diff f6bf760..6fbd3ac simplicial_volume_scalar_bound_v2.txt
diff --git a/simplicial_volume_scalar_bound_v2.txt b/simplicial_volume_scalar_bound_v2.txt
index 0bd5a75..6fe780f 100644
--- a/simplicial_volume_scalar_bound_v2.txt
+++ b/simplicial_volume_scalar_bound_v2.txt
@@ -168,11 +168,11 @@ and the last stage composes them.
In every conformal class the Yamabe minimizer has constant scalar
curvature. If |M|>0, the vanishing theorem forces that constant to
be negative, and a constant rescaling normalizes it to -1.
- H"older's inequality then bounds the volume of the normalized metric
+ H"older's ineuality then bounds the volume of the normalized metric
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) whenever Scal_gequiv-1,
+ |M|le C_nVol_g(M) whenever Scal_geuiv-1,
which is Theorem~thm:scalar-volume.
@@ -230,7 +230,7 @@ and the last stage composes them.
The last step contains the only place where the exponent of E matters:
the tangent directions of the comparison family contribute the volume
L_T^m, the normal directions contribute the reciprocal
-L_T^{-m}L_N^{-n} of the degree denominator, and the entropy inequality
+L_T^{-m}L_N^{-n} of the degree denominator, and the entropy ineuality
contributes E^{n/2}e^{-W}; the three factors multiply to
b_*^{-n}q^{-n/2}rho, in which no power of E appears.
@@ -295,7 +295,7 @@ In particular, the hypothesis permits scalar curvature to vanish
identically. Theorem~thm:vanishing-input is the only external result
invoked below without proof.
-The Yamabe reformulation of the scalar curvature inequality is
+The Yamabe reformulation of the scalar curvature ineuality is
standard; cite[Lemma 2.1]{MinZhengZhu2026} records the equivalence of
its uniform pointwise and integral forms. Include the short
classwise reduction that identifies the use of the vanishing input.
@@ -339,7 +339,7 @@ constant function for this minimizing metric gives
[
-Y(M,[g])=Vol_h(M)^{2/n}.
]
-For each positive (u), H"older's inequality gives
+For each positive (u), H"older's ineuality gives
[
int_MScal_g u^2,dV_g
ge
@@ -467,7 +467,7 @@ Write (ε(v)=sum_ell v_ell) for the mass of a finitely
supported vector. Multilinear extension of the cocycle equation gives
[
0=sum_{j=0}^{n+1}(-1)^jε(w_j)
- c(w_0,...,widehatwidehat w_j,...,w_w_j,...,w_{n+1}).
+ c(w_0,...,what_j,...,w_w_j,...,w_{n+1}).
]
Choose (w_0) of mass one and let (w_1,...,w_{n+1}) have mass
zero. It follows that (c(w_1,...,w_{n+1})=0). All tangent vectors
@@ -790,7 +790,7 @@ Consequently
At (A=0), (t=0) and
((X_{∥}-t)^2+N^2|II^{rem}|^2=N^2|II|^2); every other
-direction-dependent term vanishes. Thus the inequality is independent
+direction-dependent term vanishes. Thus the ineuality is independent
of the arbitrary direction at such points.
This calculation is algebraic and assumes no equation for (D).
@@ -1109,7 +1109,7 @@ support (K), and let (alphage0) be a uniformly smooth
function. Fix
[
ε>0, N^2=lε,
- a=tfrac12+theta, 0<lle1, llethetaletfrac1{16}.
+ a=1/2+theta, 0<lle1, llethetale1/{16}.
]
There is a radius (R_{loc}), depending only on the
dimension and a fixed sufficiently small numerical constant
@@ -1207,7 +1207,7 @@ is at most
For example one can use the primitive of the reciprocal of a smooth
positive function comparable to
(thetaε^{3/2}+t^{3/2}) as the new profile coordinate,
-and compose with fixed smooth flattenings. The last inequality
+and compose with fixed smooth flattenings. The last ineuality
uses (llethetale1/16).
One may use this profile in a smooth approximation of
@@ -1230,7 +1230,7 @@ one may subtract the distance at the center, leaving a uniformly
bounded oscillation on its support. These derivative bounds hold in
the common buffered charts, since the (h_{ref}) exponential
maps and kernels have uniformly bounded derivatives there. The inverse metric
-inequality (h^{-1}le h_{ref}^{-1}) then bounds the
+ineuality (h^{-1}le h_{ref}^{-1}) then bounds the
gradient of the composed profile in (h).
Choose this raw profile (b) to vanish near (K), to equal
@@ -1298,7 +1298,7 @@ superpolynomial decay uniformly as (H_*) grows.
There is first a global angle bound
- vgefrac{c_1}{1+√(H_*)}
+ vgec_1/1+√(H_*)
for the uncut graph. In fact, at a recentered infimum with
@@ -1390,7 +1390,7 @@ Also (p_v=-v/(Nwidetilde f)) and
(|T(v)|le Cv/xi), so that right side is at most
[
-(c)/(widetilde f^2)
- +frac{C√(H_*),v}{widetilde f}.
+ +C√(H_*),v/widetilde f.
]
Combining the two bounds and multiplying by (widetilde f^2)
would give
@@ -1543,8 +1543,8 @@ graphs before asserting that the sequence can be constructed. The
pass parameters are
ε>0, N^2=lε,
- a=tfrac12+theta, lambda=tfrac12-2theta,
- 0<lle1, llethetaletfrac1{16},
+ a=1/2+theta, lambda=1/2-2theta,
+ 0<lle1, llethetale1/{16},
with (N>0). On one leaf, start with ((h_0,phi_0,F)), where
@@ -1800,7 +1800,7 @@ adds exactly
Since (L_ige1), this is at least
- B_i:=calB_i:=
+ B_i:=B_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+B_i]calB_i]
+ +ε N^2|dF_i|_{h_0}^2+B_i]B_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_iB_i-calB_i-delta_0).
+ +frac rε(sum_iB_i-B_i-delta_0).
Indeed (rF_i^2/f_i^2=sigma_i). The terms with (B_i) and
@@ -1907,12 +1907,12 @@ Completing its square therefore gives the identity
&+sum_i A_i^2[(2a-4a^2)sigma_i+2ar-1] notag
&=
ε(M_0^2+E_nu V_i^2+
- (tfrac14-lambda^2)Var_nu u)
+ (1/4-lambda^2)Var_nu u)
-4theta^2m .
Here
-(tfrac14-lambda^2=2theta-4theta^2ge7theta/4).
+(1/4-lambda^2=2theta-4theta^2ge7theta/4).
This supplies a reserve for all three squares in the
proposition. Finish by paying the terms in
(B_i from fixed fractions of this reserve and of
@@ -1975,7 +1975,7 @@ Cauchy--Schwarz applied to
For any fixed small (zeta>0), the derivative bound in
-(eq:sweep-boost-profile) and Young's inequality give
+(eq:sweep-boost-profile) and Young's ineuality give
[
2NA_if_i|db_i|
le4ε A_if_i√(R_i),|psi'(R_i)|
@@ -1986,7 +1986,7 @@ The first term is a small fraction of (beta_iA_i^2).
For the other costs, use
[
sum_i f_i^2R_i
- =tfrac12(r^2-sum_i f_i^4)le r^2/2,
+ =1/2(r^2-sum_i f_i^4)le r^2/2,
sum_ipsi(R_i)F_i^2le Ckappa r^2sum_iF_i^2=Ckappa r^2.
]
@@ -2055,7 +2055,7 @@ 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.
-This proves (eq:sweep-inequality).
+This proves (eq:sweep-ineuality).
Finally, the choices of errors impose no estimate on the
growth of analytic constants from one color to the next.
@@ -2068,12 +2068,12 @@ value error at most (eta_0/k) per color. Later choices
of (H_*) may be arbitrarily larger. The summed errors
are therefore precisely those asserted in
(eq:sweep-prescription-error) and
-(eq:sweep-inequality).
+(eq:sweep-ineuality).
This section turns a root probability map for a metric of scalar curvature
(-1) into a map with a large energy coefficient in a weighted scalar
-inequality. The accompanying volume term is the logarithm of the actual
+ineuality. The accompanying volume term is the logarithm of the actual
volume expansion from a density no larger than the original volume density.
Also make the gradient absolutely small and control all label
overlaps by one graph. The last two properties allow the energy coefficient
@@ -2081,16 +2081,16 @@ 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
-(B:[1,∞)calB:[1,∞)to[0,∞)) elementary if, for some fixed
+(B:[1,∞)B:[1,∞)to[0,∞)) elementary if, for some fixed
(HinNcup{0}) and (A,B>0),
[
- B(q)<= exp^{[H]}(B(1+q)^A)(qgeq1).
+ B(q)<= exp^{[H]}(B(1+q)^A)(qge1).
]
When the function is prescribed in a fixed dimension, its parameters
(H,A,B) are allowed to depend on that dimension.
[Entropy metrics]
-Let ((M^n,g)) be a closed connected Riemannian manifold, where (ngeq3)
+Let ((M^n,g)) be a closed connected Riemannian manifold, where (nge3)
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.
@@ -2099,7 +2099,7 @@ Suppose (L_0 is a countable set with finitely many free
[
F^{in}:widetilde MtimesΩ_0→
ell^2(L_0),
- F^{in}_ellgeq0,
+ F^{in}_ellge0,
sum_{ellinL_0}{in}_ell)^2=1
]
is equivariant, Borel in the transverse variable, and smooth along
@@ -2121,12 +2121,12 @@ 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 (L There are also (E>E_{min}), an
-integer (qgeq n+1), and a constant (C=C(n,B)<∞calB)<∞) such
+integer (qge n+1), and a constant (C=C(n,B)<∞B)<∞) such
that, on an invariant conull set of transverse parameters,
- &rholeq dV_g,
- W:=log(dV_h)/(rho)geq0,
+ &rhole dV_g,
+ W:=log(dV_h)/(rho)ge0,
&wscal(h,phi)>=
-C+2(W-frac n2log E)+E e_h(F),
e_h(F)^{1/2}<= E^{-1/100},
@@ -2151,7 +2151,7 @@ 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,#(GammabackslashL),Delta+1}
+ qge max{n+1,#(GammabackslashL),Delta+1}
]
and a Borel equivariant coloring
(chi:LΩto{1,...,q}) such that labels of
@@ -2166,8 +2166,8 @@ will fix the density (rho), which then remains fixed through all later
sweeps and conformal changes. For
(W=log(dV_h/rho)), the balance propagated after the bootstrap is
[
- wscal(h,phi)-ε e_h(F)-2W+nlogεgeq-C+P,
- Wgeq0, Pgeq0.
+ wscal(h,phi)-ε e_h(F)-2W+nlogεge-C+P,
+ Wge0, Pge0.
]
First raise an initially small energy coefficient to a dimensional
positive value. Subsequent sweeps have a summable scalar loss and
@@ -2227,8 +2227,8 @@ Proposition~prop:sweep. In particular, its input is
((h_0,phi_0,F)), its output is ((h_+,phi_+,Y)), and
[
- N^2=lε, a=tfrac12+theta,
- lambda=tfrac12-2theta,
+ N^2=lε, a=1/2+theta,
+ lambda=1/2-2theta,
r=sum_i f_i^2>0, Y=f/sqrt r,
W_p=sum_ilog L_i, m=sum_i A_i^2.
@@ -2257,8 +2257,8 @@ The graph operations give
N^2 e_{h_+}(Y)<= frac mr.
The added graph tensors are nonnegative. Thus, as quadratic forms,
-(h_0leq h_{i-1}<= h_ileq h_+); the inverse forms reverse these
-inequalities. In particular, for every covector (alpha),
+(h_0le h_{i-1}<= h_ile h_+); the inverse forms reverse these
+ineualities. In particular, for every covector (alpha),
[
|alpha|_{h_+}<=|alpha|_{h_i}
<=|alpha|_{h_{i-1}}<=|alpha|_{h_0}.
@@ -2267,7 +2267,7 @@ The positive reserve (Pi_p) is defined in
(eq:sweep-positive-reserve), with coefficient exactly one on
(T_0). In a sweep without boost its final term is omitted; in a
boosted sweep its constants (c_*,C_*) may depend on the fixed lower
-threshold for (ε). The sweep inequality reads
+threshold for (ε). The sweep ineuality reads
Q&:=(rN^2)/(ε)
@@ -2314,16 +2314,16 @@ The same reset contracts the volume density:
=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
+that, whenever (2le C_0le4), (0<εleε_b), and
(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).
+ Scal_kge-(C_0+C_b lε)+(1+l)ε e_k(Y).
First derive the budget needed for this step. The reset gives
(e_k(Y)=e^{2W_p/(n-1)}e_{h_+}(Y)). Substituting the definition of
-(Q) and the input scalar inequality into
+(Q) and the input scalar ineuality into
(eq:entropy-bootstrap-reset), and dropping its nonnegative
gradient term, shows that it suffices to have
[
@@ -2335,21 +2335,21 @@ Multiply by (rN^2), and use
(N^2 e_{h_+}(Y)<= m/r) and (N^2=lε). A sufficient budget
for (eq:entropy-bootstrap-step) is therefore
- ε Qgeq
+ ε Qge
lε m+C_0lε r(1-e^{-2W_p/(n-1)})
-C_b l^2ε^2r e^{-2W_p/(n-1)}.
To pay this budget when (ε) is small, one needs control of both
-(m) and the deviation of (r) from one. The sweep inequality implies
+(m) and the deviation of (r) from one. The sweep ineuality implies
- ε Qgeq
+ ε Qge
c_n(ε^2(1-r)^2+m^2)
-C_nε l^2m-rdelta'_0 ,
where (delta'_0>0) can be chosen arbitrarily small. Here is a proof.
-Fix a sufficiently small dimensional (m_b>0). If (mgeq m_b) and
-(ε_b) is small enough, then (barbar u=m/εu=m/εgeq4a). Since
+Fix a sufficiently small dimensional (m_b>0). If (mge m_b) and
+(ε_b) is small enough, then (barbar u=m/εu=m/εge4a). Since
[
-M_0=ar+(barbar u)/(2)-a,u)/(2)-a,
]
@@ -2361,14 +2361,14 @@ Suppose next that (m<m_b). The prescription term (D_i^0) of
Proposition~prop:sweep satisfies
[
ND_i^0+t_i=(ε f_i)/(r)
- [V_i+M_0+(lambda+tfrac12)barbar u+u+
- (tfrac32-lambda)u_i].
+ [V_i+M_0+(lambda+1/2)barbar u+u+
+ (3/2-lambda)u_i].
]
Both displayed coefficients of (barbar uu) and (u_i) are positive and
bounded below. Moving (V_i+M_0) to the other side, taking positive
parts, and summing squares therefore yields
- sum_i t_i^2leq
+ sum_i t_i^2le
(Cε^2)/(r)(E V_i^2+M_0^2)
+Csum_i(ND_i^0+t_i)_+^2 .
@@ -2383,14 +2383,14 @@ longitudinal component of (II_i) is (-X_i/N), so
The terms ((X_i-t_i)^2), (N^2|II_i^{rem}|^2), and
(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}<= C A_i|II_i|).
+(L_ile 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
&<= C m^2sum_jN^2|II_j|^2
- &<= C' m^2(X+calX+sum_jt_j^2).
+ &<= C' m^2(X+X+sum_jt_j^2).
]
For example, the first line follows by bounding
@@ -2400,9 +2400,9 @@ For example, the first line follows by bounding
Choosing (m_b) small absorbs the last sum of (t_j^2) in
(eq:entropy-bootstrap-t). Since
[
- m^2=(sum_i f_i t_i)^2leq rsum_i t_i^2,
+ m^2=(sum_i f_i t_i)^2le rsum_i t_i^2,
- ε^2(1-r)^2leq C(ε^2M_0^2+m^2),
+ ε^2(1-r)^2le C(ε^2M_0^2+m^2),
]
fractions of the (M_0^2,E V_i^2,X terms in
(εPi_p) control both required squares. The arbitrarily
@@ -2412,27 +2412,27 @@ The term (-4theta^2m) in (eq:entropy-sweep-form) is
Now compare the right side of
(eq:entropy-bootstrap-coercivity) with the required budget.
-When (mgeq m_b), the first two terms on the right of
+When (mge m_b), the first two terms on the right of
(eq:entropy-bootstrap-budget) are at most
(lε(m+4r)). The squares
(m^2+ε^2(1-r)^2) dominate this expression with a fixed
fraction to spare after (l) is chosen small and then (ε_b)
-is decreased if needed. One may check this separately for (rleq2),
-where (mgeq m_b) controls the constant part, and for (r>2), where
-Young's inequality controls (ε r) by
+is decreased if needed. One may check this separately for (rle2),
+where (mge m_b) controls the constant part, and for (r>2), where
+Young's ineuality controls (ε r) by
(ε^2(r-1)^2) and a sufficiently small fixed part of (m^2).
The term (C_nε l^2m) is covered at the same time.
When (m<m_b), the identity
-(L_i=(1-A_i^2)^{-1/2}) gives (W_pleq C m).
-For (rnotin(1/2,2)), Young's inequality applied to
+(L_i=(1-A_i^2)^{-1/2}) gives (W_ple C m).
+For (rnotin(1/2,2)), Young's ineuality applied to
(lε m(1+r)) again gives the required domination by the two
squares, with (l) small. For (1/2<r<2), the positive terms on the
right of (eq:entropy-bootstrap-budget), together with the term
(C_nε l^2m) from the coercivity estimate, are at most
(C_nlε m). Here
[
- C_nlε mleq (c_n)/(2)m^2+C'_n l^2ε^2.
+ C_nlε mle (c_n)/(2)m^2+C'_n l^2ε^2.
]
On this region (r e^{-2W_p/(n-1)}) has a dimensional positive lower
bound. Therefore choose (C_b) dimensionally to cover the
@@ -2447,17 +2447,17 @@ lower bound for fixed (l,ε). Consequently choose
The input map has a finite uniform energy bound in (g). Thus for
some positive, possibly input-dependent (ε_{in})
small enough,
-(Scal_g=-1geq-2+ε_{in}e_g(F^{in})).
+(Scal_g=-1ge-2+ε_{in}e_g(F^{in})).
Choose a dimensional (ε_*>0) below (ε_b), small
enough that (2+2C_bε_*<4), and decrease
(ε_{in}) further so that
(0<ε_{in}<ε_*). Iterate
(eq:entropy-bootstrap-step) until the coefficient first reaches
(ε_*). The output coefficient is at most
-((1+l)ε_*leq2ε_*), and
+((1+l)ε_*le2ε_*), and
[
sum_{bootstrap sweeps}lε
- =ε_0-ε_{in}leq2ε_*.
+ =ε_0-ε_{in}le2ε_*.
]
This bounds the total scalar cost by (2C_bε_*), independently
of the number of sweeps. By (eq:entropy-bootstrap-density),
@@ -2478,11 +2478,11 @@ the reserve remains after paying the budget, and a region near
(m=0,r=1), where the exact quadratic coefficient will matter.
[Entropy increment]
-There are dimensional constants (m_0>0), (D_*geq1), (l_0>0),
+There are dimensional constants (m_0>0), (D_*ge1), (l_0>0),
(gamma_n>0), and (C_n<∞) with the following property.
-For (εgeqε_*), take a boosted sweep with
+For (εgeε_*), take a boosted sweep with
[
- 0<lleq l_0, theta=D_*lleqtfrac1{16},
+ 0<lle l_0, theta=D_*lle1/{16},
N^2=lε.
]
The errors can be chosen so that
@@ -2496,21 +2496,21 @@ The errors can be chosen so that
where the nonnegative, pointwise surplus is
P_p=gamma_n(ε)/(rN^2)Pi_p,
- indic_{{mgeq m_0},cup,{|1-r|geq1/4}}.
+ indic_{{mge m_0},cup,{|1-r|ge1/4}}.
In particular, on the indicated region (P_p) retains a fixed fraction
-of every positive term in the sweep inequality.
+of every positive term in the sweep ineuality.
Proof.
The output energy bound in (eq:entropy-sweep-density) shows that,
in the units of (Q), it suffices to prove
- Qgeq lm+2lrW_p-nrllog(1+l)-C_nr l^3
+ Qge lm+2lrW_p-nrllog(1+l)-C_nr l^3
+(rN^2)/(ε)P_p .
The term (lm) pays for the additional (lε e_{h_+}(Y)).
-Choose (delta_0leqε l^3) in
+Choose (delta_0leε l^3) in
(eq:entropy-sweep-form). It remains to compare the other terms
there with the budget in (eq:entropy-budget).
@@ -2525,7 +2525,7 @@ As matrices,
m_f=n-tr(h_+^{-1}h_0)
<= n(1-e^{-2W_p/n}).
-The inequality is arithmetic--geometric mean applied to the positive
+The ineuality is arithmetic--geometric mean applied to the positive
eigenvalues of (h_+^{-1}h_0), whose product is (e^{-2W_p}).
There also holds
@@ -2535,53 +2535,53 @@ There also holds
First choose a small dimensional (m_0>0), in particular
(m_0llε_*) and (m_0<1/4). Claim that
- T_0+ε M_0^2geq c_n(m+rW_p)
- quadif mgeq m_0
- or if m<m_0, |1-r|geqtfrac14.
+ T_0+ε M_0^2ge c_n(m+rW_p)
+ quadif mge m_0
+ or if m<m_0, |1-r|ge1/4.
-For (mgeq2n), the (C_i) terms of (T_0) give
-(T_0geq m-m_fgeq m-ngeq m/2).
-For (m_0leq mleq2n), first suppose that every (A_i^2) is at most
+For (mge2n), the (C_i) terms of (T_0) give
+(T_0ge m-m_fge m-nge m/2).
+For (m_0le mle2n), 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)<=(1+Ceta)m.
]
By (eq:entropy-trace),
[
- m-m_fgeq m-n(1-e^{-(1+Ceta)m/n})>= c_n m.
+ m-m_fge m-n(1-e^{-(1+Ceta)m/n})>= c_n m.
]
-For the last inequality, the same expression with (eta=0) divided
+For the last ineuality, the same expression with (eta=0) divided
by (m) has a positive minimum on ([m_0,2n]), and then (eta)
is chosen small. If instead some (A_i^2>eta), the self term of
that index in (T_0) is at least (c_n(r+u_i)). It gives a
dimensional positive bound if (r) is bounded below. If (r) is
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)>= c_n), since (nu_ileq1).
-Each bound controls (mleq2n). This proves the (m) part for
-(mgeq m_0).
+case (ε M_0^2ge c_nε_*). In the latter,
+(ε=m/barbar uule C_n) and
+(u_i=A_i^2/(εnu_i)>= c_n), since (nu_ile1).
+Each bound controls (mle2n). This proves the (m) part for
+(mge m_0).
-For the volume part, indices with (A_igeq1/4) satisfy
+For the volume part, indices with (A_ige1/4) satisfy
[
- A_i^2(L_i-1)2argeq c,r(L_i-1)>= c,rlog L_i.
+ A_i^2(L_i-1)2arge 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)
+For all other indices, (log L_ile C A_i^2), so their contribution
+to (rW_p) is at most (Crm). If (rle2), the bound for (m)
already covers this. If (r>2), then
(|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
+(ε M_0^2ge cε rbarbar u=cu=c,rm).
+This proves (eq:entropy-coarse-coercivity) when (mge m_0).
+When (m<m_0), one has (barbar uule m_0/ε_*) small and
[
W_p=frac m2+O(sum_iA_i^4)<= C m.
]
-If (|1-r|geq1/4), choosing (m_0) still smaller gives
+If (|1-r|ge1/4), choosing (m_0) still smaller gives
(|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
+(εgeε_*) and (mle m_0), and for (rge2)
+use ((1-r)^2ge r^2/4). This proves the claim in the remaining
case.
On the region of (eq:entropy-coarse-coercivity), the positive
@@ -2595,9 +2595,9 @@ remains. Choose (gamma_n) below that fraction. The error term
It remains to prove the budget when
- m<m_0, |1-r|<tfrac14.
+ m<m_0, |1-r|<1/4.
-Here (T_0geq c_nm^2). Indeed, expanding the exponential in
+Here (T_0ge c_nm^2). Indeed, expanding the exponential in
(eq:entropy-trace) and using
(W_p=m/2+O(sum_iA_i^4)) gives
[
@@ -2606,16 +2606,16 @@ Here (T_0geq c_nm^2). Indeed, expanding the exponential in
]
after decreasing (m_0). The self terms
(sum_i A_i^2(L_i-1)2ar) are at least
-(c_nsum_iA_i^4), by (r>3/4) and (L_i-1geq A_i^2/2).
+(c_nsum_iA_i^4), by (r>3/4) and (L_i-1ge A_i^2/2).
Adding a sufficiently small fraction of the first estimate to the
-second proves (T_0geq c_nm^2).
-Since (W_pleq Cm) on this region, this controls the budget whenever
-(mgeq Kl), for a fixed sufficiently large dimensional (K).
+second proves (T_0ge c_nm^2).
+Since (W_ple Cm) on this region, this controls the budget whenever
+(mge Kl), for a fixed sufficiently large dimensional (K).
In making this conclusion, first fix (K); after (D_*) is chosen
below, take (l) small enough that (4D_*^2l^2m) is also a small
fraction of (m^2).
-What remains is the critical regime (mleq Kl), still under
+What remains is the critical regime (mle Kl), still under
(eq:entropy-small-region). Here
(W_p=m/2+O(m^2)) and (3/4<r<5/4), so the budget expands as
[
@@ -2625,19 +2625,19 @@ What remains is the critical regime (mleq Kl), still under
The target quadratic term is covered exactly by
[
(m^2)/(n)-2lm+nl^2
- =(frac m{sqrt n}-sqrt n,l)^2geq0.
+ =(frac m{sqrt n}-sqrt n,l)^2ge0.
]
It remains to recover (m^2/n) from (T_0), while paying the
displayed (r-1) error and the errors in that recovery. This is where
the coefficient one on (T_0) in (Pi_p) is needed.
In particular (e^{W_p}=1+O(m)). Metric addition gives
-(h_0leq h_jleq h_+) as quadratic forms.
+(h_0le h_jle h_+) as quadratic forms.
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_+<= e^{2W_p}h_0=(1+O(m))h_0.
+ h_0le h_jle 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
@@ -2668,7 +2668,7 @@ in the exact formulas differs from its (h_0) expression by
Inserting (eq:entropy-BC-expansion) into (T_0) yields
- T_0geq
+ T_0ge
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)_+).
@@ -2676,9 +2676,9 @@ Inserting (eq:entropy-BC-expansion) into (T_0) yields
To see the leading coefficient, the (B_i) and (C_i) terms each
give one copy of
(sum_{i<j}A_i^2A_j^2xi_{ij}). In the (B_i) terms use
-(4a^2geq1), (B_igeq0); replacing (sigma_i) by one costs at most
+(4a^2ge1), (B_ige0); replacing (sigma_i) by one costs at most
(Cmsum_iA_i^2(1-sigma_i)_+). The self terms use
-(L_i-1geq A_i^2/2) and (2ageq1). They give
+(L_i-1ge A_i^2/2) and (2age1). They give
(sum_iA_i^4), up to the displayed (r-1) and (sigma_i-1) costs.
The errors in (eq:entropy-BC-expansion) sum to (O(m^3)).
This proves (eq:entropy-sharp-T). The trace in that estimate
@@ -2688,7 +2688,7 @@ with trace (m).
The last error in (eq:entropy-sharp-T) is paid for by the
centered squares. From
(a(sigma_i-1)=V_i-lambda(u_i-barbar u)u)), Cauchy--Schwarz and Young's
-inequality give, for every fixed (c'>0),
+ineuality give, for every fixed (c'>0),
msum_iA_i^2(1-sigma_i)_+
@@ -2700,21 +2700,21 @@ inequality give, for every fixed (c'>0),
Choose (c') to use a small fraction of the squares in (Pi_p).
The coefficient of (εVar u) in the last term, relative to
(thetaεVar u), is at most (C_{c'}K^2/D_*^2), because
-(mleq Kl) and (theta=D_*l). Choose (D_*) sufficiently large
+(mle Kl) and (theta=D_*l). Choose (D_*) sufficiently large
after (K) so that this term is absorbed as well. The remaining term
is
[
C_{c'}(m^2)/(theta)ε barbar u^2u^2
=C_{c'}(m^4)/(thetaε)=O_n(l^3).
]
-Here (εgeqε_*), and (K,D_*,ε_*) are already
+Here (εgeε_*), and (K,D_*,ε_*) are already
fixed dimensionally. Also
[
|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
-(l^2|M_0|), and use (barbar uuleq Kl/ε_*) on the other part.
+of (ε M_0^2) plus (O_n(l^3)): use Young's ineuality on
+(l^2|M_0|), and use (barbar uule Kl/ε_*) on the other part.
The errors (m^3), (m^2|r-1|), and (4D_*^2l^2m) in the present
region are therefore covered with a remaining error (O_n(l^3)).
@@ -2727,7 +2727,7 @@ terms are of the form (C_nr l^3). This proves
The choices just made have a definite order: first (m_0) after
(ε_*), then (K), then (D_*), and finally (l_0>0)
-small enough for all the comparisons and for (D_*l_0leq1/16).
+small enough for all the comparisons and for (D_*l_0le1/16).
The errors in each sweep are chosen afterward. Thus every constant
in (eq:entropy-increment) is dimensional.
@@ -2735,10 +2735,10 @@ in (eq:entropy-increment) is dimensional.
[Bounds supplied by the surplus]
For every sweep of Lemma~lem:entropy-increment,
- W_pleq C_nε^{1/4}(ε+P_p)^{3/4}.
+ W_ple C_nε^{1/4}(ε+P_p)^{3/4}.
There is also a dimensional (ε_{exit}) such that,
-if (εgeqε_{exit}) and (rleq1/4), then
+if (εgeε_{exit}) and (rle1/4), then
P_p&>= c_n(ε)/(lr)
@@ -2748,44 +2748,44 @@ if (εgeqε_{exit}) and (rleq1/4), then
The errors in the prescription may simultaneously be chosen to have
-(sum_i|err_i|^2leq1).
+(sum_i|err_i|^2le1).
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|>= c(r+barbar u)u)) in this range and (lleq1),
+(mge m_0), so the surplus retains the mean square and boost.
+Since (|M_0|>= c(r+barbar u)u)) in this range and (lle1),
[
- P_pgeq c_nε,
- frac{(r+barbar u)^2+c_*r^6u)^2+c_*r^6indic_{{rgeq C_*}}}{r}
+ P_pge c_nε,
+ frac{(r+barbar u)^2+c_*r^6u)^2+c_*r^6indic_{{rge C_*}}}{r}
>= 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)
+For the last ineuality, if (rle barbar u^u^{1/3}) use (barbar u^2/ru^2/r);
+if (rge barbar u^u^{1/3}) use (r^5), taking the threshold for (barbar uu)
large enough that (barbar u^u^{1/3}>= C_*). It follows in all cases that
- mleq C_nε(1+P_p/ε)^{3/5}.
+ mle C_nε(1+P_p/ε)^{3/5}.
If any (A_i>1/2), then (m>1/4>m_0), and the self terms in (T_0)
give
[
- P_pgeq c_nsum_{A_i>1/2}(L_i-1).
+ P_pge c_nsum_{A_i>1/2}(L_i-1).
]
Indeed their contribution to (T_0) is at least
(c rsum_{A_i>1/2}(L_i-1)), and
-(ε/(rN^2)=1/(lr)geq1/r).
-For (A_ileq1/2), (log L_ileq C A_i^2).
+(ε/(rN^2)=1/(lr)ge1/r).
+For (A_ile1/2), (log L_ile C A_i^2).
There are at most (4m) indices with (A_i>1/2). Concavity of
(log(1+s)) on those indices, together with the last estimate,
therefore gives
- W_pleq C_n m(1+log(1+P_p/m))
+ W_ple C_n m(1+log(1+P_p/m))
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)<= C(1+t)^{1/4}) for (tgeq0), applied with
+(1+log(1+t)<= C(1+t)^{1/4}) for (tge0), applied with
(t=y/x), makes the right side of
(eq:entropy-log-surplus), divided by (ε), at most
[
@@ -2793,20 +2793,20 @@ Put (x=m/ε), (y=P_p/ε). The elementary bound
<= C_n(1+y)^{7/10}
<= C_n(1+y)^{3/4},
]
-where (eq:entropy-m-surplus) was used in the middle inequality.
+where (eq:entropy-m-surplus) was used in the middle ineuality.
This proves (eq:entropy-pass-volume).
-Now suppose (rleq1/4). The surplus retains all positive terms
-because (|1-r|geq3/4). Outside a fixed compact subinterval of
+Now suppose (rle1/4). The surplus retains all positive terms
+because (|1-r|ge3/4). Outside a fixed compact subinterval of
(barbar uuin(0,∞)), the expression (M_0=a(1-r)-barbar u/2u/2) satisfies
-(M_0^2geq c(1+barbar u^2)u^2)). On that compact interval, for (ε)
+(M_0^2ge c(1+barbar u^2)u^2)). On that compact interval, for (ε)
sufficiently large,
[
- T_0geq m-m_fgeqε barbar u-nu-n
+ T_0ge m-m_fgeε barbar u-nu-n
>= 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
+term by (gamma_n/(lr)) proves the first ineuality in
(eq:entropy-last-surplus).
For the second, the retained derivative terms imply
@@ -2819,11 +2819,11 @@ Also
[
sum_i t_i^2=(ε^2)/(r)(barbar u^2+u^2+Var u).
]
-The first inequality of (eq:entropy-last-surplus) controls
+The first ineuality 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 (X contains
+(sum_i t_i^2/N^2le 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).
@@ -2831,20 +2831,20 @@ For (D_i), use
[
ND_i^0=(ε f_i)/(r)
(a(sigma_i-r)+(u_i)/(2)),
- E sigma_i^2leq C(1+E V_i^2+Var u).
+ E sigma_i^2le C(1+E V_i^2+Var u).
]
-Here the second inequality follows by solving the definition of (V_i)
-for (sigma_i-1). Since (rleq1/4),
+Here the second ineuality follows by solving the definition of (V_i)
+for (sigma_i-1). Since (rle1/4),
[
- sum_i(D_i^0)^2leq
+ sum_i(D_i^0)^2le
(Cε^2)/(rN^2)
(1+barbar u^2+u^2+E V_i^2+Var u)
<= 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^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).
+estimate: (sum_ibeta_i^2f_i^2leX, and one chooses (sum_i|err_i|^2le1). Its contribution (1/N^2) is
+bounded by (C_nP_p) using the first ineuality of
+(eq:entropy-last-surplus), (εge1), and (rle1/4).
@@ -2862,29 +2862,29 @@ The fixed exponents in the next two lemmas leave room, respectively,
for the number of sweeps, the conformal scalar cost, the outgoing
volume estimate, and the final gradient target:
[
- 0.26=tfrac14+0.01,
+ 0.26=1/4+0.01,
0.26+0.04<(1+0.32)/(4),
- 0.26+tfrac34(0.04)<0.32,
- 0.02>tfrac1{100}.
+ 0.26+3/4(0.04)<0.32,
+ 0.02>1/{100}.
]
The estimates below show how each margin is used.
[Accumulation of entropy passes]
-Suppose at parameter (ε_{in}geqε_*) that
+Suppose at parameter (ε_{in}geε_*) that
[
wscal(h,phi)>=-C+2W-nlogε_{in}
+ε_{in}e_h(F)+P_{in},
- W=log(dV_h/rho)geq0, P_{in}geq0.
+ W=log(dV_h/rho)ge0, P_{in}ge0.
]
Successive sweeps of Lemma~lem:entropy-increment, with
(ε_{new}=(1+l(ε))ε), preserve this
-form of inequality: at the final parameter (E) the surplus is
+form of ineuality: at the final parameter (E) the surplus is
(P=P_{in}+sum_pP_p), the volume logarithm is
(W_{new}=W+sum_pW_p), and the increase of the scalar
constant is (C_nsum_p l(ε_p)^2). Along the entire increasing
sequence of parameters,
- sum_p l(ε_p)^2leq C_n,
+ sum_p l(ε_p)^2le C_n,
#{p:ε_p<E}<= C_n(1+log^3(1+E)).
If the initial volume logarithm is zero, then, for all sufficiently
@@ -2895,10 +2895,10 @@ large dimensional (E),
The same assertion holds if the initial data have parameter (E_0),
surplus (P_0), and
[
- Wleq E_0^{0.32}(E_0+P_0)^{0.75},
+ Wle E_0^{0.32}(E_0+P_0)^{0.75},
]
-provided (Egeq E_0) is sufficiently large dimensionally and
-(E_0^{0.32}leqtfrac12E^{0.26}).
+provided (Ege E_0) is sufficiently large dimensionally and
+(E_0^{0.32}le1/2E^{0.26}).
Proof.
@@ -2910,7 +2910,7 @@ The density identity in (eq:entropy-sweep-density) gives the
sum of the (W_p).
Put (t_p=log(1+ε_p)). Since
-(ε_pgeqε_*>0) and (l_*) is fixed small,
+(ε_pgeε_*>0) and (l_*) is fixed small,
[
t_{p+1}-t_p
=log(1+l(ε_p)(ε_p)/(1+ε_p))
@@ -2928,15 +2928,15 @@ its endpoints differ by a uniformly bounded small number.
Let (k) be the number of sweeps in the group under consideration.
By Lemma~lem:entropy-surplus, since
-(ε_pleq E) and (P_pleq P),
+(ε_ple E) and (P_ple P),
[
- sum_pW_pleq C_n k E^{1/4}(E+P)^{3/4}.
+ sum_pW_ple C_n k E^{1/4}(E+P)^{3/4}.
]
For all sufficiently large (E),
-(C_n kleq tfrac12 E^{0.01}) by
+(C_n kle 1/2 E^{0.01}) by
(eq:entropy-summability). This proves
(eq:entropy-phase-volume) for zero initial volume logarithm.
-In the other case (Pgeq P_0) and (Egeq E_0), so the initial
+In the other case (Pge P_0) and (Ege E_0), so the initial
term is at most (E_0^{0.32}(E+P)^{0.75}). The stated condition on
(E_0) pays for this with the other half of
(E^{0.26}(E+P)^{0.75}).
@@ -2948,7 +2948,7 @@ 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)}<=...<= h_+,
+ h^0le h^{(1)}<=...<= h_+,
(h_+)^{-1}<=...<=(h^{(1)})^{-1}<=(h^0)^{-1}.
]
In particular, the starting metric is an admissible reference metric
@@ -2968,29 +2968,29 @@ Consider a phase starting with root map (Y^0) and metric (h^0), and
let (K_0) be a finite uniform bound for (e_{h^0}(Y^0)^{1/2}).
Suppose its last sweep has input parameter (ε), output
parameter (E=(1+l(ε))ε), and output
-((h_+,phi_+,Y)). Suppose, with (Cgeq0) and (Pgeq P_p) including the
+((h_+,phi_+,Y)). Suppose, with (Cge0) and (Pge P_p) including the
surplus from that last sweep, that
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}.
+ W&=log(dV_{h_+}/rho)ge0,
+ Wle E^{0.26}(E+P)^{0.75}.
If (E) is sufficiently large in terms of (n,K_0), there are a
-metric (widehatwidehat hh), a weight (widehatphi), and a root map (Z)
+metric (hhhat), a weight (phihat), and a root map (Z)
on two copies of the labels such that
- wscal(widehatwidehat h,h,widehatphi)
- &>=-(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}&<= E^{-1/100}.
+ wscal(hhat,h,phihat)
+ &>=-(C+C_n)+2What-nW-nlog E
+ +E e_{hhat}h(Z)+P',
+ What&=W&=log(dV_{hhat}h/rho)ge0,
+ WWlehat E^{0.32}(E+P')^{0.75},
+ e_{hhat}h(Z)^{1/2}&<= E^{-1/100}.
-Here (P'geq0), and the added scalar cost (C_n) is independent
+Here (P'ge0), and the added scalar cost (C_n) is independent
of (K_0) and of all other instance constants. The changes preserve
the uniform smoothness and support properties of
Theorem~thm:entropy.
@@ -3001,7 +3001,7 @@ In this proof all derivatives, unless otherwise indicated, use (h_+).
Write (r=sum_i f_i^2) for the last sweep, set (s=sqrt r), and put
[
c_E=frac1{sqrt E(1+K_0)},
- Z=frac{(f,c_EY^0)}{√(r+c_E^2)}.
+ Z=(f,c_EY^0)/√(r+c_E^2).
]
The two vectors in the numerator use disjoint copies of the labels.
The map is a smooth root probability map because its denominator is
@@ -3012,8 +3012,8 @@ copy is supported on the last bumps and the second on the starting
map. This parent assignment will be used to compare support overlaps
between phases. By (eq:entropy-trace),
[
- N^2sum_i|df_i|_{h_+}^2=m_fleq n,
- |ds|_{h_+}^2leq n/N^2.
+ N^2sum_i|df_i|_{h_+}^2=m_fle n,
+ |ds|_{h_+}^2le n/N^2.
]
Monotonicity of the phase metrics also gives (e_{h_+}(Y^0)<= K_0^2).
@@ -3024,7 +3024,7 @@ derivatives are perpendicular to their respective vectors. Differentiating
e(Z)=frac r{r+c_E^2}e(Y)+(c_E^2)/(r+c_E^2)e(Y^0)
+(c_E^2)/((r+c_E^2)^2)|ds|^2.
]
-For (rgeq1/4), it follows that
+For (rge1/4), it follows that
[
E(e(Z)-e(Y))<=
(Ec_E^2K_0^2)/(r+c_E^2)
@@ -3032,53 +3032,53 @@ For (rgeq1/4), it follows that
<= 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
+(eq:entropy-step-schedule). For (rle1/4), the norm of the
derivative of normalization is at most (1/√(r+c_E^2)), so
e(Z)<=(n/N^2+c_E^2K_0^2)/(r+c_E^2).
-The first inequality of (eq:entropy-last-surplus) now implies
+The first ineuality of (eq:entropy-last-surplus) now implies
[
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
-({rgeq1/4}) and an arbitrarily small fraction of (P_p) on
-({rleq1/4}), once (E) is dimensionally large. In particular,
+({rge1/4}) and an arbitrarily small fraction of (P_p) on
+({rle1/4}), once (E) is dimensionally large. In particular,
no factor (K_0) occurs in this scalar cost.
-Choose a smooth function (u=u(r)geq0), supported in
-({rleq E^{-0.32}}), with
+Choose a smooth function (u=u(r)ge0), supported in
+({rle E^{-0.32}}), with
e^uasymp
min(E^{0.02},max(1,E^{-0.16}/sqrt r)),
- |(du)/(dr)|leqfrac C r.
+ |(du)/(dr)|lefrac C r.
For example, smooth, with bounded-width transitions on the logarithmic
scale, the function obtained by truncating
-(-tfrac12log r-0.16log E) to the interval
+(-1/2log r-0.16log E) to the interval
([0,0.02log E]). The smoothing may vanish whenever the untruncated
function is nonpositive, and changes its value by a bounded amount;
this gives both the support assertion and the comparison in
(eq:entropy-u-choice). Define
[
- widehatwidehat h=e^h=e^{2u}h_+,
- widehatphi=phi_+-(n-1)u,
- widehatwidehat W=W+nu.W=W+nu.
+ hhat=e^h=e^{2u}h_+,
+ phihat=phi_+-(n-1)u,
+ What=W+nu.W=W+nu.
]
-In particular (dV_{widehatwidehat h}h=e^{nu}dV_{h_+}), so this is the
+In particular (dV_{hhat}h=e^{nu}dV_{h_+}), so this is the
claimed volume logarithm. Differentiating (r) gives
[
- |dr|leq2sqrt r(sum_i|df_i|^2)^{1/2}
- leqfrac{2√(nr)}N,
+ |dr|le2sqrt r(sum_i|df_i|^2)^{1/2}
+ lefrac{2√(nr)}N,
|du|<=(C_n)/(Nsqrt r).
]
The coefficient (n-1) in the new weight cancels the Laplacian of
(u) in the weighted conformal formula. With
(b=nabla_{h_+}phi_+), that formula is
- wscal(widehatwidehat h,h,widehatphi)
+ wscal(hhat,h,phihat)
=e^{-2u}[wscal(h_+,phi_+)+2b(u)-(n-1)|du|^2].
@@ -3086,17 +3086,17 @@ With the fixed density (rho), a graph sweep changes (W) by
(+W_p) and (phi) by (-W_p), so it preserves (phi+W).
The present exit instead satisfies
[
- widehatphi+widehatwidehat W=W=phi_++W+u.
+ phihat+What=W=phi_++W+u.
]
Thus after an exit (W) need not equal (-phi), and its volume
bound supplies no bound for the gradient of the weight. Need a
bound on (b(u)) which uses no bound on the size of (b).
It is enough to estimate this on the support of (u), which lies in
-({rleq1/4}) for large (E). At such a point view all the graphs
+({rle1/4}) for large (E). At such a point view all the graphs
of the last sweep in the common
Euclidean extension of ((T_xwidetilde M,h_0)) by their graph
directions, one for each active color at that point. Write
-(mathbf n_ibfn_i) for the successive unit normals. They are
+(n_i) for the successive unit normals. They are
orthonormal: a later
normal lies in the preceding graph plus its new vertical direction
and is perpendicular to every earlier normal. In this space, each
@@ -3105,12 +3105,12 @@ unused later directions. Writing (b_i) for the embedded gradient
of (phi_i), orthogonal projection to the new graph gives the exact
recurrence
[
- b_i=b_{i-1}-< b_{i-1},mathbf n_i>bfn_i>mathbf n_ibfn_i
+ b_i=b_{i-1}-< b_{i-1},n_i>n_i>n_i
-nabla_{h_i}log L_i,
]
where the last gradient is embedded in the same space. The normal
coefficient is
-(< b_{i-1},mathbf n_i>=-U_i(bfn_i>=-U_i(phi_{i-1})), and
+(< b_{i-1},n_i>=-U_i(n_i>=-U_i(phi_{i-1})), and
its magnitude satisfies
[
|U_i(phi_{i-1})|<= |D_i|+sqrt n,|II_i|,
@@ -3121,7 +3121,7 @@ the sum of these normal corrections is at most
(C_n√(P_p/theta)).
For the subtracted tangent gradients, split the indices according as
-(A_ileq1/2) or (A_i>1/2). In the first set
+(A_ile1/2) or (A_i>1/2). In the first set
(|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
@@ -3132,36 +3132,36 @@ while the vertical component of the final gradient is
((Ndf_i(b))_i).
The normal and tangent bounds therefore prove
[
- sum_i(Ndf_i(b))^2leq C_n(1+m)P_p/theta.
+ sum_i(Ndf_i(b))^2le C_n(1+m)P_p/theta.
]
It follows that
[
- |b(u)|leqfrac C r|b(r)|
+ |b(u)|lefrac C r|b(r)|
<= C_n(((1+m)P_p)/(theta N^2r))^{1/2}
- <= frac{C_nP_p}{√(theta E)}.
+ <= C_nP_p/√(theta E).
]
-For the last inequality use (Easympε), (N^2=lε),
+For the last ineuality use (Easympε), (N^2=lε),
and
[
- P_pgeq(c_n)/(lr)
+ P_pge(c_n)/(lr)
(ε+(m^2)/(ε))
>= (c'_n(1+m))/(lr)
]
-for (εgeq1). The same first surplus bound gives
+for (εge1). The same first surplus bound gives
- |du|^2leq(C_n)/(N^2r)<=(C_nP_p)/(E^2).
+ |du|^2le(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
+({rle1/4}) for large (E). Since
(theta Easymp_n E/(1+log E)^2) tends to infinity, the two
derivative terms in (eq:entropy-conformal-exit) cost an
arbitrarily small fraction of (P_p).
There is enough surplus on this region also for the change in the
-entropy term. From (rleq E^{-0.32}) and
+entropy term. From (rle E^{-0.32}) and
(eq:entropy-last-surplus),
[
- Pgeq P_pgeq c_n E^{1.32}.
+ Pge P_pge c_n E^{1.32}.
]
Using (eq:entropy-before-exit) and (e^{2u}<= C E^{0.04}),
one obtains there
@@ -3171,7 +3171,7 @@ one obtains there
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)
+In the first bound use was made of (E+Ple C_nP), valid for large (E)
on this region. Both quantities tend to zero with dimensional
constants.
@@ -3181,29 +3181,29 @@ dimensionally. Put (B_u=2|b(u)|+(n-1)|du|^2). Substituting in
(eq:entropy-conformal-exit) and moving the new entropy term
to the left gives
- &wscal(widehatwidehat h,h,widehatphi)
- -E e_{widehatwidehat h}h(Z)-2(W+nu)+nlog E
- &quadgeq -(C+C_n)
+ &wscal(hhat,h,phihat)
+ -E e_{hhat}h(Z)-2(W+nu)+nlog E
+ &quadge -(C+C_n)
+e^{-2u}[alpha_n P-B_u
-2(e^{2u}-1)W-2nu e^{2u}]
+(1-e^{-2u})nlog E .
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
+for (Ege1). 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
volume and derivative costs vanish. Thus set
[
P'=alpha'_n e^{-2u}P
]
-for a fixed (alpha'_n>0), proving the scalar inequality in
+for a fixed (alpha'_n>0), proving the scalar ineuality in
(eq:entropy-after-exit). The only scalar constant charged was
the dimensional replacement cost.
The volume estimate at exit follows as well. Since
-(Pleq C_ne^{2u}P'<= C_nE^{0.04}P') and (nuleq C_nlog E),
+(Ple C_ne^{2u}P'<= C_nE^{0.04}P') and (nule C_nlog E),
[
- widehatwidehat WWleq
+ WWlehat
C_n E^{0.26+0.04(0.75)}(E+P')^{0.75}+C_nlog E
<= E^{0.32}(E+P')^{0.75}
]
@@ -3212,24 +3212,24 @@ for all sufficiently large dimensional (E).
Finally, (eq:entropy-Z-energy) and the conformal scaling of
energy give
[
- e_{widehatwidehat h}h(Z)^{1/2}
- <= e^{-u}frac{sqrt n/N+c_EK_0}{√(r+c_E^2)}.
+ e_{hhat}h(Z)^{1/2}
+ <= e^{-u}sqrt n/N+c_EK_0/√(r+c_E^2).
]
The choice (eq:entropy-u-choice) implies
[
frac{e^{-u}}{√(r+c_E^2)}
<= 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
+Using (sqrt E/Nle C_n(1+log E)), (c_EK_0le E^{-1/2}),
+and (Nlesqrt E), one concludes that
- e_{widehatwidehat h}h(Z)^{1/2}
+ e_{hhat}h(Z)^{1/2}
<= 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
(O_n((1+log E)E^{-0.34})). Taking (E) large enough in terms of
-(n,K_0) proves the final inequality in
+(n,K_0) proves the final ineuality in
(eq:entropy-after-exit). The functions (u) and the normalizing
denominator are smooth with finite bounds in each fixed construction.
The supports of (Z) are those of the last bumps and of (Y^0).
@@ -3282,7 +3282,7 @@ It follows that the graph joining every pair of labels whose possible
enlarged supports in the first phase meet, or come within a required
buffer, has degree at most
- Delta_1+1leq
+ Delta_1+1le
exp(C_{inst}(1+log E_1)^3).
Here (C_{inst}) is finite and fixed before the
@@ -3295,7 +3295,7 @@ Take (E_1) sufficiently large for
Lemma~lem:entropy-accumulation and
Lemma~lem:entropy-phase-exit; in the latter the initial (K_0)
is the finite input-dependent gradient bound of (F_*).
-Now there holds the scalar inequality, the volume bound with exponent
+Now there holds the scalar ineuality, 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,G_1
@@ -3335,15 +3335,15 @@ and call the first parameter reaching it (E_{j+1}). Thus
[
T_{j+1}<= E_{j+1}<=(1+l_*)T_{j+1}.
]
-The choice (1/1000) leaves the strict inequality
+The choice (1/1000) leaves the strict ineuality
(3/1000<1/100): the cubic logarithmic cost of support enlargement
will be smaller than the gain from the preceding gradient bound.
For large (E_j) this target satisfies
-(E_j^{0.32}leqtfrac12 E_{j+1}^{0.26}). The outgoing bound with
+(E_j^{0.32}le1/2 E_{j+1}^{0.26}). The outgoing bound with
exponent (0.32) from phase (j) and
Lemma~lem:entropy-accumulation therefore give the bound with
exponent (0.26) before the next exit. The next initial gradient
-bound is (K_0leq E_j^{-1/100}leq1), so all further thresholds
+bound is (K_0le E_j^{-1/100}le1), so all further thresholds
for Lemma~lem:entropy-phase-exit are dimensional. Increasing
the first target once ensures these conclusions in every one of a
prescribed finite number of phases.
@@ -3378,7 +3378,7 @@ are equal or at distance at most two gives a fixed graph
(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.
+ Delta_{j+1}+1le2(Delta_j+1)^2.
It also doubles the number of label orbits. The preliminary coloring
construction applies to these graphs before each sweep.
@@ -3393,7 +3393,7 @@ 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
- log qleq C_{inst,J,n}(1+log E_1)^3.
+ log qle 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 (G_J and this coloring have exactly
@@ -3409,9 +3409,9 @@ elementary bound. Equation (eq:entropy-q-growth) implies
(1+log E_1)^3)
]
for large (E_1). On the other hand, the recurrence
-(eq:entropy-phase-targets) gives, for each fixed (Jgeq2),
+(eq:entropy-phase-targets) gives, for each fixed (Jge2),
- E_Jgeqexp^{[J-1]}(c_J E_1^{1/1000})
+ E_Jgeexp^{[J-1]}(c_J E_1^{1/1000})
for some (c_J>0) and all sufficiently large (E_1).
One proves this by induction: raising a fixed-height exponential
@@ -3441,19 +3441,19 @@ at most (2C_bε_*), independently of its input-dependent
number of sweeps. The initial logarithmic normalization has a
dimensional cost because (ε_0in[ε_*,2ε_*]).
All entropy sweep losses together are bounded by
-(C_nsum l^2leq C_n). Each exit charges at most (C_n), and their
+(C_nsum l^2le C_n). Each exit charges at most (C_n), and their
number (J=H+2) depends only on the prescribed elementary function
and (n). The input gradient bound (K_0), the initial small
(ε_{in}), the compact support radii, and all metric
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
+the remaining ineuality is (eq:entropy-contract) with
(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
-increases by (W_pgeq0) and (nugeq0). Finitely many sweeps and
+(rhole dV_g). The volume logarithm starts at zero and only
+increases by (W_pge0) and (nuge0). Finitely many sweeps and
exits preserve the asserted smoothness, uniformity, compact supports,
and finite number of free label orbits. The transverse spaces were
enlarged only by the probability products used for the colorings,
@@ -3674,7 +3674,7 @@ It is odd and increasing and has poles at pm b, where
b=frac1{L_1}int_0^∞(dy)/(psi(y))
lefrac{C(1+B)^{2/3}}{L_1}.
]
-For the inequality, split the integral at (1+B)^{2/3} and use the
+For the ineuality, split the integral at (1+B)^{2/3} and use the
|y|^{3/2} bound on the tail. Choose the absolute C_0 in
(eq:degree-horizon) at least this C. Thus
@@ -3689,17 +3689,17 @@ be a smooth cooriented local hypersurface, with unit normal nu, and set
D=H+∂_nuphi,
H=trII,
II(Y,Z)=<nabla_Ynu,Z>,
- P=|II|^2+calP=|II|^2+
+ P=|II|^2+P=|II|^2+
(Ric_h-nabla^2phi)(nu,nu).
]
-Put mathfrak a=L_1fraka=L_1psi(D)>0. When D=P_0(s-G_1), the derivatives
-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
+Put a=L_1fraka=L_1psi(D)>0. When D=P_0(s-G_1), the derivatives
+of the raw pressure are ∂_sP_0(s-G_1)=a and
+d_xP_0(s-G_1)=-a,dG_1. The following calculation uses
only a positive smooth function v satisfying the local equation
-(Delta_{Σ,phi}+P
- -mathfrak afraka,dG_1(nu))v=mathfrak a.fraka.
+ -a,dG_1(nu))v=a.a.
Here Delta_{Σ,phi} is weighted by
e^{phi|_Σ}dA_h, and slice norms use the induced metric. The
@@ -3711,26 +3711,26 @@ 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^2-2P+|II|^2.calP+|II|^2.
+ =wscal(h,phi)+D^2-2P+|II|^2.P+|II|^2.
]
Adding ln v to the weight changes this by
-2Delta_{Σ,phi}v/v+|dln v|^2. Substitution of
(eq:degree-local-lapse) yields the exact identity
- S_Σ=wscal(h,phi)+D^2-2mathfrak afraka,dG_1(nu)
- +|II|^2+(2mathfrak a)/(v)+|dfraka)/(v)+|dln v|^2.
+ S_Σ=wscal(h,phi)+D^2-2fraka,dG_1(nu)
+ +|II|^2+(2a)/(v)+|dfraka)/(v)+|dln v|^2.
This is also the hypersurface calculation behind
Lemma~lem:graph-identity; it is valid for p=0 with left side zero.
Set t=L_1|dG_1|_hle B and x=v/L_1. Since gamma(D)le D^2,
[
- -2mathfrak afraka,dG_1(nu)
+ -2fraka,dG_1(nu)
ge-2t(1+(gamma(D))/(8(1+B))),
(tgamma(D))/(4(1+B))le(D^2)/(4).
]
Discard the remaining nonnegative part of D^2, and use
-2mathfrak a/vfraka/vge2/x. Finally,
+2a/vfraka/vge2/x. Finally,
tle1+(t-1)_+ and x^{-1}+ln xge1 for x>0. Thus
(eq:degree-slice-identity) implies
@@ -3752,12 +3752,12 @@ controlled-band and degree conditions needed to use this comparison.
For a small eta>0, prolong P_0 smoothly and strictly increasingly
past the points pm(b-eta), keeping it unchanged on
[-b+eta,b-eta]. Denote the resulting smooth function by
-widehatwidehat PP and set
+PPhat and set
[
- mathfrak h_s(x)=frakh_s(x)=widehatwidehat P(s-G_1(x)),P(s-G_1(x)), -tau<s<tau.
+ h_s(x)=h_s(x)=Phat(s-G_1(x)),P(s-G_1(x)), -tau<s<tau.
]
Choose eta sufficiently small for the given band and then keep
-it fixed. On every boundary portion used below, widehatwidehat PP will agree
+it fixed. On every boundary portion used below, PPhat will agree
with P_0 and hence obey (eq:degree-pressure).
Least minimizers, including on noncompact leaves.:
@@ -3765,7 +3765,7 @@ Write P_phi(E;U)=int_{∂^*Ecap U}e^phi,dA_h for the weighted
perimeter of a set of locally finite perimeter. Use local differences
of the functional
- P_phi(E)-int_Emathfrak h_sfrakh_s e^phi,dV_h.
+ P_phi(E)-int_Efrakh_s e^phi,dV_h.
This is a weighted mu-bubble functional. For soap bubbles in Gromov's
terminology, see cite[Section~5frac56]{Gromov1996}; for the weighted
@@ -3809,17 +3809,17 @@ L^1 set; call it the least minimizer in K.
parnoindent{footnotesizehttps://doi.org/10.1017/CBO9781139108133}
Order and exhaustion.
-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
+The same ineuality proves both monotonicities one needs. If Ksubset K'
+and two forcings satisfy h',h', a minimizer for
+(K,h)h) and one for (K',h')h') obey
[
- F_{mathfrak h}frakh(Ecap F)+F_{mathfrak h'}frakh'(Ecup F)
- le F_{mathfrak h}frakh(E)+F_{mathfrak h'}frakh'(F),
+ F_{h}h(Ecap F)+F_{h'}h'(Ecup F)
+ le F_{h}h(E)+F_{h'}h'(F),
]
-where F denotes (eq:degree-functional); the inequality for
+where F denotes (eq:degree-functional); the ineuality 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
+ h'.h'. Minimality in
+the two domains makes both ineualities equalities, and leastness in K
implies Esubset F. Exhaust a leaf by increasing finite unions of
compact coordinate subboxes whose interiors exhaust it. The least
minimizers increase with this exhaustion. Their union, interpreted in
@@ -3941,16 +3941,16 @@ which fills B_t, and perimeter slicing, give for almost every t<r
[
P_{norm}(E_s;B_t)+H V(t)le C V'(t).
]
-The chart isoperimetric inequality applied to that complementary phase,
+The chart isoperimetric ineuality applied to that complementary phase,
extended by zero outside B_t, gives, when V(t)>0,
[
V(t)^{(d-1)/d}le C(P_{norm}(E_s;B_t)+V'(t)).
]
-If the complementary phase occurs in B_{r/4}, the two inequalities first
+If the complementary phase occurs in B_{r/4}, the two ineualities first
give V'ge cV^{(d-1)/d} and hence V(r/2)ge cr^d. They also give
V'ge cHV on [r/2,r]. It follows that
V(r)ge cr^dexp(cHr), contrary to V(r)le Cr^d when Hr is large.
-This argument includes d=1, where the first differential inequality
+This argument includes d=1, where the first differential ineuality
has exponent zero. Since Hrasymp_{inst}eta^{-1}, small
eta excludes the complementary phase in the smaller ball. Negative
forcing gives the empty phase by the same argument. Applying these
@@ -3973,10 +3973,10 @@ relatively compact chart. To place the functional under the standard
codimension-one theorem, normalize phi by a constant value phi_0
in the chart and use the smooth metric
[
- widehatwidehat h=e^h=e^{2(phi-phi_0)/p}h.
+ hhat=e^h=e^{2(phi-phi_0)/p}h.
]
Its area is the normalized weighted area. Its volume forcing is
-mathfrak h_sfrakh_s e^{-(phi-phi_0)/p}, bounded smoothly in a smaller chart.
+h_s e^{-(phi-phi_0)/p}, bounded smoothly in a smaller chart.
Minimality implies that its perimeter is minimizing up to a fixed
multiple of the volume of the symmetric difference for compact
variations. The regularity theorem for such almost-minimizing boundaries
@@ -4019,15 +4019,15 @@ center. The common bounds on all derivatives, and the metric and
inverse-metric bounds, make the normalized metric, weight, and forcing
coefficients precompact in C^∞ on every smaller fixed box, by
diagonal Arzelà--Ascoli. Pass to such a convergent subsequence; in
-particular the absorbed metrics widehatwidehat h_jh_j converge smoothly to a
-positive metric widehatwidehat h_∞.h_∞. Ball comparison gives the local
+particular the absorbed metrics hhat_jh_j converge smoothly to a
+positive metric hhat_∞.h_∞. Ball comparison gives the local
perimeter bounds used above, so BV compactness gives a further
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;,·,) and
-mu_∞=P_{widehatwidehat h_∞}h_∞(E;,·,). To prove convergence
+Write mu_j=P_{hhat_j}h_j(E_j;,·,) and
+mu_∞=P_{hhat_∞}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
@@ -4044,7 +4044,7 @@ Here the perimeter of the fixed set E in the varying metric converges
by uniform convergence of the metric integrands on unit conormals.
Shrinking the belt makes the right side tend to mu_∞(V), because
mu_∞(∂ V)=0. Lower semicontinuity gives the reverse
-inequality. Such coordinate continuity domains determine weak local
+ineuality. Such coordinate continuity domains determine weak local
convergence of Radon measures; the local mass bounds give compactness,
and the equality just proved identifies every subsequential limit.
Consequently mu_j converge weakly locally to mu_∞, and their
@@ -4060,24 +4060,24 @@ from a delta-collar of ∂ U. Local weak BV convergence
handles this cutoff test. On the compact chart uniform ellipticity
gives |Dmathbf1_bf1_{E_j}|le Cmu_j. For a compact closed
delta-collar F_delta, weak perimeter convergence and the closed-set
-inequality give
+ineuality give
limsup_jmu_j(F_delta)lemu_∞(F_delta), which tends to zero
as delta↓0 because mu_∞(∂ U)=0. The same
bound for the limiting vector measure lets the collar shrink and proves
the restriction convergence. The perimeter masses on U also converge.
In coordinates the perimeter
integrand for a metric k is
-mathfrak p_k(x,z)=√(frakp_k(x,z)=√(det k(x)),|z|_{k^{-1}(x)}.
+p_k(x,z)=√(p_k(x,z)=√(det k(x)),|z|_{k^{-1}(x)}.
Uniform convergence of this integrand on unit conormals and the local
-mass bound replace widehatwidehat h_jh_j by widehatwidehat h_∞h_∞ in that mass
+mass bound replace hhat_jh_j by hhat_∞h_∞ in that mass
convergence. This is convergence for one continuous positively
one-homogeneous integrand with Reshetnyak's strict convexity property:
-equality in its triangle inequality occurs only for vectors on a common
+equality in its triangle ineuality occurs only for vectors on a common
nonnegative ray. Reshetnyak's continuity theorem
cite[Section~4, Theorem~3]{Reshetnyak1968} consequently gives convergence
for every continuous positively one-homogeneous integrand with linear
growth. For a continuous compactly supported varifold test a(x,P), use
-mathfrak p_frakp_{widehatwidehat h_∞}h_∞(x,z)a(x,ker z) for zne0 and zero
+p_{hhat_∞}h_∞(x,z)a(x,ker z) for zne0 and zero
at z=0. This gives precisely the local varifold convergence; the
uniform convergence of the metric integrands gives it also for the
original varying metrics.
@@ -4086,7 +4086,7 @@ original varying metrics.
parnoindent{footnotesizehttps://doi.org/10.1007/BF02196453}
To prove the support and graph assertions, the first variation in
-the metric widehatwidehat hh is, by BV Gauss--Green, integration of the bounded
+the metric hhhat is, by BV Gauss--Green, integration of the bounded
volume forcing against the reduced-boundary conormal. It therefore has
bounded mean curvature and no singular part. The monotonicity formula
gives a common positive lower mass bound in a small ball about a support
@@ -4096,10 +4096,10 @@ the limit support. Conversely, every ball about a point of the limit
support has positive limit mass, so weak convergence forces nearby
support points. Near a fixed smooth multiplicity-one sheet of the
limit, Allard's interior regularity theorem gives single graphs with a
-common C^{1,alpha_{mathrm A}rmA} bound for some
-alpha_{mathrm A}rmAin(0,1) on a smaller cylinder [cite].
+common C^{1,alpha_{A}rmA} bound for some
+alpha_{A}rmAin(0,1) on a smaller cylinder [cite].
Compactness and identification of the already known varifold limit give
-C^{1,beta} graph convergence for every 0<beta<alpha_{mathrm A}rmA.
+C^{1,beta} graph convergence for every 0<beta<alpha_{A}rmA.
The prescribed equation, with the smoothly converging coefficients,
then gives C^∞ convergence on still smaller cylinders by
interior elliptic estimates. The cylinder here may depend on the fixed
@@ -4165,10 +4165,10 @@ 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,·)-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
+D_t=h_t>h_t>h_s=D_sfrakh_s=D_s at the same point. This is
impossible. Use the conventions for D and II from the
local slice calculation, with nu the exterior unit normal. The
-equation D=mathfrak h_sfrakh_s is the first variation of
+equation D=h_s is the first variation of
(eq:degree-functional) on a regular portion.
Choose countably many possible graph cylinders in the chart enumerations,
@@ -4192,8 +4192,8 @@ In positive graph dimension the w_t solve linear elliptic equations on
each smaller cylinder. This follows by subtracting the two prescribed
graph equations and integrating their linearizations between the two
graphs. Smooth graphical convergence gives common smooth coefficient
-bounds and ellipticity; the explicit t-dependence of mathfrak h_tfrakh_t gives a
-smooth bounded source. Harnack's inequality with a bounded source,
+bounds and ellipticity; the explicit t-dependence of h_t gives a
+smooth bounded source. Harnack's ineuality with a bounded source,
applied using any one of the bounded test values, bounds w_t on a
smaller patch. Interior elliptic estimates then give smoothly convergent
subsequences there; these are the standard local Harnack and interior
@@ -4206,22 +4206,22 @@ component is a smooth function vge0 on the regular portion. For
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.
+Differentiation of D=h_s now identifies this normal speed.
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
+a=∂_sfraka=∂_sh_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
+D-h_s=0frakh_s=0 cancelling, is
- -(Delta_{Σ,phi}+P+∂_calP+∂_numathfrak h_s)vfrakh_s)v
- =mathfrak a.fraka.
+ -(Delta_{Σ,phi}+P+∂_calP+∂_nuh_s)vfrakh_s)v
+ =a.a.
The convention for Delta_{Σ,phi} is weighted by
e^{phi|_Σ}dA_h. The variation of ∂_nuphi adds
vnabla^2phi(nu,nu)-< dphi,dv>_Σ to the ordinary
mean-curvature variation, giving the signs in
(eq:degree-lapse). A nonnegative v cannot vanish at an interior
-point: there the left side is nonpositive, contrary to mathfrak a>0.fraka>0.
+point: there the left side is nonpositive, contrary to a>0.a>0.
Thus v>0. In graph dimension zero the differential terms vanish and
the same equation rules out v=0.
@@ -4247,7 +4247,7 @@ In a graph cylinder, normal speed times area has the expression
v,dA_h=J_h(y,r(s,y)),∂_s r(s,y),dy,
]
where J_h(y,z),dy,dz is ambient Riemannian volume in the coordinates.
-The one-dimensional change-of-variables inequality for a monotone
+The one-dimensional change-of-variables ineuality for a monotone
function says that integration against its almost-everywhere derivative
counts at most integration in z. A jump or singular part of the
monotone function only removes mass from the former integral. Apply this
@@ -4269,11 +4269,11 @@ Require one more consequence of (eq:degree-lapse). On a
family of regular graph patches with common buffers and graph estimates,
all positive-order derivatives of ln v have uniform bounds on smaller
patches. Indeed the coefficients in (eq:degree-lapse) and the
-derivatives of mathfrak afraka are uniformly bounded there, and
-mathfrak afraka has a uniform positive lower bound. Confinement provides
+derivatives of a are uniformly bounded there, and
+a has a uniform positive lower bound. Confinement provides
the finite margin from the pressure poles needed for these bounds. If
the value of v could tend to zero at an interior point of such patches,
-Harnack's inequality with bounded source would bound v on a smaller
+Harnack's ineuality with bounded source would bound v on a smaller
patch, and elliptic compactness would give a nonnegative limit with an
interior zero satisfying (eq:degree-lapse), an impossibility.
Thus v has a positive local lower bound. Harnack then bounds its local
@@ -4285,8 +4285,8 @@ of ln v, which is not required by Definition~def:degree-band.
For p=0 the assertion about positive-order derivatives is empty.
The local slice calculation now applies to the actual boundaries.
-Confinement gives D=mathfrak h_s=P_0(s-G_1)frakh_s=P_0(s-G_1) and hence
-∂_numathfrak h_s=-frakh_s=-mathfrak afraka,dG_1(nu) on every retained
+Confinement gives D=h_s=P_0(s-G_1)h_s=P_0(s-G_1) and hence
+∂_nuh_s=-h_s=-a,dG_1(nu) on every retained
regular portion. Equation~(eq:degree-lapse) therefore becomes
exactly (eq:degree-local-lapse). With
phi_Σ=phi|_Σ+ln v, the scalar bound
@@ -4406,7 +4406,7 @@ relative to the original induced metric. The scalar and volume target is
dV_{h^+}=W,dA_h,
wscal(h^+,phi^+)gewscal(h,phi)-H_1(G)-2ln(v/L_1)
- +2lnW-calW-frac1q.
+ +2lnW-W-frac1q.
The coefficient 2 makes the volume factor cancel after exponentiation,
@@ -4585,7 +4585,7 @@ The constants may depend on the fixed local controls and on p, but are
independent of r_i and of the eventual number of balls. To check the
first estimate before cutoff, put u=dist_h(x,c_i)^2.
The normal-coordinate Hessian bound, the bounded mean curvature
-H=mathfrak h_s-∂_frakh_s-∂_nuphi, and the bounded weight gradient give
+H=h_s-∂_frakh_s-∂_nuphi, and the bounded weight gradient give
[
Delta_{Σ,phi}uge2p-Csqrt u, |du|_Σ^2le4u.
]
@@ -4635,11 +4635,11 @@ delta.
At each point the u_j are the moments of positive atoms: one atom at
ε_{base} of mass one, and atoms at K_i of masses
M_*r_i^alpha.
-Make their total active mass at most two. Moment inequalities give
+Make their total active mass at most two. Moment ineualities give
frac{u_{alpha+6}}{u_{alpha+4}}geΛ^2
gefrac{u_{alpha+2}}{u_alpha},
- |dlnΛ|^2le(L)/(8),calL)/(8),
+ |dlnΛ|^2le(L)/(8),L)/(8),
|dln u_alpha|le C(Λ+1),
Λge c' u_alpha^{1/alpha}.
@@ -4647,17 +4647,17 @@ 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
E Zle(E Z^2)^{1/2} and
-E Z^3ge(E Z^2)^{3/2} give the two inequalities. The
+E Z^3ge(E Z^2)^{3/2} give the two ineualities. 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
|du_alpha| by Csum_iM_*r_i^alpha(K_i^{alpha+1}+1).
The first part divided by u_alpha is at most Λ by H"older's
-inequality for those probability weights; the second is at most one after
+ineuality for those probability weights; the second is at most one after
the same small active-mass choice. Finally, monotonicity of the
L^r norms for a measure of mass at most two gives
u_{alpha+4}/u_alphage(u_alpha/2)^{4/alpha}, proving the last
-inequality. These constants are independent of M_* and k'.
+ineuality. These constants are independent of M_* and k'.
At a point of D_{k'} some ball has distance at most its radius, so its
term in u_alpha is at least c''M_*. Hence
@@ -4674,20 +4674,20 @@ regular boundary. Put Q_1=deta_0. The weight-change formula used in
Delta_{Σ,phi_Σ}=Delta_{Σ,phi}
+< dln v,d,·,>, yields from
(eq:degree-potential-laplacian) the scalar gains
-cΛ^2+L/2-2<calL/2-2< dln v,Q_1>-|Q_1|^2-delta.
+cΛ^2+L/2-2<L/2-2< dln v,Q_1>-|Q_1|^2-delta.
Here the high moment in (eq:degree-moments) supplies Λ^2.
-Since |Q_1|^2leL/8,calL/8,
+Since |Q_1|^2leL/8,L/8,
[
(L)/(2)+|dcalL)/(2)+|dln v|^2
-2< dln v,Q_1>-|Q_1|^2
- ge(L)/(2)-2|Q_1|^2calL)/(2)-2|Q_1|^2ge(L)/(4).calL)/(4).
+ ge(L)/(2)-2|Q_1|^2calL)/(2)-2|Q_1|^2ge(L)/(4).L)/(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+(L)/(4)-calL)/(4)-delta.
+ +cΛ^2+(L)/(4)-L)/(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
@@ -4723,7 +4723,7 @@ Also
dV_{k_f}=L_f,dV_k, L_f-L_f^{-1}geln L_f.
-The last inequality holds for every L_fge1 by differentiation. These
+The last ineuality holds for every L_fge1 by differentiation. These
relations will make the scalar cost and the volume change use the same
factor L_f.
@@ -4745,7 +4745,7 @@ Differentiating the prescribed D_f gives the exact terms
The final term is at least 2ln L_f by
(eq:degree-graph-elementary). For the derivative of C_f, Young's
-inequality and (eq:degree-moments) give
+ineuality and (eq:degree-moments) give
[
2|D_fU_f(ln C_f)|
le0.8D_f^2+1.25|dln C_f|^2
@@ -4761,7 +4761,7 @@ 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 L/4calL/4 covers
-1.25L/8.calL/8. Combining (eq:degree-weight-gain) and
+1.25L/8.L/8. Combining (eq:degree-weight-gain) and
(eq:degree-graph-identity), and discarding nonnegative terms, thus
gives
@@ -4850,11 +4850,11 @@ estimates that will be used for the band sides and its gradients:
|J_j^+|&ge |G_j|-ε_{val}.
-For the gradient inequality,
+For the gradient ineuality,
|df'_j|le1 in its own graph metric and hence in h^+, and all these
metrics dominate the original induced metric and the parent metric on
tangent vectors. Thus
-|dJ_j^+|_{h^+}le|dbarbar J_j|_h+e.J_j|_h+e. For the value inequality use
+|dJ_j^+|_{h^+}le|dbarbar J_j|_h+e.J_j|_h+e. For the value ineuality use
1+Azetage1 in the first line. The quantities
ε_{grad} and ε_{val} can be as
small as prescribed by the first-derivative and value perturbation
@@ -5049,11 +5049,11 @@ On the child the initial auxiliary metric is the induced metric, so
(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-first-graph-gain) give the scalar ineuality in
(eq:degree-child-scalar) on the child.
This completes the construction in high dimensions. In the case
dle7 treated above, use W=1. The volume identity is then
-immediate, and (eq:degree-slice-bound) gives the scalar inequality
+immediate, and (eq:degree-slice-bound) gives the scalar ineuality
in (eq:degree-child-scalar). In dimension p=0 the same statements
hold on the counted points with the zero-dimensional scalar convention.
@@ -5111,7 +5111,7 @@ the child and use its containment in the parent b_d region. This is
Proof. [Proof of Theorem~thm:degree]
-The case d=0 is the signed-count inequality in the statement. Suppose
+The case d=0 is the signed-count ineuality in the statement. Suppose
the theorem is known in dimension p=d-1, with the same q,B,tau and
the residual lengths. Its hypotheses hold for the child of
Lemma~lem:degree-descent, including the unchanged strict horizon
@@ -5123,14 +5123,14 @@ condition. For almost every s the induction hypothesis and
int_{Σ_scap{|G|_∞<b_d}}v_sΘ,dA_h.
]
For p=0 this is the same use of the unsigned-count bound. Take
-C_1ge C_*. Integrate this scalar inequality over -tau<s<tau
+C_1ge C_*. Integrate this scalar ineuality over -tau<s<tau
and apply (eq:degree-coarea) with
F=mathbf1_bf1_{{|G|_∞<b_d}}Θ. The factor 2tau on the
left changes its power from p to d, and the right is bounded by
the integral in (eq:degree-estimate). This proves the induction.
There is no implicit measurable selection of children in this last step.
-For each typical s a child proves the displayed scalar inequality
+For each typical s a child proves the displayed scalar ineuality
against the same parent integral. That parent integral is a measurable
function of s by the lapse construction and is governed by
(eq:degree-coarea); it is the only function being integrated.
@@ -5144,7 +5144,7 @@ constants were used only to make those choices.
The entropy construction supplies a root probability map (F) whose
energy (e_h(F)) has a large coefficient in the weighted scalar-curvature
-inequality. Its coordinatewise square takes values in the probability
+ineuality. Its coordinatewise square takes values in the probability
simplex. Now turn its pairing with the bounded cocycle into comparison
degrees to which Theorem~thm:degree applies. A common integer
clears the rounded cubical coefficients, giving integer multiplicities;
@@ -5158,9 +5158,9 @@ The probability-map and auxiliary-degree strategy has antecedents in
quantitative measured transfer and its estimates here.
-For every integer (ngeq 3), there is a finite constant (C_n>0) such
+For every integer (nge 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
+smooth Riemannian metric (g) with (Scal_geuiv-1) satisfy
[
|M|<= C_nVol_g(M).
]
@@ -5168,11 +5168,11 @@ smooth Riemannian metric (g) with (Scal_gequiv-1) satisfy
Prepare the proof for a manifold with (|M|>0). Choose the
homogeneous alternating bounded cocycle (c), the initial probability
-map, and the number (Igeq|M|/2) provided by
-Proposition~prop:detector; in particular (|c|_∞leq1).
+map, and the number (Ige|M|/2) provided by
+Proposition~prop:detector; in particular (|c|_∞le1).
First consider conditional data having the properties asserted by
Theorem~thm:entropy, without yet invoking their existence.
-Denote the scalar constant in this contract by (C_{rm ent}).
+Denote the scalar constant in this contract by (C_{ent}).
All functions called elementary in this section have a fixed height of
exponentiation on a polynomial, with parameters depending only on (n).
@@ -5180,17 +5180,17 @@ 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
[
- mathfrak e=e_h(F),frake=e_h(F), W=log(dV_h/rho).
+ e=e_h(F),e=e_h(F), W=log(dV_h/rho).
]
Its estimates are
- rholeq dV_g, Wgeq0,
- wscal(h,phi)>=-C_{rm ent}+2W-nlog E+Emathfrak e,frake,
- mathfrak e^frake^{1/2}<= E^{-1/100}.
+ rhole dV_g, Wge0,
+ wscal(h,phi)>=-C_{ent}+2W-nlog E+Ee,e,
+ e^e^{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
-coordinates of (F). The integer (qgeq n+1) bounds the number of
+coordinates of (F). The integer (qge 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
@@ -5198,24 +5198,24 @@ Its dependence on (omegainΩ) is Borel. Set
constructed below include their transverse measures.
medskipnoindentConditional transfer target.
-For every (ngeq3), there are a constant (K_n>0) and an elementary
+For every (nge3), there are a constant (K_n>0) and an elementary
function (Psi_n:[1,∞)to(0,∞)), chosen from (n) alone,
such that every choice of detector and conditional entropy data just
-specified, including its constant (C_{rm ent}), satisfies
+specified, including its constant (C_{ent}), satisfies
- EgeqPsi_n(q)⇒
- Ileq K_n e^{C_{rm ent}/2}int_{M}
+ EgePsi_n(q)⇒
+ Ile K_n e^{C_{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
-estimates for each integer (qgeq n+1), uniformly over conditional
+instance geometry, and (C_{ent}) are chosen. Carry out the
+estimates for each integer (qge n+1), uniformly over conditional
data satisfying the stated label and coloring bounds.
The grid, the common rounding, and an upper density bound for the
resolution come first. That density bound permits a small collar;
the ensuing curvature and rotation bounds then determine one comparison
width and the required energy threshold. Only after the conditional
target is closed will Theorem~thm:entropy supply an actual pair
-((q,E)) and make (C_{rm ent}) dimension-only.
+((q,E)) and make (C_{ent}) dimension-only.
For a fixed (omega), let (K=K_omega) consist of all nonnegative
@@ -5238,7 +5238,7 @@ a standard Borel family in (omega), with countably many charts and an
equivariant (Gamma)-action. The map (F) lifts smoothly to this
family, and, writing (z_F=zcirc F),
[
- e_h(z_F)=e_h(F)=mathfrak e.frake.
+ e_h(z_F)=e_h(F)=e.e.
]
There is a closed smooth (n)-form, again denoted (eta), obtained
in these flat charts by pulling back the detector form through the cutoff
@@ -5290,9 +5290,9 @@ the nonzero rows of (dF) are merely moved to different color positions
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)}|<= C_jdelta^{1-j}) for (jgeq1). In a ball put
+which is zero for (tle12delta), equals (t) for
+(tge25delta), lies between zero and (max{t,0}), and satisfies the usual rescaled derivative bounds
+(|chi^{(j)}|<= C_jdelta^{1-j}) for (jge1). In a ball put
[
w(z)=frac{(chi(z_j))_{j=1}^q}{|(chi(z_j))_{j=1}^q|},
p_j=w_j^2.
@@ -5337,7 +5337,7 @@ already suffice), since (delta^{-1}=1000sqrt q).
Choose a rotation (Vin SO(q)) with Haar probability, independently
-of the entropy data, and put (widehatwidehat z=Vzz=Vz). In these coordinates
+of the entropy data, and put (zhat=Vzz=Vz). In these coordinates
take the cubical grid of step (r) and shift (s), where (s) is
uniform in ([0,r)^q). Fix (r=r(q)>0) so that
(rsqrt qlldelta) and so that every (n)-component of (eta)
@@ -5377,11 +5377,11 @@ facet points by at most (rsqrt q), so they have the stipulated
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
+There is an integer (D_0ge1), 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}Z such that
- E_{rm rd}u'_C=u_C, |u'_C-u_C|<= r^nq^{-n}.
+ E_{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).
@@ -5425,7 +5425,7 @@ ordered list of slots consisting of an anchor representative, an
injective color assignment on its neighborhood, an axis set and grid
index, and a witness clique in that neighborhood containing the anchor.
It is enough to use the fixed index box
-(|k_j|leqlceil3/rrceil+3), which contains every possible selected
+(|k_j|lelceil3/rrceil+3), which contains every possible selected
center for all shifts. The slot is valid when the anchor color is the
canonically chosen largest unrotated color of its center and that center
is at distance less than (4delta) from the spherical orthant of its
@@ -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 (b· v=0). There are
+(|b|_∞le2(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,
@@ -5463,7 +5463,7 @@ bound. The kernel of this
integer matrix has a real basis of integer vectors (b_1,...,b_k)
with
[
- kleq N_0,
+ kle 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
@@ -5478,10 +5478,10 @@ Write (v=sum_j t_jb_j), using the pivot solution, which is Borel.
Independently round each (t_j) to the two adjacent multiples of
(D_0^{-1}), choosing their probabilities so that its conditional
expectation is (t_j). Choose a single integer
-(D_0geq N_0K_0r^{-n}q^n), using uniform elementary upper bounds
+(D_0ge 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}Z, has expectation (v),
-and satisfies (|v'-v|_∞<= N_0K_0/D_0leq r^nq^{-n}).
+and satisfies (|v'-v|_∞<= N_0K_0/D_0le 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
@@ -5493,7 +5493,7 @@ the asserted incidence equations.
The rounded coefficients define the original oriented dual current
- Z_{rm rd}:=sum_C D_0u'_C[C^#].
+ Z_{rd}:=sum_C D_0u'_C[C^#].
Here ([C^#]) is integration over the oriented cube, and the
coefficients are integers. Verify the local finiteness implicit
@@ -5514,12 +5514,12 @@ 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|_{T_}=0.
+ ∂ Z_{rd}big|_{T_}=0.
This is a local assertion: the boundary current vanishes on test forms
compactly supported in (T_
-To resolve (Z_{rm rd}) smoothly, realize each coefficient by
+To resolve (Z_{rd}) smoothly, realize each coefficient by
(|D_0u'_C|) oriented copies of its cube, reversing orientation when
(u'_C<0). At every facet where boundary balance is required, pair
positive and negative incident copies and glue them by their common
@@ -5547,7 +5547,7 @@ Write (ε_sigmain{1,-1}) for its sign relative to that
color order, and call this doubled signed simplicial diagram
(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
+(Z_{rd}) remains the object to be represented on
(T_; the doubling supplies closed facet pairings
for the resolution.
@@ -5578,33 +5578,33 @@ 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 (T_ represented by
original chamber tiles form a Borel leaf-open subset
-(A_{rm orig}subset X^{-1}(T_ For every
+(A_{orig}subset X^{-1}(T_ For every
compactly supported smooth (m)-form (alpha) on (T_,
- int_{A_{rm orig}}X^*alpha=Z_{rm rd}(alpha).
+ int_{A_{orig}}X^*alpha=Z_{rd}(alpha).
For the full doubled diagram, measured oriented integration also gives
- X_*[A]=[calA]=[P].calP].
+ X_*[A]=[A]=[P].P].
For a parameter (0<lambda<1/2) and a choice
0<2gamma_0<(m+1)^{-(m+1)}lambda^{m(m+1)},
there are a smooth metric (k) and a closed seam set in each tile such
-that (|dX|_kleq1), and (k=X^*k_{rm flat}) off the seam.
+that (|dX|_kle1), and (k=X^*k_{flat}) off the seam.
There are buffered smooth charts with elementary overlap, metric,
inverse-metric, and fixed-order derivative bounds whenever
(lambda^{-1}) and (gamma_0^{-1}) are elementary. In the sorted
charts for each tile, the seam is contained in the union of the slabs
-(|Z_j|leqlambda), (1leq jleq m). The (k)-volume density
+(|Z_j|lelambda), (1le jle m). The (k)-volume density
in those charts has an elementary upper bound chosen before
(lambda,gamma_0); in particular this bound does not depend on
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
-(R:calR:A SO(q)) equal to (O_sigma) off the
+(R:R: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
(|dR|_k) are elementarily bounded under the same
@@ -5663,11 +5663,11 @@ of its collection of pair ratios ((x_i/x_j)_{ine j}) in the
product of copies of ([0,∞]). For each permutation
(i_0,...,i_m) use the coordinate chart
- u_j=x_{i_j}/x_{i_{j-1}}in[0,4), 1leq jleq m.
+ u_j=x_{i_j}/x_{i_{j-1}}in[0,4), 1le jle m.
All pair ratios in the chart are products of these or their reciprocals,
so the (u_j) are independent coordinates. Sorted orders have
-(u_jleq1) and cover the closure. The facet (u_j=0) has label
+(u_jle1) and cover the closure. The facet (u_j=0) has label
(S_j={i_0,...,i_{j-1}}); the labels of a corner are nested.
Glue each facet to the identical resolved coordinates in the state
obtained by (s_{S_j}). The commutation just proved makes the gluing
@@ -5766,7 +5766,7 @@ parmedskip
Verify that these are ordinary smooth charts. On an overlap of two permutations,
a cut (S_j) which is not a cut of the other permutation has
-(u_jgeq4^{-2m}). To see this, choose a pair of indices on opposite
+(u_jge4^{-2m}). To see this, choose a pair of indices on opposite
sides of that cut whose order is inverted in the other permutation.
Their ratio in the first ordering is at most (4^m u_j); its inverse
in the second ordering is at most (4^m). The asserted lower bound
@@ -5837,7 +5837,7 @@ order in its signed coordinate. The vertices in (S) are the same
on the paired chambers, and their coefficients are smooth functions
of the squares of the signed coordinates. Thus (X) is smooth
through the gluing. Its denominator is at least (e^{-(m+1)}), since
-some (x_igeq1/(m+1)), which also gives elementary upper derivative
+some (x_ige1/(m+1)), which also gives elementary upper derivative
bounds in the tile charts.
On the positive simplex the map (x|-> b) is a diffeomorphism.
@@ -5861,7 +5861,7 @@ give, for every compactly supported smooth (m)-form (alpha),
=sum_sigmaε_sigmaint_sigmaalpha.
]
This proves (eq:transfer-push). To obtain the local identity,
-define (A_{rm orig}) as the image in
+define (A_{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
@@ -5894,28 +5894,28 @@ equivariant. The same transition and product estimates, applied to
(lambda). The affine simplexes have diameter at most (rsqrt m).
Put (Θ=prod_i x_i). Choose a smooth (j=j(Θ)) equal
-to one for (Θleqgamma_0), zero for
-(Θgeq2gamma_0), and between zero and one elsewhere, with
+to one for (Θlegamma_0), zero for
+(Θge2gamma_0), and between zero and one elsewhere, with
the rescaled smooth cutoff estimates. It is smooth in the signed
charts and the same on both sides of every facet. Set
- k=X^*k_{rm flat}+j^2 k_0,
- seam={Θleq2gamma_0}.
+ k=X^*k_{flat}+j^2 k_0,
+ seam={Θle2gamma_0}.
Call the complement of this closed seam the straight part.
The metric is positive definite near the tile boundary because (j=1) there. Where
-(j<1), one has (Θ>gamma_0), hence (x_igeqgamma_0) for
+(j<1), one has (Θ>gamma_0), hence (x_igegamma_0) for
every (i), and (X) is an immersion. The formula proves
-(|dX|_kleq1) and equality of metrics off the seam.
+(|dX|_kle1) and equality of metrics off the seam.
The upper density bound requires no inverse estimates. As quadratic forms,
-(kleq X^*k_{rm flat}+k_0), so its volume density in every sorted
+(kle X^*k_{flat}+k_0), so its volume density in every sorted
box is bounded by the upper bounds for (dX,k_0) already chosen.
Call a common elementary bound (H_0(q)). In a sorted box
-(|Z_j|leq1). If all (|Z_j|>lambda), normalization of the
+(|Z_j|le1). If all (|Z_j|>lambda), normalization of the
successive products gives
[
- Θgeq(m+1)^{-(m+1)}
+ Θge(m+1)^{-(m+1)}
lambda^{2(1+2+...+m)}
=(m+1)^{-(m+1)}lambda^{m(m+1)}.
]
@@ -5929,7 +5929,7 @@ with (j<1), all positive ratios are bounded below by powers of
(gamma_0), so the product formulas control the inverse from raw
barycentric to signed coordinates. The normalized weights in
(eq:transfer-tile-map) obey
-(b_igeq e^{-1/gamma_0}/(m+1)). On a tangent vector
+(b_ige e^{-1/gamma_0}/(m+1)). On a tangent vector
(w=(w_i)), (sum_i w_i=0), their derivative is
[
db_i(w)=b_i((w_i)/(x_i^2)
@@ -5964,7 +5964,7 @@ tile
]
It is the identity in a neighborhood of each facet and thus glues
smoothly despite the change of (sigma). It equals (O_sigma)
-where (j=0), and replacing (1-j) by (t(1-j)), (0leq tleq1),
+where (j=0), and replacing (1-j) by (t(1-j)), (0le tle1),
gives the asserted homotopy. The cutoff and metric inverse estimates
give the derivative bounds. The choices can be made on the indexed
chamber data, so they are Borel and equivariant as asserted.
@@ -5981,21 +5981,21 @@ not affect any row norm or any cubical test.
The parameter selection below fixes (lambda(q)) and (gamma_0(q))
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
+envelopes (K_R(q)) and (K_{scal}(q)) such that
[
|dR|_{∞,k}<= K_R(q),
- (-Scal_k)_+<= K_{rm scal}(q)
+ (-Scal_k)_+<= K_{scal}(q)
]
for every allowed diagram and every (V,s,omega), rounding parameter,
and state. These envelopes come from the uniform chart derivative
-bounds and do not use (C_{rm ent}). Fix the single number
+bounds and do not use (C_{ent}). Fix the single number
tau=tau(q)=
frac{1}{300sqrt qmax{r^{-1},delta^{-1},1+K_R(q)}}.
It has an elementary reciprocal and is common to all those parameters.
In particular
-(3tausqrt qleqmin{r/100,delta/100,(1+K_R(q))^{-1}}).
+(3tausqrt qlemin{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 (A over
@@ -6003,9 +6003,9 @@ their shared (omega), and divide by the diagonal (Gamma)-action.
Orient the product with the (widetilde M) directions first and the
(A directions second. The resulting measured leaves have
dimension (n+m=q). Using the leaf-open set
-(A_{rm orig}) from Lemma~lem:transfer-resolution,
+(A_{orig}) from Lemma~lem:transfer-resolution,
work only on the open set of pairs ((x,a)) with
-(ainA_{rm orig}) for which (X(a)) is in the lift of the ball chart of
+(ainA_{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)=R(a)(z_F(x)-z_calR(a)(z_F(x)-z_{X(a)}).
@@ -6025,14 +6025,14 @@ 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}D=D_0I
+ E_{rd}D=D_0I
for every fixed (V,s).
Proof.
The local pushforward (eq:transfer-local-push) will express the
-degree using a smoothed form of (Z_{rm rd}). Its primal-cell periods
+degree using a smoothed form of (Z_{rd}). Its primal-cell periods
are the integer coefficients (D_0u'_C). Their expectations match the periods of
(D_0eta); a compatible piecewise cubical primitive and mollification
then give the same measured pairing with (F). First verify the
@@ -6053,7 +6053,7 @@ transverse measures and orientations.
For an inner closed comparison cube there is a uniform margin from
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
+(|z_F-z_X|_2le3tausqrt qlldelta/4). The large witness
color at (F) remains present with a uniform small buffer. In a
flat ball chart a lifted path whose projected coordinates stay in
the ball stays in that same lift, by uniqueness of lifts for a local
@@ -6095,7 +6095,7 @@ degree.
After removing the rotation, fiber integration of
((z,w)|-> z-w) against (beta) and the original current
-(Z_{rm rd}) gives a smooth (n)-form (alpha_{rm rd}) near (K).
+(Z_{rd}) gives a smooth (n)-form (alpha_{rd}) near (K).
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
@@ -6103,10 +6103,10 @@ and uses the original side. By the local identity
(eq:transfer-local-push), Fubini, and change of variables on tile
interiors, the degree is
- D={M}*alpha_{rm rd}.
+ D={M}*alpha_{rd}.
Boundary cancellation of the dual cubes shows that
-(dalpha_{rm rd}=0) on, for example, the tube of radius
+(dalpha_{rd}=0) on, for example, the tube of radius
(delta/8) around (K). To verify this directly, apply fiber
Stokes to each cube. Its boundary terms only involve facets within
the support distance of the point (z); these lie in the balanced
@@ -6115,7 +6115,7 @@ tube and cancel with their paired incidence signs.
For every primal (n)-cell sufficiently near (K) in the same
sheet,
- int_Calpha_{rm rd}=D_0u'_C.
+ int_Calpha_{rd}=D_0u'_C.
Indeed a primal (n)-cell and a complementary dual (m)-cell of
this grid meet only when their indices match, and then at their common
@@ -6137,7 +6137,7 @@ one, proving (eq:transfer-convolution-cells). The small support
and the chart buffers ensure that only the indicated lifts contribute.
medskipnoindentThe detector pairing.
-Set (alpha=E_{rm rd}alpha_{rm rd}-D_0eta). It is a smooth
+Set (alpha=E_{rd}alpha_{rd}-D_0eta). It is a smooth
closed form on that tube, and (eq:transfer-rounding),
(eq:transfer-cell-value), and
(eq:transfer-convolution-cells) give (int_Calpha=0) on all
@@ -6203,7 +6203,7 @@ output. Choose a positive constant (b_*), depending only on (n),
whose smallness will be fixed in the estimate below, and put
L_T=b_*sqrt E, L_N=b_*√(qE),
- h_{rm tot}=h+L_T^2k.
+ h_{tot}=h+L_T^2k.
Once the degree hypotheses are verified, its length denominator, the
product volume, and the entropy contribution will combine as
@@ -6218,12 +6218,12 @@ while the collar volume controls the seam.
Use the weight (phi), constant in the tile directions. The
product scalar formula and the definition of (wscal) give
-(wscal(h_{rm tot},phi)=wscal(h,phi)+L_T^{-2}Scal_k).
-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_{tot},phi)=wscal(h,phi)+L_T^{-2}Scal_k).
+By the definition of (K_{scal}(q)), the lower bound
+(Ege b_*^{-2}K_{scal}(q)) ensures
+(L_T^{-2}Scal_kge-1). By (eq:transfer-entropy),
- wscal(h_{rm tot},phi)>=-C_{rm ent}-1+2W-nlog E+Emathfrak e.frake.
+ wscal(h_{tot},phi)>=-C_{ent}-1+2W-nlog E+Ee.e.
Assign (L_i=L_T) to the first (m) comparison rows and
@@ -6231,27 +6231,27 @@ Assign (L_i=L_T) to the first (m) comparison rows and
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
(dR term has norm at most
-(3tausqrt q|dR|_kleq1) by
+(3tausqrt q|dR|_kle1) 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}}
+ L_i|dG_i|_{h_{tot}}
<= C_n(sqrt q+√(qE),E^{-1/100}).
]
Take (B) to be this upper bound increased by two. Then
(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),
+it suffices to test the smallest length (L_T), and, for (Ege1),
[
frac{C(1+B)^{2/3}}{L_T}
- <= c_{{rm hor},n}q^{1/3}E^{-13/75}
+ <= c_{{hor},n}q^{1/3}E^{-13/75}
<(tau)/(10q).
]
-Here (c_{{rm hor},n}) depends only on the absolute horizon constant
+Here (c_{{hor},n}) depends only on the absolute horizon constant
and the dimensional choice of (b_*), and
(13/75=1/2-(2/3)(1/2-1/100)>0). Thus the displayed strict horizon
-inequality follows if
-(Egeq(20c_{{rm hor},n}q^{4/3}/tau)^{75/13}).
+ineuality follows if
+(Ege(20c_{{hor},n}q^{4/3}/tau)^{75/13}).
Therefore apply Theorem~thm:degree
with (d=q).
@@ -6261,29 +6261,29 @@ normal to its dual cube. Put (A_i=(Vdz_F)_i), with covector norms
measured by (h), and define
U_{I}
- frac q{mathfrak e}frakesum_{iinI}|A_i|_h^2
- quadif mathfrak e>0,frake>0,
- U_{I} mathfrak e=0.frake=0.
+ frac q{e}esum_{iinI}|A_i|_h^2
+ quadif e>0,e>0,
+ U_{I} e=0.e=0.
-Here (sum_i|A_i|^2=mathfrak efrake). At a straight point
-(k=X^*k_{rm flat}) and (R=Q_ V) is constant.
+Here (sum_i|A_i|^2=e). At a straight point
+(k=X^*k_{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
- <= L_T^2mathfrak e=b_*^2Efrake=b_*^2Emathfrak e,frake,
+ sum_{tangent}H_i
+ <= L_T^2e=b_*^2Efrake=b_*^2Ee,e,
]
by (2(√(1+t)-1)<= t). Normal rows have no tile derivative,
so Cauchy--Schwarz gives
[
- sum_{rm normal}H_i
- leq2L_Nsum_{iinI}|A_i|
- leq2b_*sqrt n√(Emathfrak e)frake),U_{I}
+ sum_{normal}H_i
+ le2L_Nsum_{iinI}|A_i|
+ le2b_*sqrt n√(Ee)e),U_{I}
]
-Completing the square in (√(Emathfrak e)frake)), for (b_*<1),
+Completing the square in (√(Ee)e)), for (b_*<1),
shows that
- (sum_iH_i-Emathfrak e)/(2)frake)/(2)
+ (sum_iH_i-Ee)/(2)e)/(2)
<= c_n U_{I},
c_n=(n b_*^2)/(2(1-b_*^2)).
@@ -6299,12 +6299,12 @@ moments satisfy
[
E(q u^2)^j
=(q^j(1·3...(2j-1)))/(q(q+2)...(q+2j-2))
- leq2^j j!.
+ le2^j j!.
]
-Consequently (E e^{tqu^2}<=(1-2t)^{-1}) for (0leq t<1/2).
+Consequently (E e^{tqu^2}<=(1-2t)^{-1}) for (0le 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
+(q|A_i|^2/e) as a convex combination of such random
+variables, with its singular-value weights. Jensen's ineuality
gives the same exponential bound for that convex combination.
Another application of Jensen to the sum of the (n) rows in
(eq:transfer-U) bounds its exponential moment by
@@ -6313,45 +6313,45 @@ Another application of Jensen to the sum of the (n) rows in
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
+which is still independent of the instance. By the triangle ineuality
for each row, ((a-1)_+<= a), and Cauchy--Schwarz on the horizontal
rows,
[
- sum_iH_ileq 2L_N√(qmathfrak e)frake)
+ sum_iH_ile 2L_N√(qe)e)
+4q L_N/L_T
- leq2b_*q√(Emathfrak e)+4q^frake)+4q^{3/2}.
+ le2b_*q√(Ee)+4q^e)+4q^{3/2}.
]
Completing the square once more gives
- (sum_iH_i-Emathfrak e)/(2)frake)/(2)<= c_{{rm seam},n}q^2,
- c_{{rm seam},n}=2+(b_*^2)/(2).
+ (sum_iH_i-Ee)/(2)e)/(2)<= c_{{seam},n}q^2,
+ c_{{seam},n}=2+(b_*^2)/(2).
-Apply the degree inequality, then use
+Apply the degree ineuality, then use
(eq:transfer-total-scalar). Its loss is (exp(Cq/q)), an
absolute constant. The product volume form is
-(dV_{h_{rm tot}}=L_T^m dV_h,dV_k), so the measure factor is
+(dV_{h_{tot}}=L_T^m dV_h,dV_k), so the measure factor is
exactly (eq:transfer-prefactor). Together with
-(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:
+(eq:transfer-degree) and (|ED|D|leE|D|D|),
+this gives the following ineuality, with
+(C_{out}=exp(C_{deg}+(C_{ent}+1)/2)b_*^{-n}), where
+(C_{deg}) is the absolute exponent in Theorem~thm:degree:
- D_0I(2tau)^qleq C_{rm out}q^{-n/2}
+ D_0I(2tau)^qle C_{out}q^{-n/2}
int_{M},
E_{V,s,rm rd}
int_{{a: |G(x,a)|_∞<b_q}}
- B(x,a)calB(x,a),dV_k(a),
+ B(x,a)B(x,a),dV_k(a),
where the fiber integral includes its state transversals and only the
compatible original-side band, and
[
- B(x,a)=calB(x,a)=
+ B(x,a)=B(x,a)=
exp(c_nU_{I},&a straight over axes I,
- exp(c_{{rm seam},n}q^2),&a in the seam.
+ exp(c_{{seam},n}q^2),&a in the seam.
]
The entropy constant enters only through this final prefactor, not the
@@ -6377,14 +6377,14 @@ where the component is in the (V)-rotated frame.
On a straight tile the (k)-volume is exactly the flat volume of
its image in the chamber. The signed permutation in (G) does not
change its cube test, which is
-(|widehatwidehat z_y-z_y-widehatwidehat z_X|_∞<b_qz_X|_∞<b_q). Summing the chamber
+(|zhat_y-z_y-zhat_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 (I, summing over
all projected centers and averaging the shift gives exactly
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
+ indic_{{|zhat_y-c-t|_∞<b_qz_y-c-t|_∞<b_q}},dt,dc
=r^{-q}r^m(2b_q)^q.
Here (t) is placed in the complementary axes. This identity is
@@ -6396,7 +6396,7 @@ 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_{|I|=n}calI|=n
+ E_Vsum_{|I|=n}I|=n
(|eta_{I}|+2q^{-n})e^{c_nU_{I}}
&<=
(E_Vsum_{I}|eta_{I}|^2)^{1/2}
@@ -6412,20 +6412,20 @@ straight part of the inner expectation in
(eq:transfer-average) is at most
(C_nD_0q^{n/2}(2b_q)^q), because (r^{n+m-q}=1). Multiplying
by its prefactor (q^{-n/2}) bounds that part, for a fixed
-dimension-only constant (C_{{rm str},n}), by
+dimension-only constant (C_{{str},n}), by
- C_{{rm str},n}D_0(2tau)^q,
+ C_{{str},n}D_0(2tau)^q,
-since ((b_q/tau)^q=(1+3/(4q))^qleq e^{3/4}).
+since ((b_q/tau)^q=(1+3/(4q))^qle e^{3/4}).
For the seam use only the consequence
-(|z_X-z_y|_2leq b_qsqrt q) of the cube test. Discard the tile
+(|z_X-z_y|_2le b_qsqrt q) of the cube test. Discard the tile
boundaries, which have leafwise measure zero. The remaining set is
the disjoint union of tile interiors indexed by their actual
chamber (sigma) and its state arrays. Partition each such interior
into positive sorted-coordinate pieces, for example by taking the
first sorted permutation in a fixed order. On these pieces
-(0<Z_jleq1), so no signed-chart reflection changes the chamber.
+(0<Z_jle1), so no signed-chart reflection changes the chamber.
The conditional state measure over each actual chamber has mass one,
and its (k)-density is at most the elementary (H_0(q)) chosen
before (lambda).
@@ -6443,8 +6443,8 @@ centers and chamber shapes.
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_{I}
- +d_{I,shape,flag}calI,shape,flag(Z).
+ zhat_X=s+r(k+z_X=s+r(k+1/2indic_{I}
+ +d_{I,shape,flag}I,shape,flag(Z).
]
Here (indic_{I} is the coordinate indicator vector of
(I
@@ -6455,7 +6455,7 @@ the shift integral of the rough test is consequently
[
frac1{r^q}int_{[0,r)^q}sum_{kinZ^q}
- indic_{{|s+r(k+frac12indic_{I}+d(Z)-widehatwidehat z_y|_2z_y|_2
+ indic_{{|s+r(k+frac12indic_{I}+d(Z)-zhat_y|_2z_y|_2
<= b_qsqrt q}},ds
&=r^{-q}Vol_{R^q}{b_qsqrt q})
&<= r^{-q}(4tausqrt q)^q.
@@ -6467,7 +6467,7 @@ the enlargement to all centers. Tonelli's theorem applies to these
nonnegative bounds.
In each positive sorted piece the seam lies in
-(∪_j{Z_jleqlambda}), of coordinate volume at most
+(∪_j{Z_jlelambda}), of coordinate volume at most
(mlambda). A cube has (2^m m!) barycentric top chambers and
at most ((m+1)!) sorted flags per chamber. Thus the seam part of
the inner expectation is bounded by
@@ -6478,33 +6478,33 @@ where one may take the elementary envelope
[
C_*(q)=max{1,,
C_{eta,n}H_0(q)binom qn(2^m m!)(m+1)!,m,
- r^{n-q}(2sqrt q)^qexp(c_{{rm seam},n}q^2)}.
+ r^{n-q}(2sqrt q)^qexp(c_{{seam},n}q^2)}.
]
Every factor here was bounded before choosing (lambda). Thus
(C_*) is independent of (lambda,gamma_0,tau), of the bounds
-(K_R,K_{rm scal}), and of (C_{rm ent}). The powers (tau^q) in the shift
+(K_R,K_{scal}), and of (C_{ent}). The powers (tau^q) in the shift
count have already canceled those in ((2tau)^q).
The two fiber estimates now have the same normalization. Insert
(eq:transfer-straight-average) and
(eq:transfer-seam-average) into (eq:transfer-average),
-using (q^{-n/2}leq1) for the seam, and cancel the positive factor
+using (q^{-n/2}le1) for the seam, and cancel the positive factor
(D_0(2tau)^q). For the conditional data and parameter bounds used
so far, this gives
- Ileq C_{rm out}(C_{{rm str},n}+C_*(q)lambda)
+ Ile C_{out}(C_{{str},n}+C_*(q)lambda)
int_{M}
Now fix these bounds before invoking the entropy theorem. For each
-integer (qgeq n+1), the form derivative bound first gives a
+integer (qge n+1), the form derivative bound first gives a
deterministic (r(q)) with elementary reciprocal. The size bound for
the universal type list, (D_0), incidence counts, and the bounds for
(k_0,dX), including (H_0(q)), have elementary envelopes uniform
over all permitted graphs and cocycles. Fix (b_*) using
(eq:transfer-sphere); this determines the dimensional constants
-(c_{{rm seam},n}) and (c_{{rm hor},n}), independently of
-(C_{rm ent}). The displayed formula for (C_*(q)) is consequently
+(c_{{seam},n}) and (c_{{hor},n}), independently of
+(C_{ent}). The displayed formula for (C_*(q)) is consequently
available at this stage. Set
[
lambda(q)=frac1{4C_*(q)},
@@ -6513,19 +6513,19 @@ available at this stage. Set
These deterministic choices have elementary reciprocals and satisfy
(eq:transfer-gamma). The resolution estimates now give common
elementary envelopes for its fixed-order chart derivatives, hence
-(K_R(q),K_{rm scal}(q)), and then the common (tau(q)) in
+(K_R(q),K_{scal}(q)), and then the common (tau(q)) in
(eq:transfer-tau). None of these choices uses the entropy constant.
-For integer (jgeq n+1), define
+For integer (jge n+1), define
[
- Psi_n^circ(j)=max{1, b_*^{-2}K_{rm scal}(j),
- (frac{20c_{{rm hor},n}j^{4/3}}{tau(j)})^{75/13}
+ Psi_n^circ(j)=max{1, b_*^{-2}K_{scal}(j),
+ (frac{20c_{{hor},n}j^{4/3}}{tau(j)})^{75/13}
},
]
and define a majorant on all (tin[1,∞)) by
[
Psi_n(t)=1+max_{substack{jinZ
- n+1leq jleqmax{n+1,lceil trceil}}}
+ n+1le jlemax{n+1,lceil trceil}}}
Psi_n^circ(j).
]
Every expression used here has an elementary upper bound; the finite
@@ -6533,13 +6533,13 @@ maximum is bounded by an increasing elementary envelope at
(max{n+1,lceil trceil}). Thus (Psi_n) is an elementary
majorant fixed solely from (n), and it dominates both the curvature
and strict horizon requirements. Hence every conditional datum with
-(EgeqPsi_n(q)) satisfies all hypotheses used in
+(EgePsi_n(q)) satisfies all hypotheses used in
(eq:transfer-before-choices). Our choice
-(C_*(q)lambda(q)=1/4) and the definition of (C_{rm out}) prove
+(C_*(q)lambda(q)=1/4) and the definition of (C_{out}) prove
(eq:transfer-conditional-target) with
[
- K_n=e^{C_{rm deg}+1/2}b_*^{-n}
- (C_{{rm str},n}+1/4).
+ K_n=e^{C_{deg}+1/2}b_*^{-n}
+ (C_{{str},n}+1/4).
]
This closes the conditional target for all the specified data before
any entropy family is chosen.
@@ -6554,19 +6554,19 @@ the detector map to (Ω) and embedding its labels equivariantly
among the output's finite copies, the measured affine probability homotopy
of Section~2 preserves (I); the output therefore supplies the conditional
data with (E>Psi_n(q)). Its constant
-(C_{rm ent}=C_{rm ent}(n,Psi_n)) depends only on (n). Since
-(rholeq dV_g) and the transverse measure over (M) is a
+(C_{ent}=C_{ent}(n,Psi_n)) depends only on (n). Since
+(rhole dV_g) and the transverse measure over (M) is a
probability measure,
[
- Ileq K_ne^{C_{rm ent}/2}int_{M}
- <= K_ne^{C_{rm ent}/2}Vol_g(M).
+ Ile K_ne^{C_{ent}/2}int_{M}
+ <= K_ne^{C_{ent}/2}Vol_g(M).
]
-Finally (Igeq|M|/2), so the constant
-(C_n=2K_ne^{C_{rm ent}/2}) proves the theorem.
+Finally (Ige|M|/2), so the constant
+(C_n=2K_ne^{C_{ent}/2}) proves the theorem.
Proof. [Proof of Theorem~thm:main]
-Fix (ngeq3), and let (C_n>0) be the constant for the normalized
+Fix (nge3), and let (C_n>0) be the constant for the normalized
scalar-curvature estimate in Theorem~thm:scalar-volume.
Proposition~prop:yamabe-reduction gives
[
@@ -6630,7 +6630,7 @@ href{https://doi.org/10.1007/978-3-642-61798-0}
M. Gromov, Volume and bounded cohomology,
Publications Math'ematiques de l'IH'ES 56 (1982), 5--99.
-href{https://www.numdam.org/item/PMIHES_1982__56__5_0/}
+href{./pdf/unknown-nd-href.pdf}
{Numdam: PMIHES_1982__56__5_0}.
@@ -6718,7 +6718,7 @@ N. S. Trudinger, Remarks concerning the conformal deformation of
Riemannian structures on compact manifolds,
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (3)
22 (1968), no. 2, 265--274.
-href{https://www.numdam.org/item/ASNSP_1968_3_22_2_265_0/}
+href{./pdf/unknown-nd-href.pdf}
{Numdam: ASNSP_1968_3_22_2_265_0}.