:
and call the first parameter reaching it (E_{j+1}). Thus
[
- T_{j+1}leq E_{j+1}leq(1+l_*)T_{j+1}.
+ T_{j+1}<= E_{j+1}<=(1+l_*)T_{j+1}.
]
The choice (1/1000) leaves the strict inequality
(3/1000<1/100): the cubic logarithmic cost of support enlargement
@@ -3353,30 +3353,30 @@ Use (h_j), the starting metric, as the reference metric in phase
the preceding argument, including both support enlargement radii and
all separation buffers. The number of sweeps gives
[
- R_{j+1}leq C_n(1+log^3 E_{j+1})
- leq C_n E_j^{3/1000}.
+ R_{j+1}<= C_n(1+log^3 E_{j+1})
+ <= C_n E_j^{3/1000}.
]
Consequently
- R_{j+1}E_j^{-1/100}leq C_n E_j^{-7/1000}→0.
+ R_{j+1}E_j^{-1/100}<= C_n E_j^{-7/1000}→0.
For large (E_j), the overlap condition therefore holds with
(K=E_j^{-1/100}). For every relevant pair the preceding estimate
becomes
[
|F_j(x)-F_j(y)|_{ell^2}
- leq E_j^{-1/100}dist_{h_j}(x,y)<sqrt2.
+ <= E_j^{-1/100}dist_{h_j}(x,y)<sqrt2.
]
This uses only the starting metric of the phase, without a comparison
constant between that metric and (g).
It follows that every separation graph needed in phase (j+1) is
-contained in the square (mathcal G_j^2calG_j^2), after identifying labels
+contained in the square (G_j^2, after identifying labels
with their parents at the start of the phase. At exit there are two
copies of each parent. Joining two such copies whenever their parents
are equal or at distance at most two gives a fixed graph
-(mathcal G_calG_{j+1}) containing all new cooccurrences. If
-(Delta_j) is the degree of (mathcal G_jcalG_j), this construction gives
+(G_{j+1}) containing all new cooccurrences. If
+(Delta_j) is the degree of (G_j, this construction gives
Delta_{j+1}+1leq2(Delta_j+1)^2.
@@ -3388,7 +3388,7 @@ not satisfy a useful quantitative bound. They do not enter
(eq:entropy-graph-recurrence).
After (J) phases, apply the coloring argument to
-(mathcal G_J^2calG_J^2), whose maximum degree is at most (Delta_J^2).
+(G_J^2, whose maximum degree is at most (Delta_J^2).
Take (q) to be the maximum of its degree plus one, the number of
label orbits, and (n+1). Equations (eq:entropy-first-graph) and
(eq:entropy-graph-recurrence) imply
@@ -3396,15 +3396,15 @@ label orbits, and (n+1). Equations (eq:entropy-first-graph) and
log qleq C_{inst,J,n}(1+log E_1)^3.
For fixed (J,n) the coefficient here is fixed before (E_1) is
-chosen. The graph (mathcal G_JcalG_J) and this coloring have exactly
+chosen. The graph (G_J and this coloring have exactly
the properties asserted in the theorem.
It remains to make the first target large enough for the already fixed
(J=H+2). Let
-(mathcal B(q)calB(q)leqexp^{[H]}(B(1+q)^A)) be the prescribed
+(B(q){[H]}(B(1+q)^A)) be the prescribed
elementary bound. Equation (eq:entropy-q-growth) implies
[
- mathcal B(q)calB(q)leq
+ B(q)<=
exp^{[H+1]}(C'_{inst,J,n,A,B}
(1+log E_1)^3)
]
@@ -3429,7 +3429,7 @@ two sides are now the same, and
c_J E_1^{1/1000}>
C'_{inst,J,n,A,B}(1+log E_1)^3
]
-for all sufficiently large (E_1). Thus (E_J>mathcal B(q)calB(q)).
+for all sufficiently large (E_1). Thus (E_J>B(q)
Choose the first target still larger to have
(E_J>E_{min}), to satisfy the initial gradient threshold involving
(K_0), and to meet all dimensional thresholds used above.
@@ -3449,7 +3449,7 @@ and elliptic bounds occur only in the choices of the first target,
graphs, and arbitrarily small analytic errors. None contributes to
the scalar constant. Discard the final nonnegative surplus (P');
the remaining inequality is (eq:entropy-contract) with
-(C=C(n,mathcal B)calB)).
+(C=C(n,B)
The density (rho) has been unchanged since the bootstrap, so
(rholeq dV_g). The volume logarithm starts at zero and only
@@ -3478,7 +3478,7 @@ flow boxes
]
whose images cover X and which are Borel isomorphisms onto their images.
Here Q_a is an open box
-in mathbb R^d,bbR^d, T_a is standard Borel, and T_a carries a
+in R^d, T_a is standard Borel, and T_a carries a
sigma-finite measure mu_a. The plaques kappa_a({t}times Q_a)
fit together as oriented smooth d-manifolds, each Hausdorff and second
countable, and each plaque is open in its leaf. The overlap of two boxes is a countable union of pieces on
@@ -3554,7 +3554,7 @@ the indicated local bounds before using current Stokes.
If omega is a smooth, compactly supported d-form on Q_T^d with
int_{Q_T^d}omega=1, define
- mathcal D_X(G)=calD_X(G)=int_X G^*omega.
+ D_X(G)= G^*omega.
The integral is absolute: its support lies in an inner band, the
derivatives of G there are bounded, and that band has finite measured
@@ -3563,7 +3563,7 @@ zero has a compactly supported primitive. Applying (eq:degree-stokes) to the
pullback of this primitive shows that (eq:degree-definition) is
independent of omega. This is the measured degree, with the
usual signed orientation convention. For d=0 it means the measured
-signed count of the oriented points. Abbreviate it to mathcal DcalD
+signed count of the oriented points. Abbreviate it to D
when the band and map are understood.
[Measured degree estimate]
@@ -3578,7 +3578,7 @@ Suppose (X,h,phi,G) is a controlled measured d-band over Q_{T_d}^d
in the sense of Definition~def:degree-band. Let Bge1 and
L_1,...,L_d>0. Suppose throughout X that
- L_i|dG_i|_hle B-frac dqqquad(1le ile d),
+ L_i|dG_i|_hle B-frac dq (1le ile d),
and suppose that
@@ -3587,7 +3587,7 @@ and suppose that
Define H_i(G)=2(L_i|dG_i|_h-1)_+. Then
- |mathcal D_X(G)|(2calD_X(G)|(2tau)^d
+ |D_X(G)|(2calD_X(G)|(2tau)^d
le frac{exp(C_1d/q)}{prod_{i=1}^dL_i}
int_{{|G|_∞<b_d}}
exp(frac{-wscal(h,phi)+sum_{i=1}^dH_i(G)}2)dV_h.
@@ -3627,7 +3627,7 @@ saturations may be removed for each such s.
leint_XF,dV_h.
For each such s there is a controlled measured p-band
(X_s^+,h_s^+,phi_s^+,J_s^+) over Q_{T_p}^p of degree
-mathcal D_X(G).calD_X(G). Its coordinates are indexed by j=2,...,d and
+D_X(G). Its coordinates are indexed by j=2,...,d and
satisfy
[
L_j|dJ_{s,j}^+|_{h_s^+}le B-frac pq.
@@ -3689,7 +3689,7 @@ be a smooth cooriented local hypersurface, with unit normal nu, and set
D=H+∂_nuphi,
H=trII,
II(Y,Z)=<nabla_Ynu,Z>,
- mathcal P=|II|^2+calP=|II|^2+
+ P=|II|^2+calP=|II|^2+
(Ric_h-nabla^2phi)(nu,nu).
]
@@ -3698,7 +3698,7 @@ of the raw pressure are ∂_sP_0(s-G_1)=mathfrak afraka and
d_xP_0(s-G_1)=-mathfrak afraka,dG_1. The following calculation uses
only a positive smooth function v satisfying the local equation
- -(Delta_{Σ,phi}+mathcal PcalP
+ -(Delta_{Σ,phi}+P
-mathfrak afraka,dG_1(nu))v=mathfrak a.fraka.
Here Delta_{Σ,phi} is weighted by
@@ -3711,7 +3711,7 @@ S_Σ=wscal(h|_{TΣ},phi_Σ). Gauss' equation and the
normal/tangential decomposition of derivatives of phi give
[
wscal(h|_{TΣ},phi|_Σ)
- =wscal(h,phi)+D^2-2mathcal P+|II|^2.calP+|II|^2.
+ =wscal(h,phi)+D^2-2P+|II|^2.calP+|II|^2.
]
Adding ln v to the weight changes this by
-2Delta_{Σ,phi}v/v+|dln v|^2. Substitution of
@@ -3776,7 +3776,7 @@ this construction uses no global weighted-volume bound.
parnoindent{footnotesizehttps://doi.org/10.1007/978-1-4612-4098-3_1}
-parnoindent{footnotesizettfamily src-chodoshli2024.pdf}
+parnoindent{footnotesizettfamily ./pdf/src-chodoshli2024.pdf}
The output will be a canonical nested family E_s on each open leaf,
independent of the exhaustion and jointly Borel in the plaque data and
@@ -3813,10 +3813,10 @@ The same inequality proves both monotonicities one needs. If Ksubset K'
and two forcings satisfy mathfrak hfrakhlemathfrak h',frakh', a minimizer for
(K,mathfrak h)frakh) and one for (K',mathfrak h')frakh') obey
[
- mathcal F_calF_{mathfrak h}frakh(Ecap F)+mathcal F_calF_{mathfrak h'}frakh'(Ecup F)
- le mathcal F_calF_{mathfrak h}frakh(E)+mathcal F_calF_{mathfrak h'}frakh'(F),
+ F_{mathfrak h}frakh(Ecap F)+F_{mathfrak h'}frakh'(Ecup F)
+ le F_{mathfrak h}frakh(E)+F_{mathfrak h'}frakh'(F),
]
-where mathcal FcalF denotes (eq:degree-functional); the inequality for
+where F denotes (eq:degree-functional); the inequality for
the volume terms follows on E∖ F from
mathfrak hfrakhlemathfrak h'.frakh'. Minimality in
the two domains makes both inequalities equalities, and leastness in K
@@ -3862,7 +3862,7 @@ Starting at a plaque, enumerate its accessible normalized plaque
occurrences Borel measurably in slots jge1, using dummy slots when
necessary. Use the fixed Polish space
[
- mathcal Z=calZ=prod_{j,ellge1}L^1(R_ell),
+ Z={j,ellge1}L^1(R_ell),
]
with Lebesgue L^1 norms and the product topology. A set is coded by its
indicator on R_ell in occurrence j; dummy entries are zero.
@@ -3871,7 +3871,7 @@ including repeated occurrences. The closures of the first finitely
many accessible R_ell in a fixed enumeration of the pairs (j,ell)
give the compact exhaustion domains. The conditions that the code is an
indicator, agrees on overlaps, and vanishes off the specified domain are
-Borel, and for fixed plaque data are closed in mathcal Z.calZ.
+Borel, and for fixed plaque data are closed in Z.
Integrals against chartwise Borel smooth densities are Borel by Fubini.
Perimeter is Borel as well: its supremum over divergence integrals may
@@ -3887,7 +3887,7 @@ perimeter, since the forcing integral is uniformly bounded on this
fixed compact set. In every active R_ell the metric and density
have a positive lower bound on its closure, so this gives a Euclidean BV
bound there; the indicator's L^1(R_ell) norm is bounded by |R_ell|.
-Diagonal BV compactness gives convergence in mathcal ZcalZ along a
+Diagonal BV compactness gives convergence in Z along a
subsequence. The fiberwise closed indicator, overlap, and support
conditions persist. The volume term is continuous, since K is
covered by finitely many compact subboxes, and perimeter is lower
@@ -3993,7 +3993,7 @@ where the resulting set changes is at most the mass of the added current.
The set comparison therefore implies that formulation as well.
-parnoindent{footnotesizettfamily src-eichmair2009.pdf}
+parnoindent{footnotesizettfamily ./pdf/src-eichmair2009.pdf}
Local compactness claim.
Suppose dge2 and E_j minimize the absorbed-perimeter functionals
@@ -4026,8 +4026,8 @@ subsequence converging in L^1_{loc} to a set E. The belt
comparison with any compact variation, as proved above and now with the
smoothly converging coefficients, shows that E is locally minimizing.
-Write mu_j=P_{widehatwidehat h_j}h_j(E_j;,cdot,) and
-mu_∞=P_{widehatwidehat h_∞}h_∞(E;,cdot,). To prove convergence
+Write mu_j=P_{widehatwidehat h_j}h_j(E_j;,·,) and
+mu_∞=P_{widehatwidehat h_∞}h_∞(E;,·,). To prove convergence
of these measures, begin with a relatively compact coordinate domain
V from a convergence-determining family, chosen so that
mu_∞(∂ V)=0. Surround it by a thin outer belt with level
@@ -4150,7 +4150,7 @@ so it never contributes to the integrals over the boundary current.
Differentiating the ordered family.:
On each leaf the sets E_t converge to E_s in L^1_{loc} as
tto s except at countably many s. In fact, on a compact chart set
-K the function tmapstoint_Kmathbf1_bf1_{E_t} is monotone, and nestedness
+K the function t|->int_Kmathbf1_bf1_{E_t} is monotone, and nestedness
identifies its increments with the L^1(K) distance between the
indicators. It has at most countably many discontinuities. Use the
countable chart exhaustion to obtain the assertion on the leaf.
@@ -4161,9 +4161,9 @@ At such a continuity parameter, the local convergence result above makes
In a smaller graph cylinder choose coordinates (y,z) with the sets
below the graphs, and write the graph heights as z=r(t,y). They are
monotone in t. They are strictly ordered wherever two such regular
-graphs are defined: if r(t,cdot)ge r(s,cdot) for t>s and they touch,
+graphs are defined: if r(t,·)ge r(s,·) for t>s and they touch,
their tangent planes agree at contact. The comparison of the graph mean
-curvatures at that minimum of r(t,cdot)-r(s,cdot) gives
+curvatures at that minimum of r(t,·)-r(s,·) gives
D_tle D_s, whereas their equations give
D_t=mathfrak h_t>frakh_t>mathfrak h_s=D_sfrakh_s=D_s at the same point. This is
impossible. Use the conventions for D and II from the
@@ -4207,13 +4207,13 @@ p=0 the same conclusion follows from the ordinary almost-everywhere
derivative of a monotone height.
Differentiation of D=mathfrak h_sfrakh_s now identifies this normal speed.
-Use mathcal PcalP from the local slice calculation and put
+Use P from the local slice calculation and put
mathfrak a=∂_sfraka=∂_smathfrak h_s=L_1frakh_s=L_1psi(D)>0; the last equality
uses confinement, where the forcing agrees with the raw pressure.
The normal variation formula, with the tangential derivative of
D-mathfrak h_s=0frakh_s=0 cancelling, is
- -(Delta_{Σ,phi}+mathcal P+∂_calP+∂_numathfrak h_s)vfrakh_s)v
+ -(Delta_{Σ,phi}+P+∂_calP+∂_numathfrak h_s)vfrakh_s)v
=mathfrak a.fraka.
The convention for Delta_{Σ,phi} is weighted by
@@ -4313,7 +4313,7 @@ boundary where the residual form is supported. Measured current Stokes therefor
int_{∂[E_s]} chi_{deg}(G_1)J^*beta
&=int_{E_s}(a_-(G_1)-a_+(G_1)),dG_1wedge J^*beta
&=int_Xa_-(G_1),dG_1wedge J^*beta
- =mathcal D_X(G).calD_X(G).
+ =D_X(G).
All forms here are supported in an inner band. The current [E_s] has
@@ -4349,7 +4349,7 @@ The G_1 sides of the first restriction cannot be reached by confinement,
since |G_1|<tau+b<T_p. Near any other side some residual coordinate
has absolute value near t_1>T_p. Hence every inner residual subcube
has buffered graph charts in this restriction. Its degree equals
-mathcal D_X(G)calD_X(G) by (eq:degree-boundary-degree), and its coordinate gradients only
+D_X(G) by (eq:degree-boundary-degree), and its coordinate gradients only
decrease on restriction to the tangent space. Take J^+=J and use
(eq:degree-slice-bound).
@@ -4371,7 +4371,7 @@ plaque overlap remains bounded. Point boxes have the required common
buffers, and all metric, map, and positive-order
weight derivative conditions in Definition~def:degree-band are
vacuous in dimension zero, so this is a controlled 0-band. Its measured
-signed count is mathcal D_X(G)calD_X(G) by
+signed count is D_X(G) by
(eq:degree-boundary-degree); its integral is the measured unsigned
count. At each point (eq:degree-raw-integrand) reads
1le(v/L_1)Θ, and confinement places every counted point in
@@ -4389,7 +4389,7 @@ boundary current.
Construct a controlled child carried by an open regular subset
of this slice, with the inherited orientation and transverse measure,
-and with degree mathcal D_X(G).calD_X(G). For each residual row one requires |dJ_j^+|_{h^+}le |dG_j|_h+ε_{grad} and the
+and with degree D_X(G). For each residual row one requires |dJ_j^+|_{h^+}le |dG_j|_h+ε_{grad} and the
one-sided value bound
|J_j^+|ge |G_j|-ε_{val}, with
@@ -4400,18 +4400,18 @@ The gradient bound controls the residual penalties and the child's
coordinate bound.
The value bound, together with confinement of G_1, puts the child's
b_p integration region over the parent's b_d region. Write
-mathcal W=dV_calW=dV_{h^+}/dA_h for its ordinary leafwise volume density
+W=dV_{h^+}/dA_h for its ordinary leafwise volume density
relative to the original induced metric. The scalar and volume target is
- dV_{h^+}=mathcal WcalW,dA_h,
+ dV_{h^+}=W,dA_h,
wscal(h^+,phi^+)gewscal(h,phi)-H_1(G)-2ln(v/L_1)
- +2lnmathcal W-calW-frac1q.
+ +2lnW-calW-frac1q.
The coefficient 2 makes the volume factor cancel after exponentiation,
leaving the same lapse factor as in (eq:degree-raw-integrand).
-The density mathcal WcalW includes neither v nor the transverse measure.
+The density W includes neither v nor the transverse measure.
Every constant and error chosen inside this subsection may depend on this
slice and on the band, but must be common across its retained plaques.
@@ -4481,7 +4481,7 @@ After the initial homotopy on the full boundary current, this gap will
keep the homotopies over a small target cube inside a uniformly regular
region.
To prove the local assertion behind the claim, fix a positive budget for
-a constant shift in mathbb R^pbbR^p and an inner compact plaque box.
+a constant shift in R^p and an inner compact plaque box.
There are a level and a positive gap, uniform across the controlled
plaques, such that some shift in that budget gives this gap from zero on
the marked set in the box. Otherwise take a sequence of failing plaques,
@@ -4606,12 +4606,12 @@ For j=alpha,alpha+2,alpha+4,alpha+6 put
u_j=ε_{base}^j+sum_i M_*r_i^alpha K_i^j,
Λ=(frac{u_{alpha+4}}{u_alpha})^{1/4},
- mathcal L=|dcalL=|dln u_alpha|^2+|dln u_{alpha+4}|^2.
+ L=|dcalL=|dln u_alpha|^2+|dln u_{alpha+4}|^2.
These quantities have separate roles. Make u_alpha large on
D_{k'} so that its logarithm locates the region to be excluded.
The term Λ^2 will supply the positive scalar contribution that
-pays for the later coordinate graphs, while mathcal LcalL controls the
+pays for the later coordinate graphs, while L controls the
logarithmic derivatives introduced by the weight and the first graph.
Here 0<ε_{base}<1 will be small; M_* will be a
large finite number fixed before the final cover selection. At a point, the sum is over the
@@ -4639,15 +4639,15 @@ Make their total active mass at most two. Moment inequalities give
frac{u_{alpha+6}}{u_{alpha+4}}geΛ^2
gefrac{u_{alpha+2}}{u_alpha},
- |dlnΛ|^2le(mathcal L)/(8),calL)/(8),
+ |dlnΛ|^2le(L)/(8),calL)/(8),
|dln u_alpha|le C(Λ+1),
Λge c' u_alpha^{1/alpha}.
For the first pair, use the probability weights proportional to the
atoms' masses times their alpha-powers. If Z is the square of the
atom position, then
-mathbb EbbE Zle(mathbb EbbE Z^2)^{1/2} and
-mathbb EbbE Z^3ge(mathbb EbbE Z^2)^{3/2} give the two inequalities. The
+E Zle(E Z^2)^{1/2} and
+E Z^3ge(E Z^2)^{3/2} give the two inequalities. The
second estimate follows directly by differentiating
lnΛ=(ln u_{alpha+4}-ln u_alpha)/4. For the third, the
gradient bound in (eq:degree-kernel-bounds) bounds
@@ -4672,22 +4672,22 @@ Add eta_0=(ln u_alpha+ln u_{alpha+4})/4 to phi_Σ on the
regular boundary. Put Q_1=deta_0. The weight-change formula used in
(eq:degree-slice-identity), now with
Delta_{Σ,phi_Σ}=Delta_{Σ,phi}
-+< dln v,d,cdot,>, yields from
++< dln v,d,·,>, yields from
(eq:degree-potential-laplacian) the scalar gains
-cΛ^2+mathcal L/2-2<calL/2-2< dln v,Q_1>-|Q_1|^2-delta.
+cΛ^2+L/2-2<calL/2-2< dln v,Q_1>-|Q_1|^2-delta.
Here the high moment in (eq:degree-moments) supplies Λ^2.
-Since |Q_1|^2lemathcal L/8,calL/8,
+Since |Q_1|^2leL/8,calL/8,
[
- (mathcal L)/(2)+|dcalL)/(2)+|dln v|^2
+ (L)/(2)+|dcalL)/(2)+|dln v|^2
-2< dln v,Q_1>-|Q_1|^2
- ge(mathcal L)/(2)-2|Q_1|^2calL)/(2)-2|Q_1|^2ge(mathcal L)/(4).calL)/(4).
+ ge(L)/(2)-2|Q_1|^2calL)/(2)-2|Q_1|^2ge(L)/(4).calL)/(4).
]
Combining with (eq:degree-slice-bound), and decreasing c>0 if
necessary, gives for the new weight
wscal(h|_{TΣ},phi_Σ+eta_0)
gewscal(h,phi)-H_1(G)-2ln(v/L_1)
- +cΛ^2+(mathcal L)/(4)-calL)/(4)-delta.
+ +cΛ^2+(L)/(4)-calL)/(4)-delta.
A large scalar allowance has been obtained near the marked set. The next
construction uses it to move the residual map out of the target band
@@ -4704,15 +4704,15 @@ graph metric, in addition to the earlier perturbation of barbar J.J.
Their scalar gains will pay the exact graph volume factors.
For a current metric and
-weight (k,varphi), a length-one graph of a function f has
+weight (k,phi), a length-one graph of a function f has
[
L_f=(1+|df|_k^2)^{1/2}, U_f=(nabla_k f)/(L_f),
- k_f=k+df ⊗ df, varphi_f=varphi-ln L_f,
- D_f=-div_{k,varphi}U_f.
+ k_f=k+df ⊗ df, phi_f=phi-ln L_f,
+ D_f=-div_{k,phi}U_f.
]
Lemma~lem:graph-identity says
- wscal(k_f,varphi_f)-wscal(k,varphi)
+ wscal(k_f,phi_f)-wscal(k,phi)
=D_f^2-2U_f(D_f)+|II_f|^2+|dln L_f|_{k_f}^2.
Also
@@ -4749,7 +4749,7 @@ inequality and (eq:degree-moments) give
[
2|D_fU_f(ln C_f)|
le0.8D_f^2+1.25|dln C_f|^2
- le0.8D_f^2+(1.25)/(8)mathcal L.calL.
+ le0.8D_f^2+(1.25)/(8)L.
]
The f_0 term has absolute value at most
[
@@ -4760,12 +4760,12 @@ than the remaining 0.2D_f^2 on its square term. Moreover
C_f(Λ+1)le C_{B_0}(1+Λ^2), so a further small choice of
sigma spends less than half the cΛ^2 in
(eq:degree-weight-gain) and makes the remaining constant loss as
-small as prescribed. The available mathcal L/4calL/4 covers
-1.25mathcal L/8.calL/8. Combining (eq:degree-weight-gain) and
+small as prescribed. The available L/4calL/4 covers
+1.25L/8.calL/8. Combining (eq:degree-weight-gain) and
(eq:degree-graph-identity), and discarding nonnegative terms, thus
gives
- wscal(k_f,varphi_f)gewscal(h,phi)-H_1(G)-2ln(v/L_1)
+ wscal(k_f,phi_f)gewscal(h,phi)-H_1(G)-2ln(v/L_1)
+c_1Λ^2+2ln L_f-delta_1,
where c_1>0 is fixed, while delta_1 can be made arbitrarily small. This
@@ -4775,15 +4775,15 @@ induced metric for which (eq:degree-weight-gain) was proved.
For the remaining graphs, let zeta(f) be a smooth function valued in
[0,1], zero for fle K and one for fge K+1, with derivative bound
independent of the threshold K. Write
-(k^{(1)},varphi^{(1)})=(k_f,varphi_f). For row j=2,...,d, its
+(k^{(1)},phi^{(1)})=(k_f,phi_f). For row j=2,...,d, its
graph quantities use the metric and weight after all preceding rows:
[
L'_j=(1+|df'_j|_{k^{(j-1)}}^2)^{1/2},
U'_j=frac{nabla_{k^{(j-1)}}f'_j}{L'_j},
- D'_j=-div_{k^{(j-1)},varphi^{(j-1)}}U'_j,
+ D'_j=-div_{k^{(j-1)},phi^{(j-1)}}U'_j,
k^{(j)}=k^{(j-1)}+df'_j ⊗ df'_j,
- varphi^{(j)}=varphi^{(j-1)}-ln L'_j.
+ phi^{(j)}=phi^{(j-1)}-ln L'_j.
]
Prescribe successively
@@ -4793,11 +4793,11 @@ Prescribe successively
J_j^+=barbar J_j+eJ_j+e f'_j.
The constants A,w,e,B' are positive; their order of choice is given
-below. Put h^+=k^{(d)}, phi^+=varphi^{(d)}, and
+below. Put h^+=k^{(d)}, phi^+=phi^{(d)}, and
[
- mathcal W=L_fcalW=L_fprod_{j=2}^dL'_j.
+ W=L_f{j=2}^dL'_j.
]
-The successive volume identities give dV_{h^+}=mathcal WcalW,dV_k
+The successive volume identities give dV_{h^+}=W,dV_k
relative to the current base metric k. On the eventual child that
base will be the original induced metric, so this product will be the
density in (eq:degree-child-scalar). In the metric after the first graph, |df|le1 by
@@ -5017,7 +5017,7 @@ barbar JJ on the whole parent boundary current. The small value
perturbation ensures that the preimages of this support throughout the
homotopy remain in a fixed old inner band. One can justify invariance
directly by the homotopy formula. The difference of the two pullbacks
-of beta is dmathcal HcalHbeta, where mathcal HcalHbeta is its usual
+of beta is dH, where H is its usual
transgression along the homotopy. Multiply by the cutoff
chi_{deg}(G_1) used
in (eq:degree-boundary-degree). The current ∂[E_s] is
@@ -5042,24 +5042,24 @@ region and the kernel supports, and retain the same spatial and
G_1 margins. Ordinary measured Stokes on the regular buffered charts
applies to these homotopies. The final support lies in
(eq:degree-child-domain), so its integral is the measured degree
-of the child. It equals mathcal D_X(G).calD_X(G).
+of the child. It equals D_X(G).
On the child the initial auxiliary metric is the induced metric, so
(eq:degree-graph-elementary) verifies the volume identity in
-(eq:degree-child-scalar) for the product mathcal WcalW defined above.
+(eq:degree-child-scalar) for the product W defined above.
Choose all the adjustable scalar errors above with sum at most 1/q.
The absorption of the row errors and
(eq:degree-first-graph-gain) give the scalar inequality in
(eq:degree-child-scalar) on the child.
This completes the construction in high dimensions. In the case
-dle7 treated above, use mathcal W=1.calW=1. The volume identity is then
+dle7 treated above, use W=1. The volume identity is then
immediate, and (eq:degree-slice-bound) gives the scalar inequality
in (eq:degree-child-scalar). In dimension p=0 the same statements
hold on the counted points with the zero-dimensional scalar convention.
Proof. [Completion of the proof of Lemma~lem:degree-descent]
The coarea assertion is (eq:degree-coarea-proof). The constructions
-above give a controlled child of degree mathcal D_X(G)calD_X(G) for each typical
+above give a controlled child of degree D_X(G) for each typical
s. Its coordinate bound follows from
(eq:degree-output-estimates) and (eq:degree-tolerances):
[
@@ -5080,7 +5080,7 @@ For the first row confinement gives
]
The same observation covers the counted case p=0.
-The function xmapsto(x-1)_+ is one-Lipschitz, so
+The function x|->(x-1)_+ is one-Lipschitz, so
(eq:degree-output-estimates) gives
[
sum_{j=2}^dH_j^+lesum_{j=2}^dH_j(G)
@@ -5091,7 +5091,7 @@ Use this and the scalar and volume relations in
&exp(frac{-wscal(h^+,phi^+)+sum_{j=2}^dH_j^+}{2})
- mathcal WcalW
+ W
&quadle
exp(frac1{2q}+sum_{j=2}^dL_jε_{grad})
frac v{L_1}
@@ -5099,8 +5099,8 @@ Use this and the scalar and volume relations in
&quadle e^{3/(5q)}frac v{L_1}Θ.
-The factor mathcal WcalW cancels exactly against the term
-2lnmathcal WcalW in the scalar bound. The factor v/L_1 is the lapse
+The factor W cancels exactly against the term
+2lnW in the scalar bound. The factor v/L_1 is the lapse
factor in coarea. This calculation makes no bound on either factor.
For a low-dimensional child use (eq:degree-slice-bound), with no
row error; for p=0 the same estimate is the pointwise bound for a
@@ -5118,7 +5118,7 @@ Lemma~lem:degree-descent, including the unchanged strict horizon
condition. For almost every s the induction hypothesis and
(eq:degree-child-integral) give
[
- |mathcal D_X(G)|(2calD_X(G)|(2tau)^p
+ |D_X(G)|(2calD_X(G)|(2tau)^p
le frac{e^{C_1p/q+C_*/q}}{prod_{i=1}^dL_i}
int_{Σ_scap{|G|_∞<b_d}}v_sΘ,dA_h.
]
@@ -5162,7 +5162,7 @@ For every integer (ngeq 3), there is a finite constant (C_n>0) such
that every closed connected oriented smooth (n)-manifold (M) and every
smooth Riemannian metric (g) with (Scal_gequiv-1) satisfy
[
- |M|leq C_nVol_g(M).
+ |M|<= C_nVol_g(M).
]
@@ -5176,7 +5176,7 @@ Denote the scalar constant in this contract by (C_{rm ent}).
All functions called elementary in this section have a fixed height of
exponentiation on a polynomial, with parameters depending only on (n).
-Write (mathcal M=(calM=(widetilde MtimesΩ)/Gamma) for such a measured
+Write (M=( MtimesΩ)/Gamma) for such a measured
family. Denote its metric, weight, density, and
root probability map by (h,phi,rho,F), and put
[
@@ -5185,8 +5185,8 @@ root probability map by (h,phi,rho,F), and put
Its estimates are
rholeq dV_g, Wgeq0,
- wscal(h,phi)geq-C_{rm ent}+2W-nlog E+Emathfrak e,frake,
- mathfrak e^frake^{1/2}leq E^{-1/100}.
+ wscal(h,phi)>=-C_{rm ent}+2W-nlog E+Emathfrak e,frake,
+ mathfrak e^frake^{1/2}<= E^{-1/100}.
There is a fixed invariant graph on the countable label set, with finitely
many free (Gamma)-orbits, containing every cooccurrence of nonzero
@@ -5194,7 +5194,7 @@ coordinates of (F). The integer (qgeq n+1) bounds the number of
orbits and the maximum graph degree plus one. The equivariant coloring
by (q) colors separates any two labels at graph distance at most two.
Its dependence on (omegainΩ) is Borel. Set
-(m=q-n>0). Integrals over (mathcal McalM) and over the measured spaces
+(m=q-n>0). Integrals over (M and over the measured spaces
constructed below include their transverse measures.
medskipnoindentConditional transfer target.
@@ -5204,7 +5204,7 @@ such that every choice of detector and conditional entropy data just
specified, including its constant (C_{rm ent}), satisfies
EgeqPsi_n(q)⇒
- Ileq K_n e^{C_{rm ent}/2}int_{mathcal M}calMrho.
+ Ileq K_n e^{C_{rm ent}/2}int_{M}
Thus (K_n) and (Psi_n) are fixed before the manifold, cocycle,
instance geometry, and (C_{rm ent}) are chosen. Carry out the
@@ -5221,7 +5221,7 @@ target is closed will Theorem~thm:entropy supply an actual pair
For a fixed (omega), let (K=K_omega) consist of all nonnegative
unit root vectors whose support is a clique in the label graph. Thus
(K) is the union of the spherical orthants on those cliques. A vector
-(yin K) has a color image (z_yinmathbb R^qbbR^q), obtained by placing
+(yin K) has a color image (z_yinR^q, obtained by placing
each of its root coordinates at the color of its label. Distinct labels
in a clique have distinct colors. Distant cliques can reuse colors, so
the flat neighborhood below keeps distinct label configurations with the
@@ -5230,8 +5230,8 @@ Euclidean coordinates for the grid averages.
[Flat sheet neighborhood]
Put (delta=(1000sqrt q)^{-1}). There is an oriented flat smooth
-(q)-manifold (mathcal U_calU_omega) with a local isometry
-(z:mathcal U_calU_omegatomathbb R^qbbR^q), and an inclusion of (K_omega)
+(q)-manifold (U_ with a local isometry
+(z:U_ R^qbbR^q), and an inclusion of (K_omega)
into it, such that the image of every (yin K_omega) has a flat ball
chart of radius (10delta) centered at (z_y). This construction is
a standard Borel family in (omega), with countably many charts and an
@@ -5245,7 +5245,7 @@ in these flat charts by pulling back the detector form through the cutoff
probabilities constructed below. It is Borel and equivariant in the
family, and satisfies
- int_{mathcal M}calMF^*eta=I, |eta|_{HS}leq C_n.
+ int_{M}*eta=I, |eta|_{HS}<= C_n.
Here the exterior Hilbert norm uses the Euclidean metric on the charts.
For every fixed derivative order its derivatives admit an elementary
@@ -5253,7 +5253,7 @@ bound in (q).
Proof.
-Take one copy of (B(z_y,10delta)subsetmathbb R^qbbR^q) for each witness
+Take one copy of (B(z_y,10delta)subsetR^q for each witness
(yin K). At the same point (z) in two such balls, identify the two
copies if their witnesses share a label in their nonzero supports. Check that these identifications give ball charts, rather than a branched
space. For every point under consideration, some coordinate of (z)
@@ -5292,7 +5292,7 @@ by (dz_F), proving the asserted energy equality.
To construct this cutoff pullback, choose a nonnegative smooth function (chi)
which is zero for (tleq12delta), equals (t) for
(tgeq25delta), lies between zero and (max{t,0}), and satisfies the usual rescaled derivative bounds
-(|chi^{(j)}|leq C_jdelta^{1-j}) for (jgeq1). In a ball put
+(|chi^{(j)}|<= C_jdelta^{1-j}) for (jgeq1). In a ball put
[
w(z)=frac{(chi(z_j))_{j=1}^q}{|(chi(z_j))_{j=1}^q|},
p_j=w_j^2.
@@ -5325,7 +5325,7 @@ unit root vectors (w). In the Euclidean coordinate directions
Its absolute value is at most (n!2^nprod_j w_{i_j}). The sum of the
squares of these products over all increasing (n)-tuples is at most
((sum_i w_i^2)^n=1). Thus its exterior Hilbert norm is at most a
-constant depending only on (n). The map (zmapsto w(z)) has
+constant depending only on (n). The map (z|-> w(z)) has
absolutely bounded operator norm of its differential: its denominator
is bounded below, normalization is an orthogonal projection divided by
that denominator, and (chi') is absolutely bounded. Pullback
@@ -5346,7 +5346,7 @@ radius (10rsqrt q) in the inner sheet charts. The derivative bounds
of Lemma~lem:transfer-flat permit such a choice with an elementary
bound on (r^{-1}), uniformly in (V,s,omega).
-A primal (n)-cell has an axis set (mathcal IcalIsubset{1,...,q})
+A primal (n)-cell has an axis set (I{1,...,q})
of size (n), varies along those grid intervals, and has the remaining
coordinates at grid vertices. Orient it in increasing axis order.
Include each lift (C) whose center is at distance less than (4delta)
@@ -5355,7 +5355,7 @@ chart. Its dual (m)-cube (C^#) has the same center, varies along
the complementary axes by ([-r/2,r/2]), and has the other coordinates
fixed. It too fits in the chart. Orient it so that the primal oriented
basis, followed by the negatives of its dual oriented basis, is a
-positive basis of (mathbb R^qbbR^q). Give the primal cell the coefficient
+positive basis of (R^q Give the primal cell the coefficient
u_C=int_Ceta.
@@ -5379,9 +5379,9 @@ incidence: all these statements use lifts in the same flat chart.
[Finite-type unbiased rounding]
There is an integer (D_0geq1), bounded by an elementary function of
(q) and constant in (V,s,omega), and a Borel equivariant random
-choice of coefficients (u'_Cin D_0^{-1}mathbb ZbbZ) such that
+choice of coefficients (u'_Cin D_0^{-1}Z such that
- E_{rm rd}u'_C=u_C, |u'_C-u_C|leq r^nq^{-n}.
+ E_{rm rd}u'_C=u_C, |u'_C-u_C|<= r^nq^{-n}.
The expectation is conditional on the grid and entropy data. Every
cocycle equation just described is satisfied exactly by the (u'_C).
@@ -5435,7 +5435,7 @@ an invalid slot value zero. The preceding active-color observation
makes this integral unambiguous. The validity tests and integrals are
Borel in (V,s).
-This is one vector (v=v(V,s)inmathbb R^bbR^{N_0}) for the fixed graph
+This is one vector (v=v(V,s)inR^{N_0}) for the fixed graph
and cocycle, common to all (omega). Only its size and the bounds
below are uniform over all graphs and cocycles of the given complexity:
there are at most (q^q) neighborhood color assignments, (2^q)
@@ -5451,7 +5451,7 @@ different valid witness cliques have the same value: the
throughout the primal cell.
Impose on this list every integer row (b) with
-(|b|_∞leq2(n+1)) for which (bcdot v=0). There are
+(|b|_∞leq2(n+1)) for which (b· v=0). There are
finitely many candidate rows, so the resulting matrix depends Borelly
on (v). In particular it includes the difference row for any two
duplicate valid slots. It also includes every needed cubical equation,
@@ -5464,7 +5464,7 @@ integer matrix has a real basis of integer vectors (b_1,...,b_k)
with
[
kleq N_0,
- max_j|b_j|_∞leq K_0:=(C_nN_0)^{C N_0}.
+ max_j|b_j|_∞<= K_0:=(C_nN_0)^{C N_0}.
]
To see this, choose a nonsingular maximal minor and solve for its pivot
coordinates in terms of the free coordinates; multiplying each basis
@@ -5480,8 +5480,8 @@ Independently round each (t_j) to the two adjacent multiples of
expectation is (t_j). Choose a single integer
(D_0geq N_0K_0r^{-n}q^n), using uniform elementary upper bounds
for (N_0,K_0). The resulting vector (v') lies in the same
-kernel, has entries in (D_0^{-1}mathbb ZbbZ), has expectation (v),
-and satisfies (|v'-v|_∞leq N_0K_0/D_0leq r^nq^{-n}).
+kernel, has entries in (D_0^{-1}Z, has expectation (v),
+and satisfies (|v'-v|_∞<= N_0K_0/D_0leq r^nq^{-n}).
All operations, including taking floors and using independent uniform
parameters for the adjacent choices, are Borel. The fixed slot order
is used for every pivot choice. Thus one parameter (xi), independent
@@ -5498,7 +5498,7 @@ The rounded coefficients define the original oriented dual current
Here ([C^#]) is integration over the oriented cube, and the
coefficients are integers. Verify the local finiteness implicit
in this notation. The form bound and rounding give
-(|u_C|leq C_nr^n) and an elementary bound for each (|D_0u'_C|).
+(|u_C|<= C_nr^n) and an elementary bound for each (|D_0u'_C|).
Let (B) be a small flat ball in one sheet chart and (Q) one full
projected dual cube. The intersection (Qcap z(B)) is convex. Any
lift of (Q) meeting (B) agrees with the chart inverse on this
@@ -5509,15 +5509,15 @@ projected points, agree there and then agree on the connected cube
Only finitely many projected grid cubes meet (z(B)), with bounded
coefficients, proving neighborhood local finiteness.
-Let (mathcal T_calT_deltasubsetmathcal U_calU_omega) be the strict tube
+Let (T_ U_calU_omega) be the strict tube
of points at distance less than (delta) from a witness in its ball
chart. The exact incidence equations in
Lemma~lem:transfer-rounding give
- ∂ Z_{rm rd}big|_{mathcal T_calT_delta}=0.
+ ∂ Z_{rm rd}big|_{T_}=0.
This is a local assertion: the boundary current vanishes on test forms
-compactly supported in (mathcal T_calT_delta).
+compactly supported in (T_
To resolve (Z_{rm rd}) smoothly, realize each coefficient by
(|D_0u'_C|) oriented copies of its cube, reversing orientation when
@@ -5530,7 +5530,7 @@ Local axis, grid, and copy indices order the incident copies, so these
choices may be Borel and equivariant.
The doubling does not join the two sides near (K), even through a
-lower face. If a point of any face maps into (mathcal T_calT_delta),
+lower face. If a point of any face maps into (T_,
every path of facet identifications relating chambers incident to that
face preserves its image point. Each facet on the path meets the tube,
so was balanced before doubling and was not a free mirror facet. The
@@ -5545,10 +5545,10 @@ Take barycentric subdivision of the doubled cubes. Each top simplex
its barycentric faces, and every facet pairing respects these colors.
Write (ε_sigmain{1,-1}) for its sign relative to that
color order, and call this doubled signed simplicial diagram
-(mathcal PcalP). It maps facewise to (mathcal U_calU_omega), with
+(P It maps facewise to (U_, with
elementary subdivision and incidence bounds. The original current
(Z_{rm rd}) remains the object to be represented on
-(mathcal T_calT_delta); the doubling supplies closed facet pairings
+(T_; the doubling supplies closed facet pairings
for the resolution.
@@ -5567,25 +5567,25 @@ the pairing data, and supply smooth measured charts and quantitative
metric bounds.
-parnoindent{footnotesizettfamily src-gaifullin2008.pdf}
+parnoindent{footnotesizettfamily ./pdf/src-gaifullin2008.pdf}
[Measured resolution]
-For the diagram (mathcal PcalP) there is a standard Borel measured
-family (mathcal AcalA) of oriented smooth (m)-manifolds and a smooth
-map (X:mathcal AcalAtomathcal U_calU_omega), equivariant and Borel in
+For the diagram (P there is a standard Borel measured
+family (A of oriented smooth (m)-manifolds and a smooth
+map (X:A U_calU_omega), equivariant and Borel in
all the data. The integration identities below are for fixed grid,
entropy, and rounding data, including ((V,s,omega,xi)); their
integrals use the conditional state measure and depend Borelly on those
-data. The points over (mathcal T_calT_delta) represented by
+data. The points over (T_ represented by
original chamber tiles form a Borel leaf-open subset
-(mathcal A_calA_{rm orig}subset X^{-1}(mathcal T_calT_delta)). For every
-compactly supported smooth (m)-form (alpha) on (mathcal T_calT_delta),
+(A_{rm orig}subset X^{-1}(T_ For every
+compactly supported smooth (m)-form (alpha) on (T_,
- int_{mathcal A_calA_{rm orig}}X^*alpha=Z_{rm rd}(alpha).
+ int_{A_{rm orig}}X^*alpha=Z_{rm rd}(alpha).
For the full doubled diagram, measured oriented integration also gives
- X_*[mathcal A]=[calA]=[mathcal P].calP].
+ X_*[A]=[calA]=[P].calP].
For a parameter (0<lambda<1/2) and a choice
@@ -5604,10 +5604,10 @@ these two parameters.
One may prescribe for each chamber over a dual cube a rotation
(O_sigmain SO(q)). There is also a smooth Borel equivariant
-(mathcal R:calR:mathcal AcalAto SO(q)) equal to (O_sigma) off the
+(R:calR:A SO(q)) equal to (O_sigma) off the
seam in that tile, equal to the identity near all tile facets, and
homotopic to the identity. Its fixed-order chart derivatives and
-(|dmathcal RcalR|_k) are elementarily bounded under the same
+(|dR|_k) are elementarily bounded under the same
hypotheses.
@@ -5617,7 +5617,7 @@ For each nonempty proper subset
(Ssubset{0,...,m}) and each face with exactly those vertex
colors, pair the incident positive chambers with the incident negative
chambers. The two counts agree. Indeed choose a color omitted by
-(S); the given codimension-one matching in (mathcal PcalP) across
+(S); the given codimension-one matching in (P across
facets omitting this color pairs the chambers at that face with opposite
signs. All incident sets are
finite by the incidence bound above. Choose uniformly among their
@@ -5698,7 +5698,7 @@ centering
node[below right] at (v2) {(2)};
node at (-2,-0.12) {(sigma)};
node[font=footnotesize,align=center] at (-2,-1.42)
- {one barycentricchamber (sigmasimeqDelta^2)};
+ {one barycentricchamber (sigma~=Delta^2)};
draw[-{Stealth[length=2mm]},thick] (-0.25,0.12)--(-0.90,0.12)
node[midway,above] {(X)};
[shift={(1,0)}]
@@ -5749,7 +5749,7 @@ centering
caption{The reflected resolution in dimension two, after Gaifullin's
permutohedral construction [cite]. (a) A barycentric
-chamber (sigmasimeqDelta^2) and its resolved tile. The six facets
+chamber (sigma~=Delta^2) and its resolved tile. The six facets
correspond to the nonempty proper subsets of ({0,1,2}), nested when
the labels are adjacent, and (X) maps the tile interior
diffeomorphically onto the chamber interior. (b) At a corner with
@@ -5840,7 +5840,7 @@ through the gluing. Its denominator is at least (e^{-(m+1)}), since
some (x_igeq1/(m+1)), which also gives elementary upper derivative
bounds in the tile charts.
-On the positive simplex the map (xmapsto b) is a diffeomorphism.
+On the positive simplex the map (x|-> b) is a diffeomorphism.
Given positive (b_i) of sum one, its inverse is obtained from the
unique (tin(0,1/max_i b_i)) satisfying
[
@@ -5857,12 +5857,12 @@ chamber's state probability integrates to one and the tile boundaries
have leafwise measure zero. Fubini and change of variables therefore
give, for every compactly supported smooth (m)-form (alpha),
[
- int_{mathcal A}calAX^*alpha
+ int_{A}*alpha
=sum_sigmaε_sigmaint_sigmaalpha.
]
This proves (eq:transfer-push). To obtain the local identity,
-define (mathcal A_calA_{rm orig}) as the image in
-(X^{-1}(mathcal T_calT_delta)) of tile points belonging to original
+define (A_{rm orig}) as the image in
+(X^{-1}(T_ of tile points belonging to original
chambers. At an interior point this subset is leaf-open. At a lower
face, the incidence argument preceding the lemma shows that every state
in its finite reflection star is on the same side. A neighborhood in
@@ -5871,7 +5871,7 @@ avoid nonincident facets, so it stays on that side. This proves
leaf-openness and makes the definition unambiguous at glued points;
the finite indexed charts also make it Borel. Applying the same
change-of-variables calculation only to original tile interiors, for
-a form supported in (mathcal T_calT_delta), gives
+a form supported in (T_, gives
(eq:transfer-local-push): each original cube is counted with
its coefficient (D_0u'_C), and each chamber's state fiber has mass one.
@@ -5960,7 +5960,7 @@ two-dimensional angle blocks with principal angles and, on its
even dimension there follows from determinant one. Define on that
tile
[
- mathcal R=calR=exp((1-j)A_sigma).
+ R=-j)A_sigma).
]
It is the identity in a neighborhood of each facet and thus glues
smoothly despite the change of (sigma). It equals (O_sigma)
@@ -5983,8 +5983,8 @@ deterministically. For those choices the bounds in
Lemma~lem:transfer-resolution give fixed elementary numerical
envelopes (K_R(q)) and (K_{rm scal}(q)) such that
[
- |dmathcal RcalR|_{∞,k}leq K_R(q),
- (-Scal_k)_+leq K_{rm scal}(q)
+ |dR|_{∞,k}<= K_R(q),
+ (-Scal_k)_+<= K_{rm scal}(q)
]
for every allowed diagram and every (V,s,omega), rounding parameter,
and state. These envelopes come from the uniform chart derivative
@@ -5998,17 +5998,17 @@ In particular
(3tausqrt qleqmin{r/100,delta/100,(1+K_R(q))^{-1}}).
Fix (V,s) and the rounding parameter for the moment. Take the
-product of (widetilde M) and the tile family (mathcal AcalA) over
+product of (widetilde M) and the tile family (A over
their shared (omega), and divide by the diagonal (Gamma)-action.
Orient the product with the (widetilde M) directions first and the
-(mathcal AcalA) directions second. The resulting measured leaves have
+(A directions second. The resulting measured leaves have
dimension (n+m=q). Using the leaf-open set
-(mathcal A_calA_{rm orig}) from Lemma~lem:transfer-resolution,
+(A_{rm orig}) from Lemma~lem:transfer-resolution,
work only on the open set of pairs ((x,a)) with
-(ainmathcal A_calA_{rm orig}) for which (X(a)) is in the lift of the ball chart of
+(ainA_{rm orig}) for which (X(a)) is in the lift of the ball chart of
(F(x)), at distance less than (delta/4). Define there
- G(x,a)=mathcal R(a)(z_F(x)-z_calR(a)(z_F(x)-z_{X(a)}).
+ G(x,a)=R(a)(z_F(x)-z_calR(a)(z_F(x)-z_{X(a)}).
Here (z_{X(a)}=zcirc X(a)). The requirement on the lifts is part of
the domain; coincident color coordinates on incompatible sheets are
@@ -6016,7 +6016,7 @@ never compared. The map is invariant under the diagonal action.
Use the band ({|G|_∞<3tau}) in that open domain, and put
(b_q=tau(1+3/(4q))), the inner width in Theorem~thm:degree
-when (d=q). Let (mathcal DcalD) be its measured degree.
+when (d=q). Let (D be its measured degree.
[Comparison degree]
The band in (eq:transfer-comparison) satisfies the chart and
@@ -6025,7 +6025,7 @@ fixed instance, the product metric (h+L^2k) with any fixed (L>0),
and the pullback of the entropy weight (phi). Its degree is Borel
in the auxiliary parameters and satisfies
- E_{rm rd}mathcal D=D_0IcalD=D_0I
+ E_{rm rd}D=D_0I
for every fixed (V,s).
@@ -6041,7 +6041,7 @@ band control that makes these integrations valid.
The controlled comparison band.
Before taking the quotient, the product flow boxes use the entropy
transversal (omega) and the
-state transversals of (mathcal AcalA). Variation along the tile keeps
+state transversals of (A Variation along the tile keeps
(omega) constant. These transversals carry the invariant measures
constructed above. A group element that stabilizes a product leaf
acts freely and properly because it acts so on the (widetilde M)
@@ -6051,7 +6051,7 @@ leave the color and flag coordinates unchanged, and preserve the
transverse measures and orientations.
For an inner closed comparison cube there is a uniform margin from
-the boundary of the allowed sheet domain. Indeed (mathcal RcalR) is
+the boundary of the allowed sheet domain. Indeed (R is
orthogonal and (eq:transfer-tau) gives
(|z_F-z_X|_2leq3tausqrt qlldelta/4). The large witness
color at (F) remains present with a uniform small buffer. In a
@@ -6082,10 +6082,10 @@ instance. It also justifies Fubini and measured Stokes for the
integrations that follow.
medskipnoindentThe local convolution form.
-Choose a smooth unit-integral top form (beta) on (mathbb R^qbbR^q)
+Choose a smooth unit-integral top form (beta) on (R^q
supported in a Euclidean ball about zero of radius less than
(min{tau,r/10}). In calculating the degree using (beta),
-homotope (mathcal RcalR) to the identity as in
+homotope (R to the identity as in
Lemma~lem:transfer-resolution. All maps in this homotopy are
orthogonal on the difference variable. Their pullbacks of (beta)
are thus supported where the difference has this same small norm,
@@ -6094,16 +6094,16 @@ Stokes for this supported homotopy shows that it does not change the
degree.
After removing the rotation, fiber integration of
-((z,w)mapsto z-w) against (beta) and the original current
+((z,w)|-> z-w) against (beta) and the original current
(Z_{rm rd}) gives a smooth (n)-form (alpha_{rm rd}) near (K).
-More explicitly, in a flat chart put (mathsf d(z,w)=z-wsfd(z,w)=z-w). Integrate
-(mathsf d^*sfd^*beta) along each compatible (C^#) in the second
+More explicitly, in a flat chart put (d(z,w)=z-wsfd(z,w)=z-w). Integrate
+(d^*sfd^*beta) along each compatible (C^#) in the second
variable, multiply by (D_0u'_C), and sum. The sum is locally finite
and uses the original side. By the local identity
(eq:transfer-local-push), Fubini, and change of variables on tile
interiors, the degree is
- mathcal D=calD=int_{mathcal M}calMF^*alpha_{rm rd}.
+ D={M}*alpha_{rm rd}.
Boundary cancellation of the dual cubes shows that
(dalpha_{rm rd}=0) on, for example, the tube of radius
@@ -6127,9 +6127,9 @@ coordinate gives the same separation. Such pairs cannot meet the
support of (beta). For the matching pair, the difference map
from (Ctimes C^#) contains the support ball in its image with
multiplicity one. Our fiber integration convention is
-(int_Cint_{C^#}mathsf d^*sfd^*beta=
-int_{Ctimes C^#}mathsf d^*sfd^*beta) with the base-first product
-orientation. The differential of (mathsf d(z,w)=z-wsfd(z,w)=z-w) sends this
+(int_Cint_{C^#}d^*sfd^*beta=
+int_{Ctimes C^#}d^*sfd^*beta) with the base-first product
+orientation. The differential of (d(z,w)=z-wsfd(z,w)=z-w) sends this
ordered frame to the primal basis followed by the negative dual basis,
so its orientation is positive by the choice of the dual orientation.
Its integral is therefore
@@ -6153,12 +6153,12 @@ function to its linear interpolation between the endpoints and
project a one-form (f(t)dt) to the constant one-form with the same
integral. The homotopy for a one-form is the function
[
- mathsf h(f(t)dt)(t)=sfh(f(t)dt)(t)=int_a^t
+ h(f(t)dt)(t)=sfh(f(t)dt)(t)=int_a^t
(f(v)-r^{-1}int_a^{a+r}f(s),ds)dv,
]
which vanishes at both endpoints; the homotopy on functions is zero.
-Direct differentiation gives (id-mathsf PsfP
-=dmathsf h+sfh+mathsf hsfh d). Apply these operations one coordinate
+Direct differentiation gives (id-P
+=dh+sfh+hsfh d). Apply these operations one coordinate
at a time, with the ordinary exterior differential signs. The
tensor-product homotopy still commutes with tangential restriction
to a face, since the one-dimensional primitive vanishes at the
@@ -6181,7 +6181,7 @@ form (alpha), and converge to (alpha) there. The construction
is Borel and equivariant and has bounded coefficients for the instance.
Measured Stokes applied to the pullback of each smooth primitive gives
zero; convergence and the uniform chart bounds give
-(int_{mathcal M}calMF^*alpha=0). All expectations and integrals
+(int_{M}*alpha=0). All expectations and integrals
above are measurable by their locally finite chart formulas, and
their absolute bounds for the instance justify their interchange.
Using (eq:transfer-form) in
@@ -6223,28 +6223,28 @@ By the definition of (K_{rm scal}(q)), the lower bound
(Egeq b_*^{-2}K_{rm scal}(q)) ensures
(L_T^{-2}Scal_kgeq-1). By (eq:transfer-entropy),
- wscal(h_{rm tot},phi)geq-C_{rm ent}-1+2W-nlog E+Emathfrak e.frake.
+ wscal(h_{rm tot},phi)>=-C_{rm ent}-1+2W-nlog E+Emathfrak e.frake.
Assign (L_i=L_T) to the first (m) comparison rows and
(L_i=L_N) to the last (n) rows. On the band, the derivative in
tile directions of each row of (G), measured with (k), has norm
at most two. The (dX) term has norm at most one, and the
-(dmathcal RcalR) term has norm at most
-(3tausqrt q|dmathcal RcalR|_kleq1) by
+(dR term has norm at most
+(3tausqrt q|dR|_kleq1) by
(eq:transfer-tau). The horizontal derivative is a row of a
rotation of (dz_F). It follows that
[
L_i|dG_i|_{h_{rm tot}}
- leq C_n(sqrt q+√(qE),E^{-1/100}).
+ <= C_n(sqrt q+√(qE),E^{-1/100}).
]
Take (B) to be this upper bound increased by two. Then
-(L_i|dG_i|leq B-1=B-q/q). The horizon condition of
+(L_i|dG_i|<= B-1=B-q/q). The horizon condition of
Theorem~thm:degree holds after an elementary increase of (E):
it suffices to test the smallest length (L_T), and, for (Egeq1),
[
frac{C(1+B)^{2/3}}{L_T}
- leq c_{{rm hor},n}q^{1/3}E^{-13/75}
+ <= c_{{rm hor},n}q^{1/3}E^{-13/75}
<(tau)/(10q).
]
Here (c_{{rm hor},n}) depends only on the absolute horizon constant
@@ -6256,52 +6256,52 @@ Therefore apply Theorem~thm:degree
with (d=q).
Write (H_i=2(L_i|dG_i|-1)_+), as in that theorem. At a straight
-point in a tile, let (mathcal IcalI) be the (n)-set of grid axes
+point in a tile, let (I be the (n)-set of grid axes
normal to its dual cube. Put (A_i=(Vdz_F)_i), with covector norms
measured by (h), and define
- U_{mathcal I}calI^2=
- frac q{mathfrak e}frakesum_{iinmathcal I}calI|A_i|_h^2
+ U_{I}
+ frac q{mathfrak e}frakesum_{iinI}|A_i|_h^2
quadif mathfrak e>0,frake>0,
- U_{mathcal I}calI=0quadif mathfrak e=0.frake=0.
+ U_{I} mathfrak e=0.frake=0.
Here (sum_i|A_i|^2=mathfrak efrake). At a straight point
-(k=X^*k_{rm flat}) and (mathcal R=Q_calR=Q_sigma V) is constant.
+(k=X^*k_{rm flat}) and (R=Q_ V) is constant.
For a tangent comparison row corresponding to axis (i), the squared
row norm is exactly (L_T^{-2}+|A_i|^2). Thus
[
sum_{rm tangent}H_i
- leq L_T^2mathfrak e=b_*^2Efrake=b_*^2Emathfrak e,frake,
+ <= L_T^2mathfrak e=b_*^2Efrake=b_*^2Emathfrak e,frake,
]
-by (2(√(1+t)-1)leq t). Normal rows have no tile derivative,
+by (2(√(1+t)-1)<= t). Normal rows have no tile derivative,
so Cauchy--Schwarz gives
[
sum_{rm normal}H_i
- leq2L_Nsum_{iinmathcal I}calI|A_i|
- leq2b_*sqrt n√(Emathfrak e)frake),U_{mathcal I}calI.
+ leq2L_Nsum_{iinI}|A_i|
+ leq2b_*sqrt n√(Emathfrak e)frake),U_{I}
]
Completing the square in (√(Emathfrak e)frake)), for (b_*<1),
shows that
(sum_iH_i-Emathfrak e)/(2)frake)/(2)
- leq c_n U_{mathcal I}calI^2,
+ <= c_n U_{I},
c_n=(n b_*^2)/(2(1-b_*^2)).
Choose (b_*) sufficiently small that
- E_Vexp(2c_nU_{mathcal I}calI^2)leq C_n
+ E_Vexp(2c_nU_{I}<= C_n
for every fixed horizontal differential and every fixed
-(mathcal IcalI). Here is a direct verification of this choice. If
+(I Here is a direct verification of this choice. If
(u) is one coordinate of a uniform point on (S^{q-1}), its sphere
moments satisfy
[
E(q u^2)^j
- =(q^j(1cdot3...(2j-1)))/(q(q+2)...(q+2j-2))
+ =(q^j(1·3...(2j-1)))/(q(q+2)...(q+2j-2))
leq2^j j!.
]
-Consequently (E e^{tqu^2}leq(1-2t)^{-1}) for (0leq t<1/2).
+Consequently (E e^{tqu^2}<=(1-2t)^{-1}) for (0leq t<1/2).
The singular-value decomposition of (dz_F) expresses each
(q|A_i|^2/mathfrak efrake) as a convex combination of such random
variables, with its singular-value weights. Jensen's inequality
@@ -6314,7 +6314,7 @@ is used.
At a seam point the uniform tile row bound gives a rougher estimate
which is still independent of the instance. By the triangle inequality
-for each row, ((a-1)_+leq a), and Cauchy--Schwarz on the horizontal
+for each row, ((a-1)_+<= a), and Cauchy--Schwarz on the horizontal
rows,
[
sum_iH_ileq 2L_N√(qmathfrak e)frake)
@@ -6323,7 +6323,7 @@ rows,
]
Completing the square once more gives
- (sum_iH_i-Emathfrak e)/(2)frake)/(2)leq c_{{rm seam},n}q^2,
+ (sum_iH_i-Emathfrak e)/(2)frake)/(2)<= c_{{rm seam},n}q^2,
c_{{rm seam},n}=2+(b_*^2)/(2).
@@ -6332,25 +6332,25 @@ Apply the degree inequality, then use
absolute constant. The product volume form is
(dV_{h_{rm tot}}=L_T^m dV_h,dV_k), so the measure factor is
exactly (eq:transfer-prefactor). Together with
-(eq:transfer-degree) and (|Emathcal D|calD|leqE|mathcal D|calD|),
+(eq:transfer-degree) and (|ED|calD|leqE|D|calD|),
this gives the following inequality, with
(C_{rm out}=exp(C_{rm deg}+(C_{rm ent}+1)/2)b_*^{-n}), where
(C_{rm deg}) is the absolute exponent in Theorem~thm:degree:
D_0I(2tau)^qleq C_{rm out}q^{-n/2}
- int_{mathcal M}calMrho,
+ int_{M},
E_{V,s,rm rd}
int_{{a: |G(x,a)|_∞<b_q}}
- mathcal B(x,a)calB(x,a),dV_k(a),
+ B(x,a)calB(x,a),dV_k(a),
where the fiber integral includes its state transversals and only the
compatible original-side band, and
[
- mathcal B(x,a)=calB(x,a)=
+ B(x,a)=calB(x,a)=
- exp(c_nU_{mathcal I}calI^2),&a straight over axes mathcal I,calI,
+ exp(c_nU_{I},&a straight over axes I,
exp(c_{{rm seam},n}q^2),&a in the seam.
]
@@ -6369,8 +6369,8 @@ within (b_qsqrt q) of (y), and the dual and primal extents are
at most (rsqrt m/2) and (rsqrt n/2). The variation choice
for (r) and the rounding error now give the copy-count estimate
- D_0|u'_C|leq D_0r^n
- (|eta_{mathcal I}calI(y)|+2q^{-n}),
+ D_0|u'_C|<= D_0r^n
+ (|eta_{I}|+2q^{-n}),
where the component is in the (V)-rotated frame.
@@ -6380,10 +6380,10 @@ change its cube test, which is
(|widehatwidehat z_y-z_y-widehatwidehat z_X|_∞<b_qz_X|_∞<b_q). Summing the chamber
pieces in a copy is thus bounded by the (m)-area of its whole dual
cube inside that test; the state probabilities for each chamber
-integrate to one. For a fixed axis set (mathcal IcalI), summing over
+integrate to one. For a fixed axis set (I, summing over
all projected centers and averaging the shift gives exactly
- r^{-q}int_{mathbb R^q}bbR^qint_{[-r/2,r/2]^m}
+ r^{-q}int_{R^q}{[-r/2,r/2]^m}
indic_{{|widehatwidehat z_y-c-t|_∞<b_qz_y-c-t|_∞<b_q}},dt,dc
=r^{-q}r^m(2b_q)^q.
@@ -6396,13 +6396,13 @@ The form bound holds in every rotated frame. Using Cauchy--Schwarz
jointly in (V) and the (binom qn) axis sets, and then
(eq:transfer-sphere), one obtains
- E_Vsum_{|mathcal I|=n}calI|=n
- (|eta_{mathcal I}calI(y)|+2q^{-n})e^{c_nU_{mathcal I}calI^2}
- &leq
- (E_Vsum_{mathcal I}calI|eta_{mathcal I}calI(y)|^2)^{1/2}
- (E_Vsum_{mathcal I}calIe^{2c_nU_{mathcal I}calI^2})^{1/2}
+ E_Vsum_{|I|=n}calI|=n
+ (|eta_{I}|+2q^{-n})e^{c_nU_{I}}
+ &<=
+ (E_Vsum_{I}|eta_{I}|^2)^{1/2}
+ (E_Vsum_{I}{2c_nU_{I}})^{1/2}
+C_nq^{-n}binom qn
- &leq C_nq^{n/2}.
+ &<= C_nq^{n/2}.
This use of Cauchy--Schwarz allows the form components and row norms
@@ -6434,31 +6434,31 @@ Retain the compatible-sheet restriction at this point. For a fixed
projected grid index the uniqueness proved before
(eq:transfer-copies) allows at most one compatible geometric
lift. Apply the rough copy bound
-(D_0|u'_C|leq C_{eta,n}D_0r^n) from that equation, where
+(D_0|u'_C|<= C_{eta,n}D_0r^n) from that equation, where
(C_{eta,n}) depends only on the form norm, and use the uniform
density bound. Only after these bounds may the restrictions be dropped
on available sheets, roundings, and chambers and sum over all projected
centers and chamber shapes.
-For a fixed actual chamber shape, normal axis set (mathcal IcalI), and
+For a fixed actual chamber shape, normal axis set (I, and
positive sorted flag, (eq:transfer-tile-map) has rotated position
[
- widehatwidehat z_X=s+r(k+z_X=s+r(k+tfrac12indic_{mathcal I}calI)
- +d_{mathcal I,shape,flag}calI,shape,flag(Z).
+ widehatwidehat z_X=s+r(k+z_X=s+r(k+tfrac12indic_{I}
+ +d_{I,shape,flag}calI,shape,flag(Z).
]
-Here (indic_{mathcal I}calI) is the coordinate indicator vector of
-(mathcal IcalI).
+Here (indic_{I} is the coordinate indicator vector of
+(I
The displacement (d) is independent of the pairing and bit arrays,
of (lambda,gamma_0), and of the shift; it is determined by the
fixed chamber shape and the barycentric formula. For each fixed (Z)
the shift integral of the rough test is consequently
[
- frac1{r^q}int_{[0,r)^q}sum_{kinmathbb Z^q}bbZ^q
- indic_{{|s+r(k+frac12indic_{mathcal I}calI)+d(Z)-widehatwidehat z_y|_2z_y|_2
- leq b_qsqrt q}},ds
- &=r^{-q}Vol_{mathbb R^q}bbR^q(B_{b_qsqrt q})
- &leq r^{-q}(4tausqrt q)^q.
+ frac1{r^q}int_{[0,r)^q}sum_{kinZ^q}
+ indic_{{|s+r(k+frac12indic_{I}+d(Z)-widehatwidehat z_y|_2z_y|_2
+ <= b_qsqrt q}},ds
+ &=r^{-q}Vol_{R^q}{b_qsqrt q})
+ &<= r^{-q}(4tausqrt q)^q.
]
This calculation is valid despite the shift dependence of the actual
@@ -6493,7 +6493,7 @@ using (q^{-n/2}leq1) for the seam, and cancel the positive factor
so far, this gives
Ileq C_{rm out}(C_{{rm str},n}+C_*(q)lambda)
- int_{mathcal M}calMrho.
+ int_{M}
Now fix these bounds before invoking the entropy theorem. For each
@@ -6524,7 +6524,7 @@ For integer (jgeq n+1), define
]
and define a majorant on all (tin[1,∞)) by
[
- Psi_n(t)=1+max_{substack{jinmathbb ZbbZ
+ Psi_n(t)=1+max_{substack{jinZ
n+1leq jleqmax{n+1,lceil trceil}}}
Psi_n^circ(j).
]
@@ -6544,7 +6544,7 @@ and strict horizon requirements. Hence every conditional datum with
This closes the conditional target for all the specified data before
any entropy family is chosen.
-Now apply Theorem~thm:entropy with (mathcal B=calB=Psi_n),
+Now apply Theorem~thm:entropy with (B=,
(E_{min}=1), the detector root map as (F^{in}), and the
one-point transverse space (Ω_0). The detector construction uses
finitely many compactly supported bump representatives and their deck
@@ -6558,8 +6558,8 @@ data with (E>Psi_n(q)). Its constant
(rholeq dV_g) and the transverse measure over (M) is a
probability measure,
[
- Ileq K_ne^{C_{rm ent}/2}int_{mathcal M}calMrho
- leq K_ne^{C_{rm ent}/2}Vol_g(M).
+ Ileq K_ne^{C_{rm ent}/2}int_{M}
+ <= K_ne^{C_{rm ent}/2}Vol_g(M).
]
Finally (Igeq|M|/2), so the constant
(C_n=2K_ne^{C_{rm ent}/2}) proves the theorem.
@@ -6570,7 +6570,7 @@ Fix (ngeq3), and let (C_n>0) be the constant for the normalized
scalar-curvature estimate in Theorem~thm:scalar-volume.
Proposition~prop:yamabe-reduction gives
[
- |M|leq C_nint_M(Scal_g^-)^{n/2},dV_g
+ |M|<= C_nint_M(Scal_g^-)^{n/2},dV_g
]
for every metric in the theorem. Taking (a_n=C_n^{-1}>0) proves
the stated integral bound.