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ircam
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ideo
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vier
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mat
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tere
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ruct
ures
“C
ateg
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appr
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ishe
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ricM
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mov
, « In
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earc
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pre
prin
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ly 2
012
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ww
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ov/P
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ctre
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ch-
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etric
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n En
trop
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resc
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g, b
eing
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on-tr
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met
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s a
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oup
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esca
lings
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ads
the
amaz
ing
orth
ogon
al s
ymm
etry
of t
he F
ishe
r met
ric
M. G
rom
ov, «
Con
vex
sets
and
Käh
ler m
anifo
lds»
,in
Adv
ance
s in
J. D
iffer
entia
l Geo
m.,
F. T
ricer
ri ed
., W
orld
Sci
., Si
ngap
ore,
pp.
1-3
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H. S
him
a, “
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Geo
met
ry o
f Hes
sian
Str
uctu
res”
, Wor
ld
Scie
ntifi
c, 2
007
http
://w
ww
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ldsc
ient
ific.
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ldsc
iboo
ks/1
0.11
42/6
241
dedi
cate
d to
Pro
f. Je
an-L
ouis
Kos
zul (
«th
e co
nten
t of t
he
pres
ent b
ook
finds
thei
r orig
in in
his
stu
dies
»)H
. Shi
ma,
Sym
met
ric s
pace
s w
ith in
varia
nt lo
cally
Hes
sian
st
ruct
ures
, J. M
ath.
Soc
. Jap
an,,
pp. 5
81-5
89.,
1977
H. S
him
a, «
Hom
ogen
eous
Hes
sian
man
ifold
s»,
Ann
. Ins
t. Fo
urie
r, G
reno
ble,
pp.
91-
128.
, 198
0H
. Shi
ma,
«Va
nish
ing
theo
rem
s fo
r com
pact
Hes
sian
m
anifo
lds
», A
nn. I
nst.
Four
ier,
Gre
nobl
e, p
p.18
3-20
5., 1
986
H. S
him
a, «
Har
mon
icity
of g
radi
ent m
appi
ngs
of le
vel s
urfa
ces
in a
real
a� ���
ne s
pace
», G
eom
etria
e D
edic
ata,
pp.
177
-184
., 19
95H
. Shi
ma,
«H
essi
an m
anifo
lds
of c
onst
ant H
essi
an s
ectio
nal
curv
atur
e»,
J. M
ath.
Soc
. Jap
an, p
p. 7
35-7
53.,
1995
H. S
him
a, «
Hom
ogen
eous
spa
ces
with
inva
riant
pro
ject
ivel
y fla
t a� ���
ne c
onne
ctio
ns»,
Tra
ns. A
mer
. Mat
h. S
oc.,
pp. 4
713-
4726
, 19
99
Hiro
hiko
Shim
a, E
mer
itus
Prof
esso
r of Y
amag
uchi
Uni
v.Ph
D fr
om O
saka
Uni
vers
ityIn
terp
lay
betw
een
the
Geo
met
ry o
f Hes
sian
Str
uctu
res
and
Info
rmat
ion
Geo
met
ry
Jean
-Lou
is K
oszu
l, Fr
ench
Sci
ence
s Ac
adem
yPh
Dst
uden
tof H
enri
Car
tan,
Bou
rbak
i mem
ber
Intr
oduc
tion
of K
oszu
lfor
ms,
Kos
zul-V
inbe
rgch
arac
teris
ticfu
nctio
n&
met
ricJ.
L. K
oszu
l, «
Sur l
a fo
rme
herm
itien
ne c
anon
ique
des
es
pace
s ho
mog
ènes
», c
ompl
exes
, Can
ad. J
. Mat
h. 7
, pp
. 562
-576
., 19
55J.
L. K
oszu
l, «
Dom
aine
s bo
rnée
s ho
mog
ènes
et
orbi
tes
de g
roup
es d
e tr
ansf
orm
atio
ns a
ffine
s»,
Bul
l. So
c. M
ath.
Fra
nce
89, p
p. 5
15-5
33.,
1961
J.L.
Kos
zul,
«O
uver
ts c
onve
xes
hom
ogèn
es d
es
espa
ces
affin
es»,
Mat
h. Z
. 79,
pp.
254
-259
., 19
62J.
L. K
oszu
l, «
Varié
tés
loca
lem
ent p
late
s et
co
nvex
ité»,
Osa
ka J
. Mah
t. 2,
pp.
285
-290
., 19
65J.
L. K
oszu
l, «
Déf
orm
atio
ns d
es v
arié
tés
loca
lem
ent
plat
es»,
.Ann
Inst
Fou
rier,
18 ,
103-
114.
, 196
8Se
e:M
. N. B
oyom
, «C
onve
xité
loca
le d
ans
l’esp
ace
des
conn
exio
ns s
ymét
rique
s. C
ritèr
e de
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para
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tatis
tique
s»,
, Mar
ch 2
012,
IHP,
Par
isht
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ld S
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tific,
200
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tp://
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orld
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ntifi
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orld
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ooks
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ted
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rof.
Jean
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is K
oszu
l («
the
cont
ent o
f the
pre
sent
boo
k fin
ds th
eir o
rigin
in h
is s
tudi
es»)
A pa
ir (D
; g) o
f a f
lat c
onne
ctio
n D
and
a H
essi
an m
etric
gis
cal
led
a H
essi
an s
truc
ture
.
J.
L. K
oszu
lstu
died
a f
lat m
anifo
ld e
ndow
ed w
ith a
clo
sed
1-fo
rm α ααα
such
that
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ositi
ve d
efin
ite,
whe
reup
on D
α αααis
a H
essi
an m
etric
. Thi
s is
the
ultim
ate
orig
in o
f the
not
ion
of H
essi
an s
truc
ture
sA
Hes
sian
str
uctu
re (D
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s sa
id to
be
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oszu
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ther
e ex
ists
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lose
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form
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ch th
at g
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α ααα
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seco
nd K
oszu
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m β βββ
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impo
rtan
t rol
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r to
the
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leria
nm
etric
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vbe
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ent
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etric
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lled
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iclo
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eans
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ompl
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ifold
s:
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uene
ss a
nd a
ppro
xim
atio
n”, h
ttp://
arxi
v.or
g/ab
s/12
07.3
232
M. A
rnau
don,
C. D
ombr
y, A
. Pha
n, L
e Ya
ng, «
Stoc
hast
ic
algo
rithm
s fo
r com
putin
g m
eans
of p
roba
bilit
y m
easu
res.
»,
Stoc
hast
ic P
roce
sses
and
thei
r App
licat
ions
122,
pp.
143
7-14
55,
2012
M. A
rnau
don,
A. T
halm
aier
, “B
row
nian
mot
ion
and
nega
tive
curv
atur
e”, B
ound
arie
s an
d Sp
ectr
a of
Ran
dom
Wal
ks, P
rogr
ess
in P
roba
bilit
y, V
ol. 6
4, 1
45--1
63, S
prin
ger B
asel
,201
1M
. Arn
audo
n, F
. Bar
bare
sco,
Le
Yang
, ”M
edia
ns a
nd m
eans
in
Rie
man
nian
geo
met
ry: e
xist
ence
, uni
quen
ess
and
com
puta
tion”
Mat
rix In
form
atio
n G
eom
etry
, Nie
lsen
, Fra
nk; B
hatia
, Raj
endr
a (E
ds.),
Spr
inge
r, ht
tp://
arxi
v.or
g/pd
f/111
1.31
20v1
Le Y
ang,
«M
édia
nes
de m
esur
es d
e pr
obab
ilité
dan
s le
s va
riété
s rie
man
nien
nes
et a
pplic
atio
ns à
la d
étec
tion
de c
ible
s ra
dar»
, PhD
with
adv
isor
s M
. arn
audo
n &
F. B
arba
resc
oht
tp://
tel.a
rchi
ves-
ouve
rtes
.fr/d
ocs/
00/6
6/41
/88/
PDF/
Dis
sert
atio
n-Le
_YAN
G.p
df, T
HAL
ES P
hD A
war
d 20
12
Mar
c Ar
naud
onPh
D w
ith M
. Em
ery,
Bor
deau
x U
nive
rsity
P-M
eans
Com
puta
tion
on R
iem
anni
an M
anifo
ldSt
ocha
stic
Flo
w o
n R
iem
anni
an M
anifo
ld
Mic
hel E
mer
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MA
Lab,
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asbo
urg
Uni
vers
ityPr
obab
ility
on R
iem
anni
anM
anifo
ldSt
ocha
stc
Cal
culu
son
Man
ifold
sM
. Ém
ery,
G. M
okob
odzk
i, «
Sur l
e ba
ryce
ntre
d'u
ne
prob
abili
té d
ans
une
varié
té»,
Sém
inai
re d
e pr
obab
ilité
s de
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IHP,
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39)
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eq
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e di
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ry»
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ench
el-L
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gi
ves
that
loga
rithm
of g
ener
atin
g fu
nctio
n ar
e du
al to
Kul
lbac
k D
iver
genc
e :
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[]
[]
[]
[]
)(
(.)
)(
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log
)(
),
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)(
log
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),
(
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p
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)(
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11
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ms
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12
21Σ
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),
(x
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s
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dd
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varia
nce
by in
vers
ion
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ric fo
r Gau
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log
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Elie
Car
tan
(cla
ssifi
catio
n in
6
type
s of
sym
met
ric
hom
ogen
eous
bo
unde
d do
mai
ns )
n<=3
Erne
st V
inbe
rg(li
nk w
ith h
omog
eneo
us c
onve
xe c
ones
,
S
iege
l do
mai
ns o
f 2nd
kin
d)
Fine
stru
ctur
e of
In
form
atio
n G
eom
etry
(Hes
sian
Geo
met
ry,
Käh
leria
n G
eom
etry
)
« Il
est c
lair
que
si l’
on p
arve
nait
à dé
mon
trer
que
tous
le
s do
mai
nes
hom
ogèn
es d
ont l
a fo
rme
est d
éfin
ie p
ositi
ve s
ont s
ymét
rique
s, to
ute
la th
éorie
de
s do
mai
nes
born
és h
omog
ènes
ser
ait é
luci
dée.
C
’est
là u
n pr
oblè
me
de g
éom
étrie
her
miti
enne
ce
rtai
nem
ent t
rès
inté
ress
ant »
Der
nièr
e ph
rase
de
Elie
Car
tan,
dan
s «
Sur l
es
dom
aine
s bo
rnés
de
l'esp
ace
de n
var
iabl
es c
ompl
exes
»,
Abh
. Mat
h. S
emin
ar H
ambu
rg, 1
935
()
ji
ji
ji
zddz
zz
zz
K�
∂∂
∂=
,
2,
log
Φ
Jean
-Lou
is K
oszu
l(c
anon
ical
her
miti
an fo
rm o
f co
mpl
ex h
omog
eneo
us s
pace
s,
a co
mpl
ex h
omog
eneo
us s
pace
w
ith p
ositi
ve d
efin
ite c
anon
ical
he
rmiti
an fo
rm is
isom
orph
ic to
a
boun
ded
dom
ain,
,Stu
dy o
f A
ffine
Tra
nsfo
rm G
roup
s o
f lo
cally
flat
man
ifold
s)
Car
l Lud
wig
Sie
gel
(Sie
gel d
omai
ns in
fram
ewor
k of
Sym
plec
tic G
eom
etry
)
Look
eng
Hua
(Ber
gman
, Cau
chy
and
Poi
sson
K
erne
ls in
Sie
gel d
omai
ns)
31/
31/ 2� ���1���7
������
��� ��
����
�7��
��������
�)�)
�:��
����
{}
()
()2
*
**
11
111
11
,,
1/
zww
zK
zzC
z
IV
IIIII
I ,−
=<
∈=
==
=
Hen
ri Po
inca
ré(n
=1)
()
()
()
()
()(
)(
)do
mai
n. th
eof
volu
me
eucl
idea
n is
w
here
, :
IV Ty
pefo
r 2
ZZ1
1,
1 ,
: II
I
Type
1 ,
: II Ty
pe
, : I
Type
for
det
1,
-*
*t
**
Ωμ
νΩ
μ
ννν
Ωμ
ν
ν
n
ZWW
WW
ZK
p
p
qp
ZWI
WZ
K
IV n
III
pII pI p,q
=−
+=
� �
−=
+=
+=
−=
+−+
Look
eng
Hua
02
ZZ1,1
ZZ
:lin
e 1
and
row
sn
with
m
atric
esco
mpl
ex
: IV
Type
por
der
of m
atric
essy
mm
etric
skew
com
plex
:
III
Type
por
der
of m
atric
essy
mm
etric
com
plex
: II
Type
row
s q
and
lines
p w
ith
mat
rices
com
plex
: I
Type
):
(
Mat
rixr
Rec
tang
ula
Com
plex
:
2t
t>
−+
<
−<
+
++
ZZ
co
njug
ate
tran
spos
edI
ZZZ
IV nIII
pII pI p,q
Car
l Lud
wig
Sie
gel
Elie
Car
tan
(n<=
3)
32/
32/
Air
Sys
tem
s D
ivis
ion
��9�
�����������
� .��9
���
+�
���
�$�
��
+�
���
�$�
��
+�
���
�$�
��
+�
���
�$�
��
4 444G
�
��
��
����
� 1
����-
��
�� �
�-��
�G
�
��
��
����
� 1
����-
��
�� �
�-��
�G
�
��
��
����
� 1
����-
��
�� �
�-��
�G
�
��
��
����
� 1
����-
��
�� �
�-��
�
�M
öbiu
s Tr
ansf
orm
is a
tran
sitiv
e ac
tion
that
tran
sfor
ms
uppe
r hal
f-pl
ane
to it
self
(hom
ogen
eous
spa
ce) :
�M
and
–Mha
ve s
ame
actio
n th
en w
e co
nsid
er th
e qu
otie
nt G
roup
:
�C
ompl
ex u
nit d
isk
is li
nk to
upp
er h
alf p
lane
by
Cay
ley
trans
form
:
{} 0
)Im
(:
an
d
)(
,
2>
∈=
∈++
=∈ �� ��
�� ��=
zC
zH
zd
czb
azz
MR
SLd
cb
aM
22
2/
IR
SLR
PSL
±= {
}
11et
1:
zzi
z
HD
iz
iz
z
DH
zC
zD
−+→
+−→
<∈
=
��
1-1
i -i
22
2
22
2
ydzy
dydx
ds=
+=
x
y
H
D (
)222
2
14
zdzds
−=
Car
l Lud
wig
Sie
gel C
ontr
ibut
ion
in h
is s
emin
al b
ook
«Sy
mpl
ectic
Geo
met
ry»
: a
gene
raliz
atio
n of
Poi
ncar
é Sp
ace
33/
33/
�Si
egel
Met
ric fo
r the
Sie
gel U
pper
-Hal
f Pla
ne:
�U
pper
-Hal
f Pla
ne :
�Is
omet
ries
of
a
re g
iven
by
the
quot
ient
gro
up:
with
the
Sym
plec
tic G
roup
:
�U
niqu
e M
etric
inva
riant
e by
:
�� �
����
����
�3����
� 9�
���������7�
��$�� ��
������ �
nSH {
} nI
Rn
SpR
nPS
p2
/),
()
,(
±≡
),
(F
nSp
()(
)1)
(−
++
=� �� ��
�� ��=
DC
ZB
AZZ
MD
CB
AM
� �
=−
⇔∈ �� ��
�� ��=
nT
T
TT
IB
CD
AD
BC
AF
nSp
DC
BA
Msy
mm
etric
et
),
(
{}
),
2(0
0 ,
/),
2()
,(
Rn
SLI
IJ
JJM
MF
nG
LM
Fn
Spn
nT
∈ �� ���� �� −
==
∈≡
)(Z
M(
)(
)(
)Zd
YdZ
YTr
dsSi
egel
11
2−
−=
�� ==
nRYX
0(
)(
)(
)2
12
nn
dRR
Trds
−=
iYX
Z+
=
{} 0
Im/),
(>
=∈
+=
=Y
(Z)
Cn
Sym
iYX
ZSH
n
34/
34/
���7
���A99
��1������
����
����
��� �
{} 0
Im/),
(>
=∈
+=
=Y
(Z)
Cn
Sym
iYX
ZSH
n
Sieg
el U
pper
Hal
f Spa
ce
X
0>
Y
()
()
()
ZdY
dZY
Trds
11
2−
−=
()
()(
)(
)� =
−+
=n i
ii
ZZ
d1
22
12
1/1
log
,λ
λ
kk
kYi
XZ
.+
=1=
k
2=
k
()
[]
21
2dR
RTr
ace
ds−
=
()
kk
kk
RN
WRi
Z,0
if
.≡
=
1=
k2=
k
()
()
� =
=n k
kR
Rd
1
22
12
log
,λ
()
()(
)()(
)12
12
11
21
21
21,
−−
−−
−−
=Z
ZZ
ZZ
ZZ
ZZ
ZR
()
0.
..
det
2/11
22/1
1=
−−
−I
RR
Rλ
()
()
0.
,de
t2
1=
−I
ZZ
Rλ
35/
35/
BA3
0'�C
3'�!
0��0
�2!$�/-D
��E$
$-'(�2���3
2-�
F. B
erez
in
{}
()(
)(
)(
)(
)(
)(
)(
)(
)B
BI
trac
eB
BI
gF
AB
g
ZF
Zg
FZ
BA
trac
eZ
FZ
gF
ZZ
Itr
ace
ZZ
Iz
F
AZ
BB
AZZ
gB
AA
BI
BB
AA
II
JJ
Jgg
BB
BA
g
IZZ
ZSD
tt
t
n
++
−
++
−
++
+
+=
+=
�=
∂∂=
∂∂�
++
=−
−=
−−
=
++
= � �
=−
=−
� ��� �� −
==
� ��� ��
=
<=
log
det
log
))0((
)0(
)(
)(
log
Re
2)
())
((
log
det
log
)(
: po
tent
ial
Käh
ler
)(
w
ith
0
whe
re
00
with
an
d
with
/
1*
**
**
1*
***
**
ZgZZ
gj
ZZ
KZ
gj
zg
jgZ
gZK
ZZ
dK
ZZ
Kh
c
ZZ
dK
ZZ
KZ
gZ
fh
cg
f
h
h
∂∂=
=� ��
� ��=
� ��� ��
=
−
−
−
),
(, )
,(
),
()
,(
),
(
with
)
,(
)0,0()
,(
)(
),
()0,0()
,(
)(
)(
)(
,
**
**
/1*
1
*/1
*
μ
μ
()
()
()
()
()ν
βα
αββ
α
βα
βα
πμ
μ
−+
+=
∂∂
∂−
==
=
�W
WI
WW
FW
WW
WF
gdW
dWg
ds
WW
dW
WF
WW
dn
L
det
),
(
w
here
,lo
g w
ith ,
,,
**
*2
,
*,
2
**
*
36/
36/
�In
form
atio
n G
eom
etry
for M
ultiv
aria
te G
auss
ian
Law
of z
ero
Mea
n an
d in
trin
sec
Geo
met
ry o
f Her
miti
an P
ositi
ve D
efin
ite M
atric
es
(par
ticul
ar c
ase
of S
iege
l Upp
er-H
alf P
lane
) pro
vide
the
sam
e m
etric
�In
form
atio
n G
eom
etry
:
�G
eom
etry
of S
iege
l Upp
er-H
alf P
lane
:
()
()
()
21
2n
ndR
RTr
ds−
=
[]
()(
)[
]n
n
nn
nn
n
RR
Trn
nn
n
RR
E
mZ
mZ
R
eR
RZ
pn
n
=
−−
==+
−−
−−
ˆ
and
.ˆ
with
..
)(
)/
(1
.ˆ
1π
�� ��
�� ��−
=*
2
.)
/(
ln)
(j
i
nn
ijZ
pE
g∂θ
∂θθ
∂θ
0=
nm
with
���7
�����:
�$�� �5�$
�� ��
�F���
�$�� ��
����$
���� ���
��������/�"
{} 0
Im/),
(>
=∈
+=
=Y
(Z)
Cn
Sym
iYX
ZSH
n
()
()
()
iYX
ZZd
YdZ
YTr
dsSi
egel
+=
=−
−
with
1
12
�� ==
nRYX
0(
)(
)(
)2
12
nn
dRR
Trds
−=
37/
37/
�Si
egel
Dis
tanc
e:�
Par
ticul
ar C
ase
(X=0
) and
Gen
eral
Cas
e:
�Pa
rticu
lar C
ase
(pur
e im
agin
ary
axis
) :
�G
ener
al C
ase
of S
iege
l Upp
er-H
alf P
lane
Dis
tanc
e:
:�����
������"��
����
:�$�� ��
����9
� ���
��� ������������7�
�
()
()
()
� =
−−
==
n kk
RR
RR
Rd
1
22
2/11
22/1
12
12
log
..
log
,λ
()
0de
t1
2=
−R
Rλ
with
0
with
≠
∈+
=X
SHiY
XZ
n
()
n
n kkk
Sieg
elSH
ZZ
ZZ
d∈
�� ��
�� ���� ��
�� ��
−+=� =
21
1
22
12
, w
ith
11lo
g,
λλ
with
()
0.
),
(de
t2
1=
−I
ZZ
Rλ
()
()(
)()(
)12
12
11
21
21
21,
−−
−−
−−
=Z
ZZ
ZZ
ZZ
ZZ
ZR
0
avec
>
=R
iRZ
38/
38/
Air
Sys
tem
s D
ivis
ion
Sie
gel h
as d
educ
ed a
n ot
her d
ista
nce
from
:
3��
��� �)�����
���)�)
���)
�� ��
����7�
��=� #
()
[]
21
2.
ds− Σ
Σ=
dTr
()(
)12
122 12
11 2
1 11 2
1 112
.T
=R
an
d
.T
TT
+−
−−
−−
=Σ
+Σ
Σ−
Σ=
R is
her
miti
an p
ositi
ve d
efin
ite m
atrix
with
eig
en-v
alue
s : )
..
(
and
)(
r
with
11
2/1)1(1
)2(2/1)1(
kk
2
nn
nkk
kR
RR
Rr
−=
=�� ��
�� ��+−
=σ
λσ
λλ
We
dedu
ce th
at :
()
� =�� ��
�� ��−+
= � ��� ��
�� ���� ��
−+=
ΣΣ
n kkk
nn
rrR
IR
ITr
d1
2/1
2/12
2/1
2/12
21
2
11ln
ln,
beca
use
:
[]
[]�
�=
∞ =
−=
�� ���� ��
+=
= �� ���� ��
−+n k
j kk
k
nnr
akR
RR
RI
RI
1
j2
0
22/1
12/1
2/12
RTr
nd
1.2
..4
tanh
.4ln
39/
39/
��:�$�� ��
����
����� �
�
�
��
��
$��
� �
����
��-
���
�8�
�7�
$��
���
�3�
� ��
���"
�8�
��
1�
�G
eode
sic
:
-
��
���
��
$���
����
��
��
�Sy
met
ric S
pace
as s
tudi
ed b
y El
ie C
arta
n : E
xist
ence
of b
iject
ive
geod
esic
isom
etry
�B
ruha
t-Tits
Spa
ce: s
emi-p
aral
lelo
gram
ineq
ualit
y
�C
arta
n-H
adam
ard
Spac
e(C
ompl
ete,
sim
ply
conn
ecte
d w
ith
nega
tive
sect
iona
l cur
vatu
re M
anifo
ld)
[] 10
with
)
,(
.))
(,
(,
tR
Rdt
tR
dY
XX
∈=
γ ()
()
YX
YX
Xt
XY
XX
XR
RR
tX
RR
RR
RR
RR
RR
eR
tX
YX
�=
==
==
−−
−−
)2/1(
and
)1(
,
)0()(2/1
2/12/1
2/12/1
log
2/12/1
2/1
γγ
γγ
()
()
2/12/1
2/12/1
2/11-
),
(
avec
)(
XA
BAA
AB
AB
AB
AX
GB
A−
−=
=�
��
Xx
)d(
x,x
)d(
x,x
d(x,
z))
,xd(
xx
zx
x∈
∀+
≤+
∀∃
∀
2
24
que
tel
,
22
21
22
21
21
40/
40/
x1x
2x
z
U
VU
+V
U−
V
2� ���*���
����� ��
��9�
��
�Sy
met
ric S
pace
as s
tudi
ed b
y El
ie C
arta
n : E
xist
ence
bije
ctiv
e ge
odes
ic
isom
etry
�B
ruha
t-Tits
Spa
ce: s
emi-p
aral
lelo
gram
ineq
ualit
y
22
22
22
VU
VU
VU
+=
++
−
),
(A
GB
BA
=
BG
AB
A)
,(
=(
))
(X
-1)
,(
BA
BA
XG
BA
��
=
X
()
2/1/21
2/12/1
2/1A
BAA
AB
A−
−=
�
()
[] 1,0)(
2/12/1
2/12/1
∈=
−−
tA
BAA
At
tγ
A= )0(γ
B= )1(γ
22
21
22
21
22
4)
d(x,
x)
d(x,
xd(
x,z)
),x
d(x
+≤
+
E. J
. Car
tan
M. B
erge
r
J. T
its
41/
41/
Air
Sys
tem
s D
ivis
ion
�Th
is is
omet
ry fo
r met
ric s
pace
:
�Is
an
exte
nsio
n of
this
one
:
�To
be
com
pare
d w
ith E
uclid
ean
«sy
mm
etric
» sp
ace
������ ��
��9�
��
()
()
()
()
� �
=
=•
••
=−
−
−−
22/1
2/12
2/12/1
2/12/1
2/11-
),
(lo
g,
w
ith
)(
XBA
AB
A
ABA
AA
BA
BA
BA
XG
BA
δ
()
()
()
()
� �
=
==
−2
12
1
log
,
with
ab
ba
abb
ab
ax
ba
xG
-(a
,b)
δ
��
�
()
� �
−=+
=� ��
� ��+
+� ��
� ��+
=2
2,
2
with
2
2b
ab
a
ba
ba
ba
-xb
ax
G(a
,b)
δ
�
42/
42/
Air
Sys
tem
s D
ivis
ion
�is
the
only
fixed
poin
t bec
ause
:
�du
e to
trac
e pr
oper
tyof
:
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7��
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�In
195
7, M
auric
e R
ené
Fréc
het i
ntro
duce
d th
is d
ista
nce,
bas
ed o
n a
prev
ious
Pau
l Lev
y’s
pape
r fro
m 1
950
()
[]
−=
−=
),
()
(,
22
,
2y
xH
dd
yx
YX
EIn
fG
Fd
yx
YX
58/
58/
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�9� ���� .�
���1=���� ������:�����
��
��
8�>6
��
��
��7
����
��
�
�19
57 F
réch
et p
aper
in C
RA
S :
�Le
tter f
rom
Pau
l Lev
y to
Mau
rice
Fréc
het (
2 A
pril
1958
)�
… J
’ai a
insi
pu
appr
écie
r ce
que
vous
avi
ez fa
it, e
n pr
enan
t com
me
poin
t de
dépa
rt de
vot
re
mém
oire
ce
que
vous
app
elez
ma
prem
ière
déf
initi
on d
e la
dis
tanc
e de
deu
x lo
is d
e pr
obab
ilité
(en
fait
ce n
’éta
it pa
s la
pre
miè
re).
Vou
s l’a
vez
d’ai
lleur
s gé
néra
lisée
, en
ce s
ens
que
je n
e l’a
vais
as
soci
ée q
u’à
une
de v
os d
éfin
ition
s de
deu
x va
riabl
es a
léat
oire
s. E
t j’a
i bea
ucou
p ad
miré
co
mm
ent a
vec
votre
qua
trièm
e dé
finiti
on, v
ous
arriv
ez à
faire
que
lque
cho
se d
e m
ania
ble
d’un
e id
ée q
ui p
our m
oi é
tait
surto
ut th
éoriq
ue, v
u la
diff
icul
té d
e dé
term
iner
le m
inim
um d
e la
dis
tanc
e de
deu
x va
riabl
es a
léat
oire
s ay
ant l
es ré
parti
tions
mar
gina
les
donn
ées.
59/
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xtre
me
Cop
ulas
of F
réch
et
�4t
h di
stan
ce o
f Fré
chet
()
() [
])(
),(
),
(w
ith
),
(,
1
12
2
yG
xF
Min
yx
H
yx
Hd
dy
xG
Fd
yx
=−=
[]
[] )
(),
()
,(
0,1)
()
()
,(
with
),
()
,(
),
( 10
01
yG
xF
Min
yx
Hy
Gx
FM
axy
xH
yx
Hy
xH
yx
H
=−
+=
≤≤
Extr
eme
Fréc
het
Cop
ulas
),
( 1y
xH
),
(0
yx
H
),
(y
xH
≥ ≥≥≥≥ ≥≥≥
60/
60/
�G
. Dal
l’Agl
io (U
nive
rsité
La
Sapi
enza
à R
ome)
, «Fr
éche
t Cla
sses
: th
e B
egin
ning
s»,
Adv
ance
d in
Pro
babi
lity
Dis
trib
utio
ns w
ith
Giv
en M
argi
nals
Bey
ond
the
Cop
ulas
, Rom
e 19
91, K
luw
er�
With
hel
p of
Giu
sepp
e P
ompi
lj, G
. Dal
l’Agl
io rm
et M
auric
e Fr
éche
t, 80
yea
rs
old,
in 1
956
in R
oma,
dur
ing
visi
t to
l’Ist
ituto
di C
alco
lo d
elle
Pro
babi
lita
�D
all’A
glio
met
a 2
ndtim
e Fr
éche
t in
Par
is a
nd is
in c
onta
ct w
ith P
aul L
evy
�D
all’A
glio
writ
e : «
Levy
als
o sh
owed
som
e in
tere
st in
dis
trib
utio
ns w
ith
give
n m
argi
nals
. In
a no
te in
196
0 he
refe
rs to
a fo
rmul
a by
Poi
ncar
é fo
r ut
iliza
tion
in n
-dim
ensi
onal
dis
trib
utio
ns w
ith g
iven
s m
argi
nals
, with
out
deve
lopp
ing
the
idea
»
�Pa
ul L
evy,
«Su
r les
con
ditio
ns d
e co
mpa
tibili
té d
es d
onné
es m
argi
nale
s re
lativ
es a
ux lo
is d
e pr
obab
ilité
» C
RAS
, t. 2
50 p
p.25
07-2
509,
196
0, n
ote
prés
enté
e pa
r M. J
acqu
es H
adam
ard
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9��G&�/���&�(�%%�&�:�
��*37���&��� �
����&�����J
<
61/
61/
�G
. Dal
l’Agl
io (U
nive
rsité
La
Sapi
enza
à R
ome)
, «Fr
éche
t Cla
sses
: th
e B
egin
ning
s»,
Adv
ance
d in
Pro
babi
lity
Dis
trib
utio
ns w
ith
Giv
en M
argi
nals
Bey
ond
the
Cop
ulas
, Rom
e 19
91, K
luw
er�
By
inte
grat
ion
by p
arts
:
�as
sign
s to
all
sub-
sets
of d
iago
nal y
=x, m
axim
um o
f pro
babi
lity
com
plia
nt w
ith m
argi
nals
: S. B
ertin
o, «
Su
di u
na s
otto
clas
se d
ella
cla
sse
di
Fréc
het»
, Sta
tistic
a 25
, pp.
511
-542
, 196
8
�G
ini W
ork
: in
1914
, Gin
i int
rodu
ced
«lin
ear d
issi
mila
rity
para
met
er»
(sol
utio
n in
dis
cret
cas
e fo
r α=
1, 2
)
�To
mm
aso
Sal
vem
ini (
+ Le
ti) W
ork
: con
stru
ctio
n of
«ta
bella
di c
ogra
duaz
ione
» an
d «
di c
ontro
grad
uazi
one
» (e
quiv
alen
t to
Fréc
het E
xtre
me
Rep
artit
ion
Func
tions
)
�G
ener
aliz
atio
n to
mul
tivar
iate
by
Riz
zi in
195
7, D
all’A
glio
(196
0)
I���)�����
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����&�����J
<
[]
()
()
()
� �
≥���
�� −
−
≤���
�� −
−=
−+
=−
≤≤
≤≤
yx
zF
zG
Inf
yG
yx
zG
zF
Inf
xF
yx
H
dzz
zH
zG
zF
YX
E
xz
y
yz
x
R
if
0,)(
)(
max
)(
if
0,)(
)(
max
)(
),
(
:fu
nctio
n n
Rep
artit
io g
Min
imliz
in
),
()
()
(
*
[]
()
()
()
{} )
(),
()
,(
:Fu
nctio
n n
Rep
artit
ioM
inim
izin
g
)()
,(
)(
)1(
)()
,(
)(
1
:1Fo
r 1
22
yG
xF
Min
yx
H
dudv
uv
vu
Fu
F
dudv
vu
vu
Hv
GY
XE
vuv
u
=
−−
−
+−
−−
=−
>
<
−
>
− ααα
αα
αα
α
),
(*y
xH
62/
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9� ����
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��� �� � ����!9���
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�9� �
For t
rans
port
opt
imal
:A.
Tak
atsu
. On
Was
sers
tein
Geo
met
ry o
f the
Spa
ce o
f Gau
ssia
n M
easu
res,
to a
ppea
r in
Osa
ka J
. M
ath.
, 201
1, h
ttp://
arxi
v.or
g/ab
s/08
01.2
250
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es A
ir S
yste
ms
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e
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auch
y se
ries
give
n by
har
mon
ic a
nd a
rithm
etic
mea
n :
()
() 2/
an
d
2
, w
ith 1
11
11
00
nn
nn
nn
nn
nn
BA
BB
AA
BB
AA
BLi
mA
Lim
BA
+=
+=
==
==
+−
−−
+
∞→
∞→
�
�S
olut
ion
of th
e fo
llow
ing
OD
E :
� �
∈
−=
=∈
∀++
−
∞→
++
Sym
X
XXA
BdtdX
tX
Lim
BA
Sym
BA
t)0(
with
)
(,
,1
�
�S
olut
ion
to th
e in
equa
lity
:
0
s.t.
(Loe
wne
r)m
atrix
hi
gher
,
,> �� ��
�� ��∈
∀++
BX
XA
BA
Sym
BA
�
�S
olut
ion
of th
e eq
uatio
n :
)de
t(lo
g)
( w
ith
))(
(''1
1X
XF
BAX
XA
XF
−=
==
−−
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tem
s D
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�����
�
bx
xaab
xab
x=
⇔=
⇔=
−12
�In
sca
lar c
ase,
geo
met
ric m
ean
of a
,b>0
sat
isfie
s fo
r x>=
0 :
�Th
e la
st «
sym
met
rized
» ve
rsio
n is
sui
tabl
e fo
r gen
eral
izat
ion
to n
onco
mm
utat
ive
setti
ngs
: the
geo
met
ric m
ean
is th
e un
ique
so
lutio
n of
,
if th
e un
ique
sol
utio
n ex
ists
.�
We
defin
e a
grou
p eq
uipp
ed w
ith th
e op
erat
ion
:
bx
xa=
−1
xxy
yx
1−=
•b
ax
ba
xb
xxa
�=
⇔=
•⇔
=−1
�To
sol
ve
, we
need
to
hav
e a
uniq
ue
solu
tion.
Eve
ry e
lem
ent m
ust h
ave
a un
ique
squ
are
root
:a
ex
=•
ex
xex
==
2
2/1 aa
e=
�
�E
xten
sion
for M
atrix
Geo
met
ric M
ean
:
()
()
BA
ABA
AA
X
BAA
XAA
BAA
XAA
XAA
BAA
AX
AXA
AB
XXA
�=
=�
=�
=�
=�
=
−−
−−
−−
−−
−−
−−
−−
−−
−−
−
2/12/1
2/12/1
2/1
2/12/1
2/12/1
2/1
2/12/1
2/12/1
2/12/1
2/12/1
2/12/1
2/12/1
1
))(
(
)(
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s D
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enm
an-B
eabe
rs It
erat
ion
:
�Sc
hultz
Iter
atio
n (w
ithou
t mat
rix in
vers
ion)
:
�B
jörk
Met
hod
by S
chur
Dec
ompo
sitio
n :
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���� ����$�� �5
��8�
� ��(�
��
�� ���� ��
� ��� ��
+ � ��� ��
=� ��
� ��=
−
−
+
++
00
00
210
01
1
1
11
k
k
k
k
k
kk
YZ
ZY
ZY
X
� ��� ��
=0
00
IA
Xw
ithth
en� ��
� ��=
−∞
→0
02/
1
2/1
AA
XLi
mk
k
)3(
210
02
1
11
kk
k
kk
XI
XZ
YX
−=
� ��� ��
=+
++
TAQ
Q=
+w
ith T
Upp
er tr
iang
ular
mat
rixB
y re
curr
ence
, we
com
pute
: +
=�
=Q
UQ
AT
U2/1
2
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auss
(20
year
s ol
d) &
Lag
rang
e in
depe
nden
tly p
rove
d th
at
�A
rithm
etic
-geo
met
ric m
ean
of a
and
b :
�Is
rela
ted
to e
lliptic
inte
gral
:
�La
nden
tran
sfor
mat
ion
give
s al
so :
()
()
aa
aa
bb
bb
n
ba
ba
ba
ba
ba
b
ba
ab
ab
ab
a
nn
nn
nn
nn
nn
nn
nn
n
nn
n
=≤
≤≤
≤≤
≤≤
=≥
∀�
≥� ��
� ��−
=−
� ��� ��
+=
−
� �
=
+=
=>
≥∀
++
++
++
01
10
22
21
21
110
0
......
0
02
2
2
and
,
, ,
0
()
� �
+=
==
=
∞→
∞→ 2/ 0
22
22
sin
cos
),
(
),
(
w
ith
),
(2/
,π
π
tb
ta
dtb
aI
bLi
ma
Lim
ba
M
ba
Mb
aI
nn
nn
()
()
()
()
()
()
� ��� ��
+=
+=
� �
+=
= � ��� ��
+=
tab
Arc
tu
dttb
ta
ba
Jb
aab
Ib
aJ
ba
J
ba
Iab
ba
Ib
aI
tan
tan
: ns
f.La
nden
tra
sin
cos
,
with
,
,,
2
,,
2,
2/ 0
22
22
11
11
π
4.J(
a,b)
: P
erim
eter
of E
llips
eof
hal
f-axi
s a
& b
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rithm
etic
mea
n:
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is p
oint
min
imiz
es th
e fu
nctio
n of
dis
tanc
es:
�Th
e m
edia
n (F
erm
at-W
eber
Poi
nt) m
inim
izes
:
� =
=M i
ix
cent
erx
xd
Min
x1
2)
,(
arg
1x
2x
3x
4x
5x
� =
→=
M ii
xce
nter
xxM
inx
1
arg
� =
=M i
ice
nter
xM
x1
1
{}
Mi
ix,..
.,1=
� =
=M i
ice
nter
xM
x1
1
� =
=M i
ix
cent
erx
xd
Min
x1
2)
,(
arg
2��
�� ��
��$������3 ����
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)�$�)
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� =
=M i
ix
med
ian
xx
dM
inx
1)
,(
arg
1x
2x
3x
4x
5x
� =→→
=M i
ii
xm
edia
n
xxxxM
inx
1
arg
[]
mx
EM
inm
mm
edia
n−
=[
]2
mx
EM
inm
mm
ean
−=
� =
=M i
ix
med
ian
xx
dM
inx
1)
,(
arg
� =
→=
M ii
cent
erx
x1
0� =
→→
=M i
ice
nter
ice
nter
xx
xx
10
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�El
ie C
arta
n ha
s pr
oved
that
the
follo
win
g fu
nctio
nal :
is s
tric
tly c
onve
xe a
nd h
as o
nly
one
min
imum
(ce
nter
of
mas
s of
Afo
r dis
trib
utio
n da
) for
a m
anifo
ld o
f neg
ativ
e cu
rvat
ure
��
��
� �
6�
C
�H
erm
ann
Kar
cher
has
pro
ved
the
conv
erge
nce
of th
e fo
llow
ing
flow
to th
e C
ente
r of M
ass
:
E. J
. Car
tan
H. K
arch
er
∈
A
daa
md
mf
),
(:
2�
Μ
()
)(
)0(
avec
)(
.ex
p)
(1
nn
nn
mn
nn
mf
mf
tt
mn
−∇=
∇−
==
+γ
γ�
)(
exp
1
−−
=∇
Am
daa
f
70/
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� �*���
59��
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�M
auric
e R
ené
Fréc
het,
inve
ntor
of C
ram
er-R
ao b
ound
in 1
939,
has
als
o in
trod
uced
the
entir
e co
ncep
t of M
etric
Spa
ces
Geo
met
ryan
d fu
nctio
nal
theo
ry o
n th
is s
pace
(any
nor
med
vec
tor s
pace
is a
met
ric s
pace
by
defin
ing
bu
t not
the
cont
rary
). O
n th
is b
ase,
Fré
chet
has
then
ext
ende
d pr
obab
ility
in
abst
ract
spa
ces.
�In
this
fram
ewor
k, e
xpec
tatio
n
of a
n ab
stra
ct p
roba
bilis
tic
varia
ble
w
here
lies
on
a m
anifo
ld is
intr
oduc
ed b
y Em
ery
as a
n ex
pone
ntia
l bar
ycen
ter:
�In
Cla
ssic
al E
uclid
ean
spac
e, w
e re
cove
r cla
ssic
al d
efin
ition
of E
xpec
tatio
n E[
.]:
xy
yx
d−
=),
(M
. R. F
réch
et, “
Les
élém
ents
alé
atoi
res
de n
atur
e qu
elco
nque
dan
s un
esp
ace
dist
anci
é”, A
nnal
es d
e l’I
nstit
ut H
enri
Poi
ncar
é, n�1
0, p
p.21
5-31
0, 1
948
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p,
1
M. E
mer
y &
G. M
okob
odzk
i, “S
ur le
bar
ycen
tre d
’une
pro
babi
lité
sur u
ne v
arié
té”,
Sém
inai
re d
e Pr
oba.
XXV
, Lec
ture
s no
te in
Mat
h. 1
485,
pp.
220-
233,
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r, 19
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llow
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cons
trai
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r sta
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l) :
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ctur
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truct
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ow to
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If yo
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irst e
xam
ple
on «
Rig
ht T
riang
le»,
in th
is c
ase
Pyt
hago
re c
onst
rain
t is
repl
aced
by
HP
D &
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plitz
con
stra
ints
77/
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ex a
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egre
ssiv
e m
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:
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lose
Lin
k w
ith Is
sai S
chur
’s a
lgor
ithm
(187
5-19
41)
�[A
lpay
] D. A
lpay
, « A
lgor
ithm
e de
Sch
ur, e
spac
es à
noy
au re
prod
uisa
nt e
t th
éorie
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sys
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, Pan
oram
as e
t syn
thès
e, n�6
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Mat
hém
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, 199
8
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ture
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ovar
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pute
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met
ric fo
r CA
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:�
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met
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r Com
plex
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tivar
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ssia
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ero
mea
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lock
stru
ctur
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mat
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r CA
R m
odel
:
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TrB
AA
AA
dRR
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,et
,
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12 (
)(
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et
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nn
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Inva
rianc
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truct
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:
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lock
stru
ctur
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pute
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ende
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gen-
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an
algo
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that
cou
ld b
e pa
ralle
lized
acc
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C
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exte
nded
eig
en-v
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s :
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n in k
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stric
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Ren
orm
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erat
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:
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term
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roje
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vec
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t mod
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serv
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at :
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inte
rpre
ted
with
Info
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Geo
met
ry &
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her m
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11
2
112
12
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g
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n
n
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11
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RA
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lesk
y de
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dber
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ll di
strib
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n-d
imen
sion
nal v
aria
ble
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ssoc
iate
d w
ith A
ffine
Gro
up. I
t is
the
elem
ent s
uch
that
its
actio
n on
vec
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mat
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men
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t1
,
-*
*t
**
Ωμ
νΩ
μ
ννν
Ωμ
ν
ν
n
ZWW
WW
ZK
p
p
qp
ZWI
WZ
K
IV n
III
pII pI p,q
=−
+=
� �
−=
+=
+=
−=
+−+
{}
()
()2
*
*
*1
11
11
11
,
1/
zww
zK
zzC
z
IV
III
III ,
−=
<∈
==
==
137
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�G
roup
s of
ana
lytic
aut
omor
phis
ms
of th
ese
dom
ains
are
loca
lly
isom
orph
ic to
the
grou
p of
mat
rices
whi
ch p
rese
rve
follo
win
g fo
rms:
�Th
e gr
oup
cons
its o
f blo
ck m
atric
es A
(gen
eral
izat
ion
of th
e fr
actio
nal
linea
r tra
nsfo
rmat
ion)
. For
the
first
3 ty
pes
:
�� ���� ��−
==
=
�� ���� ��
=�� ��
�� ��−
==
=
�� ���� �� −
=�� ��
�� ��−
==
=
=�� ��
�� ��−
==
n
tIV n
p
p
p
pt
III
p
p
p
p
pt
II p
p
pI p,
q
II
HH
AHA
HAH
A
II
LI
IH
LAL
AH
AHA
II
KI
IH
KAK
AH
AHA
AI
IH
HAH
A
00
,,
, :
IVTy
pe
00
,0
0,
, ,
: II
ITy
pe
00
,0
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, ,
: II Ty
pe
1de
t,
00
, ,
: I Ty
pe
2**
**
()(
)122
2112
1122
21
1211
−+
+=
� �� ���� ��
=A
ZA
AZ
AgZ
AA
AA
A
138
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ll cl
assi
cal d
omai
ns a
re c
ircul
ar fo
llow
ing
from
Car
tan’
s ge
nera
l the
ory,
an
d th
e po
int 0
is d
istin
guis
hed
for t
he p
oten
tial :
�B
erez
in q
uant
izat
ion
is b
ased
on
the
cons
truc
tion
of th
e H
ilber
t Spa
ce o
f fu
nctio
ns a
naly
tic in
Ω ΩΩΩ:
()
()
()
()ν
Φ−
+−
=� ��
� ��=
ZZI
KZ
ZK
ZZ
det
log
0,0,ln
,*
*
ZgZZ
gj
ZZ
KZ
gj
zg
jgZ
gZK
ZZ
dK
ZZ
Kh
c
ZZ
dK
ZZ
KZ
gZ
fh
cg
f
h
h
∂∂=
=
� ��� ��
=
� ��� ��
=
−
−
−
),
( w
ith
),
()
,(
),
()
,(
),
()0,0()
,(
)(
),
()0,0()
,(
)(
)(
)(
,
**
*
*/1
*1
*/1
*
μ
μ
139
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�� �%��
�B��
�������
�������������
� .�A�
���:��#
�Th
e m
ost e
lem
enta
ry e
xam
ple
of B
erez
ian
quan
tific
atio
n is
, in
the
case
of
com
plex
dim
ensi
on 1
, giv
en b
y th
e Po
inca
ré u
nit D
isk
with
vol
ume
elem
ent :
�M
ap fr
om p
ath
on D
to a
utom
orph
y fa
ctor
:
*2
2)
1.(2/1
dzdz
zi
∧−
−
{}
()
()
*
2
*
2
**
2
22
**
1
)(
)(
)(
lnR
e2
)(
1ln
)(
: po
tent
ial
Käh
ler
1
whe
re w
ith
)1,1(
/)1,1(1
/
zz
zF
zz
gzF
zF
az
bgz
Fz
zF
ba
ab
ba
gSU
g
SSU
zC
zD
∂∂∂
=∂
∂∂�
++
=�
−−
=
=−
� ��� ��
=∈
=<
∈=
()
()
()
)(
)(
1ln
1ln
))0((
)0(
1**
1
2
1
21
*1
*2
2
gF
gF
ab
ba
g
ba
bg
Fa
bg
ba
=� �� ��
�� �� −−
=
+=
� ��� ��
−−
=�
=
−−
=−
−−
140
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0/
�Ex
tens
ion
for S
iege
l Uni
t Dis
k :
�Th
e or
bit o
f the
mat
rix Z
=0 is
the
spac
e of
mat
rices
of t
he fo
rm :
{}
()(
)(
)(
)(
)(
))
()
(ln
Re
2)
())
((
lnde
tlo
g)
( :
pote
ntia
lK
ähle
r )
(
0
whe
re
00
with
an
d
with
/
**
**
1*
***
**
ZF
Zg
FZ
BA
trac
eZ
FZ
gF
ZZ
Itr
ace
ZZ
Iz
FA
ZB
BAZ
Zg
BA
AB
IB
BA
A
II
JJ
Jgg
BB
BA
g
IZZ
ZSD
tt
t
n
∂∂=
∂∂
++
=−
−=
−−
=++
=
� �
=−
=−
� ��� �� −
==
� ��� ��
=
<=
++
−
++
+
�� �%��
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�������
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()
()
()
BB
Itr
ace
BB
Ig
FA
Bg
++
−+
=+
=�
=ln
det
ln))0(
()0(
1*
141
/14
1/
�9��
������ �%���2��
)������
�Fo
r eve
ry s
ymm
etric
Rie
man
nian
spa
ce, t
here
exi
st a
dua
l spa
ce
bein
g co
mpa
ct. T
he is
omet
ry g
roup
s of
all
the
com
pact
sym
met
ric
spac
es a
re d
escr
ibed
by
bloc
k m
atric
es (t
he a
ctio
n of
the
grou
p in
te
rms
of s
peci
al c
oord
inat
es is
des
crib
ed b
y th
e sa
me
form
ula
as th
e ac
tion
of th
e gr
oup
of m
otio
ns o
f the
dua
l dom
ain)
.
�B
erez
in c
oord
inat
es fo
r Sie
gel d
omai
n :
�� ���� ��
==
=
�� ���� ��
=�� ��
�� ��=
+
++−
00
with
,
:ly
eq
uiva
lent
or
, 1
**
II
LL
LI
AB
BA
AB
BA
t
tt ΓΓ
ΓΓΓΓ
()
()
()
++
−+
=+
=�
=BB
Itr
ace
BBI
gF
AB
gln
det
ln))0(
()0(
1*
()(
)
�� ���� ��
==
++
=� �� ��
�� ��=
−
−
IiI
iII
CC
C
AW
AA
WA
WA
AA
A
21 w
ith
:Is
omet
ry
)( 1
122
2112
1122
21
1211
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ΓΓ
142
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2/
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�Le
t M b
e a
clas
sica
l com
plex
com
pact
sym
met
ric s
pace
. The
inva
riant
vo
lum
e an
d in
varia
nt m
etric
in te
rms
of s
peci
al B
erez
in c
oord
inat
es
have
the
form
:
�Li
nk w
ith :
For a
rbitr
ary
Käh
leria
n ho
mog
eneo
us s
pace
, the
loga
rithm
of
the
dens
ity fo
r the
inva
riant
mea
sure
is th
e po
tent
ial o
f the
met
ric
()
()
()
()
()ν
βα
αββ
α
βα
βα
πμ
μ
−+
+=
∂∂
∂−
==
=
�
WW
IW
WF
WW
WW
Fg
dWdW
gds
WW
dW
WF
WW
dn
L
det
),
(
whe
re
,ln
with
,,
,
*
*
*2
,
*,
2
**
*
����������� �����
Thal
es A
ir S
yste
ms
Dat
e3��
��������������� ��
�2��
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�
144
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4/
Air
Sys
tem
s D
ivis
ion
/��#�"���
�3��
��������������� ��
�2��
���;H
���� ��
�<�
Jacq
ues
Fara
ut h
as p
ublis
hed
2 bo
oks
on A
naly
sis
on S
ymm
etric
C
ones
& o
n Li
e G
roup
:�
[1] J
. Far
aut &
A. K
oran
yi, «
Ana
lysi
s on
Sym
met
ric c
ones
», C
lare
ndon
Pre
ss, O
xfor
d,
1994
�[2
] J. F
arau
t, «
Ana
lyse
sur
les
grou
pes
de L
ie»,
Cal
vage
& M
oune
t, P
aris
, 200
6
�H
arm
onic
Ana
lysi
s in
the
spec
ial c
ase
of th
e co
ne o
f pos
itive
def
inite
m
atric
es in
the
vect
or s
pace
of a
ll re
al s
ymm
etric
mat
rices
pla
ys a
fu
ndam
enta
l rol
e in
:�
Num
ber T
heor
y (M
inko
wsk
i, si
egel
, Maa
s,…
)�
Sta
tistic
s (W
isha
rt, C
onst
antin
e, J
ames
, Mui
rhea
d)�
Phy
sic
(stu
dy o
f Lor
entz
con
e)
�G
ener
al C
ase
has
been
stu
died
by
:�
Gin
diki
n�
Vin
berg
�Sy
mm
etric
Con
es &
Tub
es (o
ver
them
) are
exa
mpl
e of
Rie
man
nian
Sy
mm
etric
Spa
ces
145
/14
5/
Air
Sys
tem
s D
ivis
ion
/��#�"���
�3��
��������������� ��
�2��
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���� ��
�<
Con
vex
cone
s :
�Le
t Vbe
a fi
nite
dim
ensi
onal
real
Euc
lidea
n Sp
ace.
A s
ubse
t Cof
Vis
sa
id to
be
a co
ne if
:
�Th
e cl
osed
dua
l con
e of
any
con
e C
is d
efin
ed b
y :
�Th
e au
tom
orph
ism
gro
up
of a
n op
en c
onve
x co
ne
is d
efin
ed
by :
�is
a c
lose
d su
bgro
up o
f
,
and
henc
e is
a L
ie G
roup
. The
op
en c
one
i
s sa
id to
be
hom
ogen
eous
if
act
s on
it tr
ansi
tivel
y.
The
open
con
e is
sai
d to
be
sym
met
ric if
it is
hom
ogen
eous
and
sel
f-du
al :
Cx
Cx
∈�
���>∈
λλ
0
()
{}
()
CC
Cx
yx
Vy
C
=
∈∀
≥∈
= ###
,0/
()
()
{}
Ω=
Ω∈
=Ω
gV
GL
gG
/(
)Ω
GΩ
()
ΩG
()
ΩG
LΩ
()
ΩG
()
()
()
Ω∈
=
GL
gg
yg
xy
gx
ofel
emen
t an
of
adjo
int
the
:*
*
146
/14
6/
Air
Sys
tem
s D
ivis
ion
/��#�"���
�3��
��������������� ��
�2��
���;H
���� ��
�<
Con
vex
cone
s :
�Fo
r any
pro
per o
pen
conv
ex c
one
:
�ch
arac
teriz
es th
e sy
mm
etric
con
es
Cha
ract
eris
tic F
unct
ion
of a
Con
e :
�Le
t
be
a pr
oper
ope
n co
nvex
con
e, it
s ch
arac
teris
tic fu
nctio
n is
:
dyEu
clid
ean
Mea
sure
on
V
The
seco
nd d
eriv
ativ
e
is
pos
itive
def
inite
at e
ach
poin
t
Ω(
)(
)
()
()
Ω∈
���
Ω∈
Ω=
Ω
Ω=
Ω
Gg
Gg
GG
**
**
if
()
()
Ω=
ΩG
G* Ω
()
Ω
−=
*
)(
dye
xy
xϕ
()
)(
det
)(
, 1
xg
gxG
gϕ
ϕ−
=Ω
∈∀
)(
)(
, 0 ,
xx
�xgx
n ϕλ
λϕ
λ−
=>
=
0)
,(
)(
)(
)(
log
),
(
0
>�
� �
+==
=
uu
Gtu
xdtd
xD
xD
Dv
uG
x
tu
vu
x
φφ
ϕ)(
log
2x
Dϕ
147
/14
7/
Air
Sys
tem
s D
ivis
ion
/��#�"���
�3��
��������������� ��
�2��
���;H
���� ��
�<
�C
hara
cter
istic
Fun
ctio
n of
a C
one
:�
Rie
man
nian
stru
ctur
e g
is g
iven
by
Ω
[]
()
log
log
21lo
g
log
)(
log
w
ith)
(lo
g
2u
2
22
2
−
+==
=
dudv
dudv
dd
du
dud
dud
xd
xd
g
vu
vu
v
u
uu
u
ϕϕ
ϕϕ
ϕϕ
ϕ
ϕϕ
ϕϕ
ϕ
148
/14
8/
Air
Sys
tem
s D
ivis
ion
/��#�"���
�3��
��������������� ��
�2��
���;H
���� ��
�<
Cha
ract
eris
tic F
unct
ion
of a
Con
e :
�Th
e ad
join
t :
�Th
e m
ap
i
s a
bije
ctio
n :
a
nd h
as u
niqu
e fix
ed p
oint
Sym
met
ric C
one
as R
iem
anni
an S
ymm
etric
Spa
ce :
�Th
e bi
linea
r for
m
is p
ositi
ve d
efin
ite,
ther
efor
e it
defin
es a
Rie
man
nian
met
ric o
n
�Th
e co
ne
e
quip
ed w
ith th
is m
etric
is a
Rie
man
nian
Man
ifold
�Si
nce
the
cone
is s
ymm
etric
, the
map
i
s a
bije
ctio
n an
d an
isom
etry
(the
man
ifold
is a
Rie
man
nian
Sym
met
ric
Spac
e gi
ven
by th
is is
omet
ry)
()
)(
)(
with
)
(lo
g*
xf
Du
xf
xx
u=
∇−∇
=ϕ
**
Ω∈
Ω∈
xx
�(
)n
xx
=*
)(
log
),
(x
DD
vu
Gv
ux
ϕ=
ΩΩ
Ω)
(lo
g*
xx
xϕ
−∇=
�
()
cons
tx
xx
x=
=
)(
)(
*
**
ϕϕ
149
/14
9/
Air
Sys
tem
s D
ivis
ion
/��#�"���
�3��
��������������� ��
�2��
���;H
���� ��
�<
The
cone
of P
ositi
ve D
efin
ite S
ymm
etric
mat
rices
:
�In
ner P
rodu
ct &
qua
drat
ic fo
rm:
�Le
t
be
the
set o
f pos
itive
def
inite
sym
met
ric m
atric
es.
The
set i
s an
ope
n co
nex
cone
and
is s
elf-d
ual :
�
is h
omeg
eneo
us :
�C
hara
cter
istic
func
tion
:
()
()
()
()
0)
( , 0
,
,
,,
>=
≠∈
∀
=∈
Tn
xQ
R
xyTr
yx
Rn
Sym
yx
ξξξ
ξξ
)(R
nΠ
=Ω
Ω=
Ω*
)(R
nΠ
=Ω
� �
===
�Ω
∈T
nT
gxg
xg
Ig
ggx
x)
(
)(
ρρ
()
()
nn
Ig
xI
gx
ϕρ
ϕρ
1)
(de
t)
()
(−
=�
=(
)(
)[
](
)n
n
nI
xx
gg
gx
ϕϕ
ρ21
1
2
)de
t()
()
det(
)(
det
)de
t()
det(
+−
+=
� � �
=
=
()
()
nIx
nx
ϕϕ
log
det
log
)1(
21)
(lo
g+
+−
=
1*
1)1
(21
)de
t(lo
g−
−+
=�
=∇
xn
xx
x
150
/15
0/
Air
Sys
tem
s D
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ion
/��#�"���
�3��
��������������� ��
�2��
���;H
���� ��
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Sym
met
ric C
one
& E
xpon
entia
l Fam
illy
of p
roba
bilit
y m
easu
re
�Le
t μ μμμbe
a p
ositi
ve B
orel
Mea
sure
on
eucl
idea
n sp
ace
V. A
ssum
e th
at
the
follo
win
g in
tegr
al is
fini
te fo
r all
xin
an
open
set
:
�Fo
r
, c
onsi
der t
he p
roba
bilit
y m
easu
re (e
xpon
entia
l fam
illy)
:
Then
V⊂
Ω(
)
−=
)(
)(
yd
ex
yx
μϕ
Ω∈x
()
()
)(
)(1
,y
de
xdy
xp
yx
μϕ
−=
)(
log
),
()
(x
dyx
ypx
mϕ
−∇=
=
()
()(
))
(lo
g)
,(
)(
)(
)(
xD
Ddy
xp
vx
my
ux
my
vu
xV
vu
ϕ=
−−
=
����������� �����
Thal
es A
ir S
yste
ms
Dat
e
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� ��
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es A
ir S
yste
ms
Dat
e
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���9
� ��
������9
� �9��
�
+$$
���
����
� 1
���
���
� �
��
� �
�A
loca
lly s
tron
gly
conv
ex h
yper
surf
ace
in th
e af
fine
spac
e R
n+1
is
calle
d an
affi
ne h
yper
bolic
hyp
ersp
here
if th
e af
fine
norm
als
thro
ugh
each
poi
nt o
f the
hyp
ersu
rfac
eei
ther
all
inte
rsec
t at o
ne
poin
t, ca
lled
its c
ente
r, th
at is
on
the
conv
ex s
ide.
�Th
is c
lass
of h
yper
surf
aces
was
firs
t stu
died
sys
tem
atic
ally
by
W.
Bla
schk
ein
the
fram
e of
affi
ne g
eom
etry
. �
E. C
alab
irai
sed
a co
njec
ture
that
:�
thes
e hy
pers
urfa
ces
are
asym
ptot
ic to
the
boun
dary
of a
con
vex
cone
�ev
ery
non-
dege
nera
te c
one
V d
eter
min
es a
hyp
erbo
lic a
ffine
hyp
ersp
here
, as
ympt
otic
to th
e bo
unda
ry o
f V, u
niqu
ely
by th
e va
lue
of it
s m
ean
curv
atur
e.
He
prov
ed th
is c
onje
ctur
e fo
r hom
ogen
eous
con
vex
cone
s un
der s
ome
cond
ition
s on
the
actio
n of
the
auto
mor
phis
mgr
oup
of th
e co
ne.
153
/15
3/
Thal
es A
ir S
yste
ms
Dat
e
����7
����
���3�����
���9
� ��
������9
� �9��
�
��
"�����
+$$
�����
��� 1
���
���� �
��� �
�Th
e th
eory
of h
omog
eneo
us c
onve
x co
nes
play
s a
cent
ral r
ole.
�
Let V
be
a no
ndeg
ener
ate
open
con
vex
cone
in R
n+1 (
x)an
d V’
be
its
dual
. A(V
) mea
ns th
e gr
oup
of a
ll lin
ear t
rans
form
atio
ns w
hich
le
ave
V in
varia
nt. T
he c
hara
cter
istic
func
tion
of V
, is
give
n by
the
equa
tion:
with
the
val
ue o
f the
line
ar fu
nctio
nal �
at x
�W
e de
note
by
S cth
e le
vel s
urfa
ce o
f
whi
ch is
a
nonc
ompa
ctsu
bman
ifold
in V
, and
by �
the
indu
ced
met
ric o
n S c
�Th
e H
essi
an
defin
esth
e m
etric
on
V.
�A
ssum
ing
the
cone
V is
hom
ogen
eous
und
er A
(V),
Sasa
ki p
rove
d th
at S
cis
a h
omog
eneo
us h
yper
bolic
affi
ne h
yper
sphe
rean
d ev
ery
such
hyp
ersp
here
sca
n be
obt
aine
d in
this
way
�
Sasa
ki re
mar
ks th
at �
is id
entif
ied
with
the
affin
e m
etric
and
Sc
is
a gl
obal
Rie
man
nian
sym
met
ric s
pace
whe
n V
is a
sel
f-dua
l con
e.
Vx
de
xV
xV
∈=
− ,
)
('
,ξ
φξ
ξ,x{
} cx
VV
=)(
:φ
φ
Vd
φlo
g2
KO
SZU
L-VI
NB
ERG
C
HAR
AC
TER
ISTI
C
FUN
CTI
ON
154
/15
4/
Thal
es A
ir S
yste
ms
Dat
e
� �9
� �3�����
���9
� ��
������9
� �9��
�
-
��
�+
$$��
���
��
� 1
���
����
� �
��
� �
�Le
t S b
e a
hype
rsur
face
in R
n+1
and
be th
e im
bedd
ing
of S
. The
imbe
ddin
g
def
ines
a v
olum
e bu
ndle
val
ued
quad
ratic
fo
rm G
on
S by
the
equa
tion:
�in
term
s of
loca
l coo
rdin
ates
of S
. Thi
s is
inva
riant
un
der u
nim
odul
ar a
ffine
tran
sfor
mat
ions
in R
n+1 .
If th
is q
uadr
atic
fo
rm is
sup
pose
d to
be
non-
dege
nera
te, i
t def
ines
a
pseu
dorie
man
nian
str
uctu
re te
nsor
gw
ith c
orre
spon
ding
vol
ume
elem
ent d
v(g)
, uni
quel
y de
fined
by
the
equa
tion:
�∧
∧⊗
�� ���� ��
∂∂∂∂
∂∂∂
=j
i
nj
in
ji
dydy
dydy
yfyf
yy
fG
,
11
2
...,..
.,,
det
1:
+→
nR
Sf
f
()
n yy
,...,
1
)(g
dvg
G⊗
=
155
/15
5/
Thal
es A
ir S
yste
ms
Dat
e
� �9
� �3�����
���9
� ��
������9
� �9��
�
-
��
+
$$��
���
��
� 1
���
���
� �
��
� �
�w
e as
sum
e th
at th
e se
t S is
loca
lly s
tron
gly
conv
ex. T
hen
the
tens
or g
can
be c
hose
n to
be
posi
tive
defin
ite c
hoos
ing
the
orie
ntat
ion
of S
so
that
G is
pos
itive
val
ued.
�
With
this
Rie
man
nian
met
ric, c
alle
d th
e af
fine
met
ric, t
he a
ffine
no
rmal
is d
efin
ed to
be
the
vect
or
w
here
�is
the
Lapl
ace-
Bel
tram
i ope
rato
r with
resp
ect t
o g.
�
For a
n af
fine
hype
rbol
ic h
yper
sphe
rew
ith th
e ce
nter
at t
he o
rigin
, n
satis
fies
the
equa
tion:
whe
re H
, cal
led
the
affin
e m
ean
curv
atur
e, is
a n
onze
ro c
onst
ant.
�C
alab
ipro
ved
that
the
hype
rsur
face
Sis
a p
rope
r affi
ne
hype
rsph
ere
with
its
cent
er a
t the
orig
in a
nd th
e af
fine
mea
n cu
rvat
ure
Hif
usa
tisfie
s th
e eq
uatio
n:
fn
nΔ
=)
/1(
Hf
n−
=
()
22
)(
det
−−
= �� ���� ��
∂∂∂
n
ji
Hu
uξ
ξξ
156
/15
6/
Thal
es A
ir S
yste
ms
Dat
e
� �9
� �3�����
���9
� ��
������9
� �9��
�
@
�>�
� �
��
� ��
���
� �
���
���
����
$�����
$$��
���
��� �
��� �
�Le
t
b
e a
linea
r coo
rdin
ate
syst
em o
f Rn+
1an
d
be
the
repr
esen
tatio
n of
S a
s th
e gr
aph
of a
lo
cally
str
ongl
y co
nvex
func
tion
f
or
rang
ing
in a
dom
ain
�
Let
be
the
imag
e of
Dun
der t
he lo
cally
in
vert
ible
map
ping
�
We
defin
e th
e fu
nctio
n
by
the
equa
tion
whe
re
is th
e pa
iring
giv
ing
the
cano
nica
l dua
lity.
�
uis
the
Lege
ndre
tran
sfor
m o
f
and
als
o th
e do
mai
n �
the
Lege
ndre
tran
sfor
m o
f Sw
ith re
spec
t to
the
coor
dina
tes
()
11 ,..
.,+n x
x(
){
}n
nx
xf
x,..
.,1
1=
+
f(
)n x
xx
,...,
1=
nR
D⊂
()
nn
Rξ
ξξ
,...,
,2
1⊂
Ω(
)i
in
xff
ff
grad
f∂∂
==
=
whe
re,..
.,1
ξ(
)Ω
∈n
uξ
ξ,..
.,1
()
)(
,)
()
(x
grad
fx
xf
xgr
adf
u+
−=
.,.f
()
11 ,..
.,+n x
x
157
/15
7/
Thal
es A
ir S
yste
ms
Dat
e
2��
�� ������������
�9� ��
����������
���9
� �9��
�
�Th
eore
m[W
u an
d Sa
ckst
eder
]�
Let S
be a
clo
sed
hype
rbol
ic a
ffine
hyp
ersp
here
with
cen
ter a
t the
orig
in a
nd th
e af
fine
mea
n cu
rvat
ure
H. t
he h
yper
surfa
ceS
is c
ompl
ete
(with
resp
ect t
o th
e af
fine
met
ric a
nd w
ith re
spec
t to
the
indu
ced
met
ric o
f the
Rie
man
nian
met
ric o
f R
n+1 )
, non
com
pact
, orie
ntab
le, s
moo
th a
nd lo
cally
stro
ngly
con
vex.
In th
is
situ
atio
n w
e ha
ve:
�S
uch
a su
rface
is th
e fu
ll bo
unda
ry o
f som
e cl
osed
con
vex
body
and
is th
e gr
aph
of a
non
-neg
ativ
e sm
ooth
stri
ctly
con
vex
func
tion
defin
ed in
som
e hy
perp
lane
.
�B
y th
is th
eore
m th
e hy
pers
urfa
ceS
can
be w
ritte
n gl
obal
ly a
s th
e se
t
, w
here
is a
pos
itive
sm
ooth
str
ictly
co
nvex
func
tion
on
�Th
e ta
ngen
t pla
ne a
t any
poi
nt o
f Sca
nnot
con
tain
the
orig
in. I
n ot
her w
ords
the
affin
e no
rmal
is n
ot ta
ngen
t to
S:�
the
norm
al v
ecto
r in
Euc
lidea
n se
nse
at o
ne p
oint
in S
is p
ropo
rtion
al to
:
with
the
coo
rdin
ate
of th
e Le
gend
re tr
ansf
orm
atio
n
�Th
e af
fine
norm
al a
t tha
t poi
nt is
with
�H
ence
:
()
{}
nn
xx
fx
,...,
11
=+
f{
} 01
=+nx
() 1
,,..
.,1
−=
nEn
ξξ
iξ
�� ���� ��
∂∂+
∂∂∂∂
=� =n i
ii
n
n1
11
2,
,...,
1ξ
ξρρ
ξρξρ
ρ
()
2/1
det
+−
=n
ijfρ
0/1
,≠
−=
ρEn
n
158
/15
8/
Thal
es A
ir S
yste
ms
Dat
e
3���9��������
����
�)� ��������
�5���
��
�S
is a
sym
ptot
ic to
the
boun
dary
of a
con
vex
cone
whe
n th
e bo
unda
ry is
equ
al to
the
set o
f all
asym
ptot
ic li
nes
of S
:�
Let
then
an
d
is a
n op
en n
on-d
egen
erat
e co
nvex
con
e.
�S
is a
sym
ptot
ic to
the
boun
dary
of V
�Th
eore
m:
�E
very
clo
sed
hype
rbol
ic a
ff�ne
hype
rsph
ere
is a
sym
ptot
ic to
the
boun
dary
of a
co
nvex
con
e. C
onve
rsel
y, e
very
non
-deg
ener
ate
cone
V d
eter
min
es a
hyp
erbo
lic
affin
e hy
pers
pher
eas
ympt
otic
to th
e bo
unda
ry o
f V, a
nd u
niqu
ely
dete
rmin
ed b
y th
e va
lue
of it
s m
ean
curv
atur
e.
�Le
gend
re tr
ansf
orm
atio
n is
an
isom
etry
with
resp
ect t
o th
e LN
-m
etric
τ τττ(in
trod
uced
by
Loew
ner&
Nire
nber
g) a
nd th
e af
fine
met
ric
g: �
For a
neg
ativ
e co
nvex
sol
utio
n u
of
Loew
ner&
Nire
nber
g d
efin
ed th
e m
etric
on
a bo
unde
d co
nvex
dom
ain �
in
{}S
pR
kpS
nk
∈∈
=+
/1
'fo
r 0
'k
kS
Sk
k≠
=�
� 0>
=k
kSV
()
22
)(
det
−−
= �� ���� ��
∂∂∂
n
ji
Hu
uξ
ξξ
ud
Hu
21
=τ
)(ξ
nR
159
/15
9/
Thal
es A
ir S
yste
ms
Dat
e
/��"
�� 10� �
��� 7�$�� ��
�LN
-met
ric τ τττ
(intr
oduc
ed b
y Lo
ewne
r& N
irenb
erg)
�Th
is m
etric
has
the
proj
ectiv
e in
varia
nce
in th
e fo
llow
ing
sens
e:
�Le
t
be
a pr
ojec
tive
trans
form
atio
n w
hich
sen
ds �
onto
A�
. Th
en A
is a
n is
omet
ryw
ith re
spec
t to
LN-m
etric
s of
�an
d A�
.
�Le
gend
re tr
ansf
orm
atio
n is
an
isom
etry
with
resp
ect t
o th
e LN
-met
ric
τ τ τ τ an
d th
e af
fine
met
ric g
�Le
t be
an e
quat
ion
of a
hyp
erbo
licaf
fine
hype
rsph
ere.
Sin
ceth
e af
fine
met
ricis
writ
ten
as
but
atth
e co
rres
pond
ing
poin
ts x
and
� b
y th
e Le
gend
re tr
ansf
orm
atio
n. H
ence
:
()
22
)(
det
−−
= �� ���� ��
∂∂∂
n
ji
Hu
uξ
ξξ
ud
Hu
21
=τ
with
()
Rn
SLA
,1+∈
()
nn
xx
fx
,...,
11
=+
()
ijn
ji
ff
ff
=,..
.,,
det
1,
()
()
fd
fg
nij
22
/1de
t+
−=
)(
)(
22
ξu
dx
fd
=
gu
du
dH
u=
==
22
1ρ
τ(
)2
/1de
t+
−=
nijf
ρw
ith
160
/16
0/
Thal
es A
ir S
yste
ms
Dat
e
����7
����
�����9
� ��
����������
���9
� �9��
��
�Le
t Vbe
a n
on-d
egen
erat
e co
nvex
con
e in
Rn+
1 . Fi
rst w
e re
call
som
e pr
oper
ties
of th
e ch
arac
teris
tic fu
nctio
n:
�te
nds
to in
finity
whe
n x
appr
oach
es to
any
poi
nt o
f the
bou
ndar
y of
V
�Th
e m
easu
re
is in
varia
nt u
nder
:
�is
con
vex
on V
. Hen
ce
def
ines
a m
etric
on
V
�Th
e le
vel s
urfa
ce o
f
is
a n
onco
mpa
ctsu
bman
ifold
in V
calle
d th
e ch
arac
teris
tic s
urfa
ce o
f V.
�W
e de
note
by
the
indu
ced
met
ric o
f
o
n
Vx
de
xV
xV
∈=
− ,
)
('
,ξ
φξ
)(x
Vφdx
xV
)(
φ)
(VA)
(fo
r
)de
t(/)
()
(V
AA
Ax
AxV
V∈
=φ
φVφ
log
Vd
φlo
g2
{} c
xS
Vc
V=
=)
(:
φφ
cω
Vd
φlo
g2
cSKO
SZU
L-VI
NB
ERG
C
HAR
AC
TER
ISTI
C
FUN
CTI
ON
161
/16
1/
Thal
es A
ir S
yste
ms
Dat
e
����7
����
�����9
� ��
����������
���9
� �9��
��
�A
ssum
ing
Vis
affi
nely
hom
ogen
eous
. The
cha
ract
eris
tic s
urfa
ce
S cis
obv
ious
ly h
omog
eneo
us w
ith re
spec
t to
unim
odul
arel
emen
ts
of A
(V):
�TH
EOR
EM (S
ASA
KI):
Ever
y ch
arac
teris
tic s
urfa
ce S
cis
a c
ompl
ete
hype
rbol
ic a
ffine
hyp
ersp
here
with
mea
n cu
rvat
ure
ac2 /n
+2w
here
a
is a
neg
ativ
e co
nsta
nt d
epen
ding
onl
y on
V.
�Pr
oof:
Let
. .
c
anbe
writ
ten
loca
llyas
by a
sm
ooth
func
tion
, the
co
ordi
nate
bein
gch
osen
such
that
. Le
t
the
Lege
ndre
tran
sfor
mof
.
Sin
ceis
cons
tant
on
we
have
on
.
By
the
defin
ition
:
B
ut
Hen
ce
Ther
efor
e
)(
log
)(
xx
Vφψ
={
} cS c
log
==
ψ)
,...,
(1
1n
nx
xf
x=
+f
()
11 ,..
.,+n x
x0
1≠
+nψ
uf
ψcS
1/
+=
ni
ifψ
ψcS
11
1/
+=
+� ��
� ��+
−=
�n
n ii
in
xf
uψ
ψψ
0,
, lo
g)1
()
()
(>
∀∈
∀+
−=
kV
xk
nx
kxψ
ψ
�+ =
+−
=1 1
)1(
)(
n
nx
xα
αα ψ
1/)1
(+
+=
nn
uψ
162
/16
2/
Thal
es A
ir S
yste
ms
Dat
e
����7
����
�����9
� ��
����������
���9
� �9��
��
�C
onsi
der
�Se
t
�W
e ha
ve�
Hen
ce
by
the
hom
ogen
eity
for s
ome
cons
tant
bw
hich
dep
ends
onl
y on
Vits
elf.
This
mea
ns
�Si
nce
, th
en
()
()
0
1de
t
,1 ,
det
det
1
11
11
1
21
1
21
11
1
11
1
+
++
++
+
++
+
++
+
++
+
=−
≤≤
�� ���� ��
++
−=
−
nj
nn
nj
n
iin
ij
nij
n
n
nn
ji
n
jni
jin
ijij
n
f
nj
if
ψψ
ψψ
ψψ
ψψ
ψψ
ψψψ
ψψ
ψψ
ψψ
ψψ
1,
1 , 0
)(
+≤
≤=
Φn
xβ
αψ
ψψ
β
ααβ
()
)(
, )(
det
)(
2V
AA
AxA
x∈
Φ=
Φ ()
12
det
det
− �� ���� ��
∂∂∂
=j
iij
uf
ξξ)
()
(2
xb
xφ
=Φ (
)c
ijn
nS
bcx
fon
)(
det
21
21
=Φ
=−
++
ψψ
2
22
11
det
bcun
un
ji
+
� ��� ��
+−
= �� ���� ��
∂∂∂
ξξ
11 ++=
n
nu
ψR
emar
k:
163
/16
3/
Thal
es A
ir S
yste
ms
Dat
e
����7
����
�����9
� ��
����������
���9
� �9��
��
�TH
EOR
EM:
Let S
a co
mpl
ete
hype
rbol
ic a
ffine
hyp
ersp
here
with
its
cen
ter a
t the
orig
in w
hich
is h
omog
eneo
us u
nder
the
subg
roup
G
of th
e un
imod
ular
grou
p. L
et V
, the
con
vex
cone
to w
hose
bo
unda
ry th
e hy
pers
urfa
ceS
is a
sym
ptot
ic:
.
Let
, T
he e
lem
ent
acts
on
Vby
Vis
hom
ogen
eous
und
er
.
Then
Sis
a c
hara
cter
istic
sur
face
of V
.Pr
oof:
Let f
unct
ion
on V
by
the
equa
tion
Sinc
e
,
is
wel
l def
ined
. Th
en
for
A e
G.
Ther
efor
e, b
y th
e ho
mog
enei
ty,
for s
ome
nonz
ero
cons
tant
b.
�0>
=k
kSV
+×
=R
GG~
()
Gt
gg
~,
∈=
)(
.x
gtg
=G~
kn
Sx
kx
∈=
−−
for
)(
1γ
γ'
for
'
kk
SS
kk
≠=
∩φ
γ)
det(
/)(
)(
Ax
Axγ
γ=
Vbφ
γ=
164
/16
4/
Thal
es A
ir S
yste
ms
Dat
e
����7
����
�����9
� ��
����������
���9
� �9��
��
�2
prev
ious
Sas
aki ‘
s Th
eore
ms
prov
e th
at th
e cl
assi
ficat
ion
of
hom
ogen
eous
hyp
erbo
lic a
ffine
hyp
ersp
here
sis
redu
ced
to th
e cl
assi
ficat
ion
of h
omog
eneo
us c
onve
x co
nes:
�R
otha
us, O
. S.,
The
cons
truct
ion
of h
omog
eneo
us c
onve
x co
nes,
Ann
. of M
ath.
, 83
, pp
. 358
-376
., 19
66
�V
inbe
rg, E
. B.,
The
theo
ry o
f con
vex
hom
ogen
eous
con
es, T
rans
. Mos
cow
Mat
h.
Soc
, 12
(196
3), 3
40-4
03
�E.
B. V
inbe
rgha
s de
fined
an
indu
ctiv
e m
etho
d pr
oduc
ing
all
hom
ogen
eous
con
vex
cone
s :
�Fr
om a
giv
en c
onve
x co
ne V
iin
Rn+
1 (x)
, one
can
con
stru
ct a
noth
er
hom
ogen
eous
con
e V
in R
(t) X
Rm
(y) X
Rn+
1 (x)
by
the
equa
tion
:
whe
re h
is a
line
ar m
appi
ng o
n R
n+1
who
se v
alue
s ar
e re
al s
ymm
etric
pos
itive
-de
finite
mat
rices
of o
rder
m a
nd, c
orre
spon
ding
to e
ach
elem
ent
B o
f som
e tra
nsiti
ve s
ubgr
oup
of A
{V1)
, the
re e
xist
s a
mat
rix
such
that
. Thi
s m
etho
d ca
n be
tran
spos
ed to
obt
ain
all p
roje
ctiv
ely
hom
ogen
eous
bou
nded
con
vex
dom
ains
or a
ll ho
mog
eneo
us h
yper
bolic
a�fi
nehy
pers
pher
es
{} y
xh
yt
xy
tV
t)
(/)
,,
(1−
>=
),
(R
mG
LA
∈(
)Bx
hA
xh
At=
)(
165
/16
5/
Thal
es A
ir S
yste
ms
Dat
e
����7
����
�����9
� ��
����������
���9
� �9��
��
�TH
EOR
EM: S
uppo
se V
is h
omog
eneo
us. T
hen
the
met
ric �
cis
id
entif
ied
with
the
affin
e m
etric
gup
to a
con
stan
t fac
tor.
Proo
f :
Let
.
c
an b
e w
ritte
n lo
cally
as
by a
sm
ooth
func
tion
,
the
coor
dina
te
be
ing
chos
en s
uch
that
Si
nce
we
have
But
. Th
eref
ore
The
assu
mpt
ion
need
ed is
not
the
hom
ogen
eity
of V
but
the
cond
ition
that
the
leve
l sur
face
is a
n af
fine
hype
rsph
ere.
)(
log
)(
xx
Vφψ
={
} cS c
log
==
ψ)
,...,
(1
1n
nx
xf
x=
+(
)1
1 ,...,
+n xx
f0
1≠
+nψ
� =+
+−
=n i
i
nin
dxdx
11
1
ψψ
� ≤≤
+−
==
nj
i
ji
ijn
Sc
dxdx
fd
c,
11
2ψ
ψω
un
n/)1
(1
+=
+ψ
Hg
nc
)1(
+−
=ω
166
/16
6/
Thal
es A
ir S
yste
ms
Dat
e
����7
����
�����9
� ��
����������
���9
� �9��
��
�R
emar
k: T
he s
olut
ion
u of
the
equa
tion
is g
iven
as
a fir
st o
rder
loga
rithm
ic d
eriv
ativ
e of
the
char
acte
ristic
func
tion,
i.e.
Sinc
e th
e ch
arac
teris
tic fu
nctio
n
of t
he p
rodu
ct o
f con
vex
cone
s V
and
W is
equ
al to
, t
he d
eriv
ativ
e of
is w
ritte
n us
ing
the
deriv
ativ
es o
f an
d
.
This
giv
es th
e co
mpo
sitio
n fo
rmul
a fo
r re
duci
ble
cone
s.
WVφ
φ
()
22
)(
det
−−
= �� ���� ��
∂∂∂
n
ji
Hu
uξ
ξξ
1/)1
(+
+=
nn
uψ
φφ
VφWφ
167
/16
7/
Thal
es A
ir S
yste
ms
Dat
e
����7
����
�����9
� ��
����������
���9
� �9��
��
�C
onsi
derin
g th
e se
lf-du
al c
one,
to
form
ulat
e an
othe
r des
crip
tion
of th
e Le
gend
re tr
ansf
orm
atio
n w
e in
trod
uce
the
*-m
appi
ng d
ue to
K
oech
er ,
It is
a m
appi
ng fr
om V
to it
s du
al V
def
ined
by
the
equa
tion:
�Th
is m
appi
ng *
has
the
follo
win
g pr
oper
ties:
�*
sets
up
a on
e to
one
cor
resp
onde
nce
betw
een
Van
d V
and
hold
s fo
r eve
ry
�If
Vis
hom
ogen
eous
, the
n V
’ is
also
hom
ogen
eous
and
is c
onst
ant
for a
ll x.
We
deno
te th
is c
onst
ant b
y
�In
hom
ogen
eous
cas
e, th
e *-
imag
e of
Sc
is a
lso
a ch
arac
teris
tic
surf
ace
of V
whi
ch w
e de
note
by
S'c.
Taki
ng a
hyp
erpl
ane
Hsu
ch
that
is a
non
-em
pty
boun
ded
conv
ex d
omai
n, w
e co
rres
pond
, for
eve
ry p
oint
th
e in
ters
ectio
n po
int o
f the
lin
e th
roug
h th
e or
igin
and
x*w
ith U
. Thu
s w
e ha
ve a
map
ping
from
S'
cto
U.
�LE
MM
A: T
he m
appi
ng *
follo
wed
by
this
map
ping
is d
efin
ed o
n S c
and
proj
ectiv
ely
equi
vale
nt to
eve
ry L
egen
dre
tran
sfor
mat
ion
of S
c.
()
Vx
xx
xn
∈−
==
+fo
r )
(),.
..,(
11
*ψ
ψξ
()
*1
* )(
xA
Axt
−=
)(VA
A∈
)(
)(
*'
xx
VV
φφ
2κ
'V
HU
∩=
'*
cSx
∈
168
/16
8/
Thal
es A
ir S
yste
ms
Dat
e
����7
����
�����9
� ��
����������
���9
� �9��
��
�PR
OPO
SITI
ON
: Sup
pose
Vis
hom
ogen
eous
. The
n * i
s an
isom
etry
w
ith re
spec
t to
the
met
rics
a
nd
�Pr
oof:
Vd
φlo
g2
'2
log
Vd
φ
()(
)()
V
lk
lk
ji
lj
Vk
iV
ji
V
ji
ji
ji
VV
k ij
kj
Vj
iV
d
kk
kj
Vj
j
jj
iV
i
Vx
VV
VV
d
dxdx
xx
xx
xx
x
dd
d
xx
xx
I
ddx
dxx
xx
xd
grad
x
ψ
ψψ
ξξψ
ξξ
ξξψ
ψ
δξ
ξψψ
ξξ
ξξψ
ξψ
ξ
ψφ
ψφ
ψ
2
,,
,
22
*'
2
,
'2
**
'2
**
*'
22
**
'2
*2
*
*'
'
)(
)(
)(
)(
).(
)(
)(
,)
()
(
,lo
g,
log
=
∂∂∂
∂∂∂
∂∂∂
=
�� ���� ��
∂∂∂
=
=∂
∂∂∂
∂∂�
=
∂∂∂
−=
∂∂∂
−=
−=
==
�
�
�
��
169
/16
9/
Thal
es A
ir S
yste
ms
Dat
e
����7
����
�����9
� ��
����������
���9
� �9��
��
�N
ow a
ssum
e V
is a
sel
f-dua
l con
e. T
hen
is a
n au
tom
orph
ism
of V
and,
mor
eove
r,
�Le
t
, * /
Sis
an
invo
lutiv
eau
tom
orph
ism
of S
.
Sinc
e
, we
have
.The
refo
re K
(x)i
s an
aut
omor
phis
mof
S
�PR
OPO
SITI
ON
: If
the
hom
ogen
eous
con
e V
is s
elf-d
ual,
then
the
char
acte
ristic
sur
face
Sc
is a
glo
bally
sym
met
ric s
pace
.�
Proo
f: Fo
r one
poi
nt
d
efin
e an
aut
omor
phis
ms
of S
by
. S
ince
K(x
)is
sym
met
ric, s
is a
n in
volu
tion
by
By
For a
gen
eral
Sc,
it is
eno
ugh
to tr
ansl
ate
the
sym
met
ry o
f Sto
Sc
by th
e m
appi
ng
xx
Kx
)(
*=
xx
xK
∂−∂
=/
)(
*
*KS
S=
()
Kx
xx
Kx
VV
Vde
t)
()
()
(*
φφ
φ=
=S
xx
K∈
=fo
r 1
))(
det(
Sx
∈0
*1
0)
()
(x
xK
xs
−=
dIx
xxx
Kx
xx
s−
=∂∂
=∂∂
=−
)(
)(
)(
xs
and
)
(0
*1
00
00
Sx
cS
xx
∈→
∈κ
170
/17
0/
Thal
es A
ir S
yste
ms
Dat
e
����7
����
�����9
� ��
����������
���9
� �9��
��
�Le
t H+ (
n, K
) be
the
cone
of p
ositi
ve-d
efin
ite h
erm
itian
sym
met
ric
mat
rices
ove
r K
= fi
elds
R, C
, H (q
uate
rnio
ns) o
r the
Cay
ley
alge
bra
Ca.
The
n th
e fo
llow
ings
are
the
list o
f all
irred
ucib
le s
elf-d
ual
cone
s V
and
the
corr
espo
ndin
g gl
obal
ly s
ymm
etric
spa
ces
S
EIV
type
ofsp
ace
the
),2(
)1(
/)1,1(
)(
)(
/)2(
),
()
(/)
,(
),
()
(/)
,(
),
( *
*
=�
=
−−
=�
=
=�
=
=�
==
�=
+++
SC
aH
Vn
SOn
SOS
nC
Vn
Sn
SUS
Hn
HV
nSU
Cn
SLS
Rn
HV
nSO
Rn
SLS
Rn
HV
p
171
/17
1/
Thal
es A
ir S
yste
ms
Dat
e
����7
����
�����9
� ��
����������
���9
� �9��
��
�C
orol
lary
: Le
t be
a re
gula
r con
vex
cone
and
let
be
the
cano
nica
l Hes
sian
met
ric. T
hen
each
leve
l sur
face
of t
he
char
acte
ristic
func
tion
i
s a
min
imal
sur
face
of t
he R
iem
anni
an
man
ifold
�
Exam
ple:
Let
be
a re
gula
r con
vex
cone
con
sist
ing
of a
ll po
sitiv
e de
finite
sym
met
ric m
atric
es o
f deg
ree
n. T
hen
is a
Hes
sian
str
uctu
re o
n
, a
nd e
ach
leve
l sur
face
of
i
s a
min
imal
sur
face
of t
he R
iem
anni
an m
anifo
ld:
ψlo
gD
dg
=
ψ(
) g,Ω
() x
Dd
gD
det
log
,−
=Ω
xde
t(
) xD
dg
det
log
,−
=Ω
����������� �����
Thal
es A
ir S
yste
ms
Dat
e
��� 1(��������"
173
/17
3/
Air
Sys
tem
s D
ivis
ion
2��
9��5
�3��� �7
���� ��$�)
������
�� ��
����� �
�Er
ich
Käh
ler G
eom
etry
is g
iven
by
:�
com
plex
Man
ifold
of n
dim
ensi
ons,
com
pact
or n
ot, w
ith K
ähle
rian
met
ric, t
hat
coul
d be
loca
lly g
iven
by
posi
tive
defin
ite R
iem
anni
an F
orm
:
�K
ähle
r con
ditio
n :
Loca
l Exi
sten
ce o
f Käh
ler p
oten
tial f
unct
ion
, (a
nd P
luri-
harm
onic
equ
ival
ent)
such
that
:
�R
icci
Ten
sor i
s gi
ven
by re
mar
kabl
e E
xpre
ssio
n [E
rich
Käh
ler]
:
�A
nd s
cala
r cur
vatu
re :
� =
=n j
i
ji
jidz
dzg
ds1
,
2.
2
ji
jiz
zg
∂∂
Φ∂
=2
()
ji
lkji
zz
gR
∂∂
∂−
=de
tlo
g2 � =
=n l
klk
lkR
gR
1,
.
Φ
174
/17
4/
Air
Sys
tem
s D
ivis
ion
��� ��
�� ��
��� �2
��9��5
�3��� �7
���� ��$�)
���
Inth
e fr
amew
ork
of A
ffine
Info
rmat
ion
Geo
met
ry, m
etric
is g
iven
by
Hes
sian
of E
ntro
py :
�E
ntro
py fo
r Mul
tivar
iate
Gau
ssia
n m
odel
of z
ero
mea
n:
�In
cas
of C
ompe
x A
utor
egre
ssiv
e M
odel
of o
rder
n, E
ntro
py c
ould
be
expr
esse
d by
re
flect
ion
coef
ficie
nts
:
()
()
() e
nR
R�
πlo
gde
tlo
g~
−−
=-R
H�
��
gj
iij
=∂
∂∂
≡
and
~
2
[]
[]1 0
1 1
2..
ln.1
ln).(
~−
− =
+−
−=�
απ
μe
nk
n)
(R�
n kk
n
[]
� =
−
− −−
==
−=
n kk
nn
n
xn
P1
20
1 0
1 12
1
1w
ith
.1 α
αμ
α(
)[
]∏
∏− =
−−
− =
−−
==
1 1
20
1 0
11
det
n k
kn
kn
n kk
nRμ
αα
175
/17
5/
Air
Sys
tem
s D
ivis
ion
�W
e de
fine
«D
oppl
er»
met
ric in
cas
e of
Com
plex
Aut
oreg
ress
ive
Mod
el b
y H
essi
an o
f Käh
ler P
oten
tial,
whe
re P
oten
tial i
s gi
ven
as in
In
form
atio
n A
ffine
Geo
met
ry b
y En
trop
y :
�K
ähle
r Pot
entia
l is
give
n by
Ent
ropy
par
amet
rized
by
refle
ctio
n co
effic
ient
s :
�M
etric
can
be
expl
icitl
y co
mpu
ted
:[]
[]1 0
1 1
2..
ln.1
ln).(
~−
− =
+−
−=�
απ
μe
nk
n)
(R�
n kk
n []
[]T
n nn
Tn
nP
)(
)( 1
11
0)
(θ
θμ
μθ
��
==
−
20
2 011
−=
=nP
ng
α(
)22
1
).(
i
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in
gμ
δ
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()
�− =−
−+
�� ���� ��
=1 1
222
2
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ci T
enso
r exp
ress
ion
give
n by
Eric
h K
ähle
r in
fram
ewor
k of
Käh
ler G
eom
etry
�In
Käh
ler G
eom
etry
, Ric
ci T
enso
r is
give
n by
:
�W
e ca
n co
mpu
te R
icci
tens
or fo
r Com
plex
Aut
oreg
ress
ive
Mod
el :
�Its
neg
ativ
e sc
alar
cur
vatu
re is
giv
en b
y :(
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i
lkji
zz
gR
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tlo
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� �
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=
−=
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.,2
for
1
2
12
22
2 011
nk
R
PR
kkllk
μδ
�=
lk
lklk
Rg
R,
.∞
−→
� ��� ��
−−
=∞
→
− =�n
n jj
nR
1 0)
(1
.2
177
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s D
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3��� �7
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���L��
�� 1-��������$
�� ��
�Pr
evio
usm
etric
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t a K
ähle
r-Ei
nste
in m
etric
, but
a c
lose
mat
rix
stru
ctur
e�
A m
etric
isca
lled
Käh
ler-
Ein
stei
nm
etric
if its
Ric
ci te
nsor
ispr
opor
tiona
lto
the
met
ric:
�In
cas
e of
Käh
ler-
Ein
stei
n M
etric
, Käh
lerP
oten
tiali
sso
lutio
n M
onge
-Am
père
E
quat
ion
:
�Fo
r Com
plex
Aut
oreg
ress
ive
Mod
el, w
eha
ve :
[]
[]
[]
()
{}
,....,
2
whe
rean
d
)(
1.2
with
1)
(
1 0
)(
)(
−
− =
−−
=
� ��� ��
−−
==
=�
in
diag
B
jn
BTr
Rg
BR
n
n j
nij
nij
()
ji
ji
lklk
jiz
zk
zz
gk
gk
R∂
∂Φ
∂=
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�=
2
0
2
00
.de
tlo
gco
nsta
nt :
w
ith
.
�� =
Φ−
func
tion
holo
mor
phe
Pote
ntia
lK
ähle
r
with
)
det(
02
� :
� :
eg
klk
ψ
178
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re o
f fun
dam
enta
l int
eres
t in
Mat
hem
atic
&
Phy
sics
bas
ed o
n va
riatio
nal A
ppro
ach
:�
Käh
ler-
Ric
ci F
low
:
�C
alab
i Flo
w :
�In
egal
ity b
etw
een
both
func
tiona
ls
Hilb
ert A
ctio
nK
ähle
r-R
icci
Flo
w
=
M
M
gdV
gdV
gR
R)
(
)(
)(
M
gdV
gR
)(
)(
jiji
jig
nRR
tg+
−=
∂∂
Cal
abi A
ctio
nC
alab
i Flo
w(
)*
*2
, ji
jiz
zz
zg
∂∂∂
=φ
Cal
abi F
low
=
M
gdV
gR
gS
)(
)(
)(
2
*
2
ji
ji
zz
Rtg
∂∂∂
=∂∂
RR
t−
=∂∂φ
�� ���� ��
≥M
M
gdV
gdV
gR
gS
)(
/)
()
()
(2
179
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9/
Air
Sys
tem
s D
ivis
ion
In 2
dim
ensi
on o
f the
com
plex
var
iabl
e, C
alab
i & K
ähle
r-Ric
ci
flow
s ar
e cl
osel
y re
late
d :
�C
alab
i & K
ähle
r-R
icci
Flo
ws
can
be re
late
d by
:
�If
seco
nd d
eriv
ativ
e ac
cord
ing
to K
ähle
r-R
icci
flow
tim
e is
id
entif
icat
ed w
ith o
ppos
ite o
f sec
ond
deriv
ativ
e ac
cord
ing
to C
alab
i flo
w ti
me
, bot
h flo
ws
are
equi
vale
nt:
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�"��
��2����
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�� 1(���
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−=
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∂∂
∂=
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�
� �
∂∂
∂−
=
∂∂−
=∂∂
�−
=∂∂
�R
Rg
tg
tg
zz
tg
zz
gR
tRtg
Rtg
ji
jiji
ji
ji
ji
ji
jiji
jiji
,
*
2
2
2
*
2
2
2
)de
t(lo
g
)de
t(lo
g
)de
t(lo
g
CR
ttt
∂∂−
=∂∂
22
*
2
2
2
ji
ji
zz
Rtg
∂∂∂
−=
∂∂�
180
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Air
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s D
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Giv
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ric :
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zR zz
∂∂
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−=
()
an
d
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t(lo
g*
**
2Φ
=∂
∂∂
−=
eg
zz
gR
zzzz
*
2
*
2
*
2
..
w
ith
zz
et
zz
te
zz
teR
tgji
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∂∂∂
=Δ
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∂Φ∂�
∂∂
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=∂Φ∂
�∂
∂Φ
∂=
∂∂�
−=
∂∂
Φ−
ΦΦ
Käh
ler-
Ric
ci F
low
ΔΔΦ
−=
∂Φ∂�
∂∂
ΔΦ∂
−=
∂∂�
ΔΦ−=
∂∂
Φ∂
−=
=∂
∂∂=
∂∂
Φ
Φ−�
t
zz
te
zz
eR
gR
zz
Rtg
ji
jiji
ji
*
2
,*
2
*
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an
d
Cal
abi F
low
181
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1/
Air
Sys
tem
s D
ivis
ion
Bas
ed o
n pr
evio
us R
ao m
etric
com
puta
tion
for
Aut
oreg
ress
ive
Mod
el :
�K
ähle
r-R
icci
Flo
w o
n R
efle
ctio
n C
oeffi
cien
ts :
�C
alab
i Flo
w o
n R
efle
ctio
n C
oeffi
cien
ts :
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()
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2
00
)(
22
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)(
an
d
.)0(
1)
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nt
in
t
ii
eP
tP
et
−−
−
=−
=−
μμ
()
�− =
−
−−
+�� ��
�� ��=
1 1
)(
4
22
24
2
0
02
22
)0(1)
()0(
)(n i
in
t
i
int
ne
di
ne
PdPn
tds
μμ
()
nti
nt
ii
eP
tP
et
−−
−=
−=
−)0(
)(
an
d
.)0(
1)
(1
00
)(
22
μμ
()
�− =
−
−−
+�� ��
�� ��=
1 1
)(
2
22
22
2
0
02
)0(1)
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)(n i
in
t
i
int
ne
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ne
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184
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/������
����
J��9�
������
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����
� ��
«Le
s an
imau
x se
div
isen
t en
: a) a
ppar
tena
nt à
l’Em
pere
ur,
b) e
mba
umés
, c)a
ppriv
oisé
s, d
) coc
hon
de la
it, e
) sirè
nes,
f)
fabu
leux
, g) c
hien
s en
libe
rtés
, h)in
clus
dan
s la
pré
sent
e cl
assi
ficat
ion,
i) q
ui s
’agi
tent
com
me
des
fous
, j)
inno
mbr
able
s, k
) des
siné
s av
ec u
n pi
ncea
u tr
ès fi
n en
poi
l de
cham
eau,
l) e
t cet
era,
m) q
ui v
ienn
ent d
e ca
sser
la c
ruch
e, n
) qu
i de
loin
sem
blen
t des
mou
ches
»En
cycl
opéd
ie c
hino
ise
tiré
d’un
text
e de
J.L
. Bor
ges
(En
préf
ace
de l’
ouvr
age
«Le
s m
ots
et le
s ch
oses
» de
M
iche
l Fou
caul
t)«
La m
onst
ruos
ité q
ue B
orge
s fa
it ci
rcul
er d
ans
son
énum
érat
ion
cons
iste
en
ceci
que
l’es
pace
co
mm
un d
es re
ncon
tres
s’y
trou
ve ru
iné.
Ce
qui e
st im
poss
ible
, ce
n’es
t pas
le v
oisi
nage
des
ch
oses
, c’e
st le
site
lui-m
ême
où e
lles
pour
raie
nt v
oisi
ner.
… L
es c
hose
s y
sont
‘cou
chée
s’,
‘pos
ées’
, ‘di
spos
ées’
dan
s de
s si
tes
à ce
poi
nt d
iffér
ents
qu’
il es
t im
poss
ible
de
trou
ver p
our e
ux
un e
spac
e d’
accu
eil ,
de
défin
ir au
des
sous
des
uns
et d
es a
utre
s un
lieu
com
mun
. … S
ur q
uelle
‘ta
ble’
, se
lon
quel
esp
ace
d’id
entit
és, d
e si
mili
tude
s, d
’ana
logi
e, a
vons
nou
s pr
is l’
habi
tude
de
dist
ribue
r tan
t de
chos
es d
iffér
ente
s et
par
eille
s ?
»M
iche
l Fou
caul
t «Le
s m
ots
et le
s ch
oses
»
Jorg
e Lu
is B
orge
s
185
/18
5/
' ��
���.���
��8��
�)����� �
����
����
��
«La
tram
e sé
man
tique
de
la re
ssem
blan
ce a
u XV
Ièm
e si
ècle
, est
fort
rich
e : A
mic
itia,
Ae
qual
itas
(con
trac
tus,
con
sens
us, m
atrim
oniu
m, s
ocie
tas,
pax
et s
imili
a), C
onso
nanc
ia,
Con
cert
us, C
ontin
uum
, Par
itas,
Pro
port
io, S
imili
tudo
, Con
junc
tio, C
opul
a.
Mai
s il
y en
a q
uatr
e qu
i son
t, à
coup
sûr
ess
entie
lles
:•L
a C
ON
VEN
IEN
TIA
: est
une
ress
embl
ance
liée
à l’
espa
ce d
ans
la fo
rme
du ‘p
roch
e en
pr
oche
’. El
le e
st d
e l’o
rdre
de
la c
onjo
nctio
n et
de
l’aju
stem
ent.
Elle
app
artie
nt m
oins
au
x ch
oses
qu’
au m
onde
dan
s le
quel
elle
s se
trou
vent
.•L
’AEM
ULA
TIO
: sor
te d
e co
nven
ance
affr
anch
ie d
e la
loi d
u lie
u, q
ui jo
uera
it im
mob
ile
dans
la d
ista
nce.
Les
ann
eaux
ne
form
ent p
as u
ne c
haîn
e co
mm
e le
s él
émen
ts d
e la
co
nven
ance
, mai
s pl
utôt
des
cer
cles
con
cent
rique
s, r
éflé
chis
et r
ivau
x.•L
’AN
ALO
GIE
: affr
onte
men
t des
ress
embl
ance
s à
trav
ers
l’esp
ace.
•le
JEU
X D
ES S
YMPA
THIE
S: n
ul c
hem
in n
’est
dét
erm
iné
à l’a
vanc
e, n
ulle
dis
tanc
e n’
est
supp
osée
, nul
enc
haîn
emen
t pre
scrit
. Elle
par
cour
t en
un in
stan
t les
esp
aces
les
plus
va
stes
. Sa
figur
e ju
mel
le, l
’ant
ipat
hie
mai
ntie
nt le
s ch
oses
en
leur
isol
emen
t et e
mpê
che
les
assi
mila
tions
.En
fin, l
l n’y
a p
as d
e re
ssem
blan
ce s
ans
sign
atur
e. L
e sa
voir
des
sim
ilitu
des
se fo
nde
sur l
e re
levé
des
sig
natu
res
et s
ur le
ur d
échi
ffrem
ent.
»«
Les
mot
s et
les
chos
es»
de M
iche
l Fou
caul
t
186
/18
6/
���� �
� �
Col
loqu
e de
rent
rée
Col
lège
de
Fran
ce 2
011
: La
vie
des
form
esA
nne
Fago
t-Lar
geau
lt-L
a fo
rme
chez
Pla
ton
et A
risto
teht
tp://
ww
w.co
llege
-de-
franc
e.fr/
docu
men
ts/a
udio
/Col
rent
ree1
1/co
l-fag
ot-
larg
eaul
t-201
1101
3.m
p3ht
tp://
ww
w.co
llege
-de-
franc
e.fr/
docu
men
ts/v
ideo
/col
loqu
e11/
Fran
cais
/col
-fa
got-l
arge
ault-
2011
1013
.mp4
Mot
«Fo
rme
» en
Gre
c: M
orph
è, E
idos
& Id
eaPo
ur P
lato
n, il
s’a
git d
e «
Met
tre
en é
vide
nce
des
Rel
atio
ns e
ntre
El
émen
ts: d
es R
elat
ions
app
réhe
ndée
s pa
r l’in
telli
genc
e
et
non
des
Elé
men
ts to
uché
s pa
r les
sen
s»
D’a
bord
écr
it «
enfo
rmer
», l
e m
ot «
info
rmer
» a
ppar
aît e
n fra
nçai
s en
128
6,
empr
eint
au
latin
« in
form
are
», li
ttéra
lem
ent «
don
ner u
ne fo
rme
».
Le
mot
« in
form
atio
n »
appa
raît
conj
oint
emen
t au
XIII
ème
sièc
le.
De
l’éty
mol
ogie
gre
cque
, ����
, mor
phê
(« fo
rme
»), n
ous
est p
arve
nu le
se
ns d
e m
orph
olog
ie, l
a sc
ienc
e de
s fo
rmes
. Le
con
cept
fond
amen
tal s
embl
e do
nc ê
tre c
elui
de
« fo
rmes
» e
t de
cite
r C
icér
on: «
mat
eria
, qua
m fi
ngit
et fo
rmat
effe
ctio
»
(la
mat
ière
que
mou
le e
t met
en
form
e la
forc
e m
otric
e).
187
/18
7/
/*.
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��� �
��1�
� 7���
Lede
veni
retl
afo
rme
…O
r,la
vie
est
une
évol
utio
n.N
ous
conc
entr
ons
une
pério
dede
cette
évol
utio
nen
une
vue
stab
lequ
eno
usap
pelo
nsun
efo
rme,
et,
quan
dle
chan
gem
ent
est
deve
nuas
sez
cons
idér
able
pour
vain
cre
l'heu
reus
ein
ertie
deno
tre
perc
eptio
n,no
usdi
sons
que
leco
rps
ach
angé
defo
rme.
Mai
s,en
réal
ité,l
eco
rps
chan
gede
form
eà
tout
inst
ant.
Ou
plut
ôtil
n'y
apa
sde
form
e,pu
isqu
ela
form
ees
tde
l'im
mob
ileet
que
laré
alité
est
mou
vem
ent.
Ce
quie
stré
el,c
'est
lech
ange
men
tcon
tinue
lde
form
e:l
afo
rme
n'es
tqu'
unin
stan
tané
pris
suru
netr
ansi
tion.
…Q
u'il
s'ag
isse
dem
ouve
men
tqu
alita
tifou
dem
ouve
men
tév
olut
ifou
dem
ouve
men
tex
tens
if,l'e
sprit
s'ar
rang
epo
urpr
endr
ede
svu
esst
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«L’
évol
utio
nC
réat
rice
»H
enri
Ber
gson
,190
7
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