diff f6bf760..6fbd3ac microstates_free_entropy_separation_v2.txt
diff --git a/microstates_free_entropy_separation_v2.txt b/microstates_free_entropy_separation_v2.txt
index 997a807..c6b0d60 100644
--- a/microstates_free_entropy_separation_v2.txt
+++ b/microstates_free_entropy_separation_v2.txt
@@ -21,13 +21,13 @@ Online (publisher): https://doi.org/10.1007/BF01231539
Online (publisher): https://doi.org/10.1007/s002220050222
Online (publisher): https://doi.org/10.1112/S0024609301008992
Local PDF: ./pdf/p_guionnet-survey.pdf
-Online: https://www.emis.mi.sanu.ac.rs/EMIS/journals/PS/images/getdoc1af6.pdf?id=48&article=18&mode=pdf
+Online: ./pdf/unknown-nd-online.pdf
-The inequality chi(X) <= chi*(X) holds for every bounded self-adjoint
+The ineuality chi(X) <= chi*(X) holds for every bounded self-adjoint
tuple. Biane, Capitaine, and Guionnet proved it via large deviation
bounds for Hermitian matrix-valued Brownian motion; Jekel and Pi gave an
elementary proof via classical entropy and Fisher information, and
-extended the inequality to conditional free entropy. Equality chi =
+extended the ineuality to conditional free entropy. Equality chi =
chi* holds in several regular settings: Dabrowski's theorem covers a
class of strictly convex potentials with Lipschitz derivatives and
noncommutative polynomial approximation; Jekel's theorem covers convex
@@ -37,11 +37,11 @@ exploited below.
Online (publisher): https://doi.org/10.1007/s00222-002-0281-4
Local PDF: ./pdf/p_jekelpi.pdf
-Online: https://arxiv.org/pdf/2305.02574
+Online: ./pdf/unknown-nd-2305.02574-online.pdf
Local PDF: ./pdf/p_dabrowski.pdf
-Online: https://arxiv.org/pdf/1604.06420
+Online: ./pdf/unknown-nd-1604.06420-online.pdf
Local PDF: ./pdf/p_jekel.pdf
-Online: https://arxiv.org/pdf/1805.08814
+Online: ./pdf/unknown-nd-1805.08814-online.pdf
Finiteness of chi(X) does not force equality. Jekel and Pi recall
nonapproximable laws with chi = -infinity < chi*. Guionnet formulated
@@ -123,7 +123,7 @@ infinite when conjugate variables fail to exist. That conjugate
variables are self-adjoint and unique in L^2 is standard.
Local PDF: ./pdf/p_nica-speicher.pdf
-Online: https://rolandspeicher.files.wordpress.com/2020/06/nica-speicher-book.pdf
+Online: ./pdf/unknown-nd-online.pdf
Lemma 1 (Variational formula for Phi*). If Z has conjugate variables
xi_i in L^2, then
@@ -163,7 +163,7 @@ For a semicircular family of variances w_i the bracket vanishes and
chi* reduces to (n/2) sum_i log(2 pi e w_i), matching the free entropy
of that family. For general Z the integral is finite on every compact
interval of positive noise times once Phi*(Z + sqrt(s) S) <= n/s, an
-inequality proved below (Lemma 3).
+ineuality proved below (Lemma 3).
Matrix conventions. Write tr_d = d^{-1} Tr and
||D||_{2,d}^2 = sum_i tr_d(D_i^* D_i) for a matrix tuple D. A standard
@@ -255,7 +255,7 @@ algebras; and that a standard semicircular family has only second
cumulants, given by its covariance. All three are in Nica-Speicher.
Local PDF: ./pdf/p_nica-speicher.pdf
-Online: https://rolandspeicher.files.wordpress.com/2020/06/nica-speicher-book.pdf
+Online: ./pdf/unknown-nd-online.pdf
Lemma 3 (Wick testing). Let C be a centered bounded self-adjoint
n-tuple, S a standard semicircular n-tuple free from C, and
@@ -409,7 +409,7 @@ GUE n-tuple and select a subsequence satisfying successively more
moment tests with a single fixed operator-norm bound. The joint
limiting moments define a positive tracial functional on the variables
(C, S) with the asserted freeness. In its GNS representation, left
-multiplication by C_i is bounded because the matrix inequality
+multiplication by C_i is bounded because the matrix ineuality
tr_d(Q^* D_i^2 Q) <= R_i^2 tr_d(Q^* Q) passes to the limit. The GNS
vacuum is cyclic for right polynomial multiplication, which commutes
with the weak closure of the left algebra, so it is separating for that
@@ -417,7 +417,7 @@ closure and its vector state is faithful and normal. Faithfulness and
the semicircular distribution give ||B_i|| = ||S_i|| = 2.
Local PDF: ./pdf/p_nica-speicher.pdf
-Online: https://rolandspeicher.files.wordpress.com/2020/06/nica-speicher-book.pdf
+Online: ./pdf/unknown-nd-online.pdf
Lemma 4 (Cumulant bound). With K = 128 and C as above, for every
m >= 1,
@@ -480,7 +480,7 @@ An explicit choice is p = 2^64 and t = 2^48: then 2 K^2 sqrt(p) = 2^47
is less than t; 32 t / p = 2^{-11}; and 2 + t < 2^49 gives
2^64 / (256 (2 + t)) > 2^64 / 2^57 = 128, while
128 sqrt(2 + t) < 2^32 makes the right-hand side of the third
-inequality at most 67 log 2 + 8 < 55. From this point p, h, t are
+ineuality at most 67 log 2 + 8 < 55. From this point p, h, t are
fixed; h will be chosen small next; only matrix dimensions tend to
infinity.
@@ -643,7 +643,7 @@ trace give
= t p sum_l (r_l/d)^2 <= t p * p * (2/p)^2 = 4 t.
Markov at squared threshold p/16 bounds failure by 64 t/p < 1/4. On
-success, the triangle inequality and (18) yield
+success, the triangle ineuality and (18) yield
||D_P (A + sqrt(t) G)_[p]||_{2,d}
>= sqrt(p/2) - sqrt(p)/4 > sqrt(p)/4.
@@ -666,7 +666,7 @@ real (2 d^2)-dimensional matrix space, giving cardinality at most
(1 + 2/delta)^{2 d^2}.
Local PDF: ./pdf/p_szarek.pdf
-Online: https://arxiv.org/pdf/math/9701213
+Online: ./pdf/unknown-nd-math9701213-online.pdf
Compression and reduction. Every decomposition of a fixed rank list
(including zero ranks) is a unitary conjugate of the standard