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English Pages 400 [407] Year 2001
LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor N.J. Hitchin, Mathematical Institute, University of Oxford, 24-29 St Giles, Odord OX I 3LB. United Kingdom 1nc titles below are available from booksellers, or, in ca!li.C of difficulty, from Cambridge University Press.
46
p-adic Analysis: a short course on recent wort., N. KOBLITZ
.59 66 86 SR
Applicable differential geometry, M. CRAMPIN & F.A.E. PIRANI Several complex variables and complex manifolds II, M.J. FIELD Topological topics, I.M. JAMES (ed) FPF ring theory, C. FAITH & S. PAGE Polylopes and symmetry, S.A. ROBERTSON Diophantine equations 0'Yer function fields, R.C. MASON Varieties of constructi'Yt mathcmatk:s, D.S. BRIOOES & F. RICHMAN Methods of differential gcomc1ry in algebraic topology, M. KAROUBI & C. LERUSTE Stopping time tcchniques for anaJysts and probabilists, L. EGG HE Elliptic structures on 3-manifolds, C.B. THOMAS A local spectral theory for closed operators, I. ERDEL YI & WANG SHENGWANG Compa:tification of Siegel moduli schemes, C.·L. CHAI Diophantine analysis, J. LOXTON & A. VAN DER POORTEN (eds) Lectures on the asymptotic theory of ideals, D. REES Representations of algebras, P.J. WEBB (ed) Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL Proceedings of Groups· St Andrews 1985, E. ROBERTSON & C. CAMPBEU. (eds) Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU Model theory and modules, M. PREST Algebraic, extremal & metric combinatorics, M.-M. DEZA, P. FRANKL & I.G. ROSENBERG (eds) Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UHL (eds) Advaoccs in homotopy theory, S. SALAMON, 8. STEER & W. SUTHERLAND (eds) Ge
h
is
0,
N
q u e s t i o n s
e a c h
- »
a t
o n
h
t h a t
c o m p u t i n g
c o m p u t e d
e n d - p o i n t s
h ~ * x *, N u*(x)
i
b e
i n c r e a s i n g
t h e
i n t o
e l e m e n t
W h a t
a s
c r e d i t
{
i n
a t
e a c h c a n
i g n o r e d .
s l o p e
h a r d
o n
I ( v h )
m o n o t o n e
[0,1]
c o n t i n u o u s
s o
c o n v e r g e s
infinite
b e h a v i o u r
a r e affine
d i s c r e t e
uo
i" a m o n g
i n t e g r a n d ) ,
b e
u*h
s u b d i v i d i n g
i n t e g r a l
t h e
i n s t a n c e
t o
f i n i t e - e l e m e n t
m e s h
w h i c h
Vh
R e m a r k a b l y ,
f a c t
T h i s
(2.2)
f o r m
first
n a t u r a l
m i n i m i z e
f u n c t i o n
e x p l i c i t
m i n i m i z e r h
E
3
m i n i m i z e r s
T h u s
0
a
w h e r e
of
S u r e l y
l i m ^ o
u
I (
h )
=
d x = — h - \
x
0 ! (see
B a l l
&
K n o w l e s
[12])
t h a t
if
1
of
I
call
it
i n
( 0 , 1 )
initial t h e t h e n
0, v ( l )
m i n
1
=
=
1 }
c o n s i d e r e d
a b o v e )
t h e n
0.
A i
L a v r e n t i e v T h e
=
f u n c t i o n s
A 3 / 2
A o o
T h e
1)
different
f u n c t i o n
s p a c e s
p h e n o m e n o n
(see
c a l c u l a t i o n
a b o v e
repulsion J ( u '
J
property, ' ' )
- >
00.
c a n
L a v r e n t i e v h a s
t h a t
t h e if
u ^
b e
dif-
[37]
for
f o l l o w i n g G
A s / 2
4
J.M.
I n in
t h e
M a n i a
b u t
py
[ 1 3 , 1 4 ] , for
n o t t h e
elliptic
x , - u , p .
e x a m p l e
s t r i c t l y
e x a m p l e
=
I {(u«))
in
A
=
p
{v
i n t e g r a n d
e
s m a l l
is a
s m o o t h
h a s
d e r i v a t i v e
i n d i c a t e d
m i z e r
is
a
i n t e r v a l u ^
E
h o l d s
i n
m i z e r s ?
b y
[(x4 f[{x*
u ' f u f
: e
- u
-
t h e
v ( - l ) 2
)
2 S
p
is a n
0,
m i n / > i n f /
(2.4),
I
a t t a i n s of
its
t h e
r e p u l s i o n
- >
u *
a.e.
C o n s i d e r
n u m e r i c a l t h e
I ( u )
of
=
of
/
B a l l
/i
c o n v e x
&
in
>
0
fpp
>
for
all
i n
N o t e
2e
>
w*
t h a t
0.
of J
t h e
H e r e ,
in
A i
for t h a t
[ — 1 , 0 ) U ( 0 , 1 ] T h e
b u t
L a v r e n t i e v
(2.4)
e v e r y
e q u a t i o n
h o l d s
->
h o l d
m i n i m i z i n g
1 } .
^ 3 , a n d
a l s o
i n
i n
t h e
s u c h
m i n i -
t h e
w h o l e
f o r m
t h a t
if
00.
s i n g u l a r
for
M i z e l
c a n
/ ( « * ) .
I ( u ^ )
m e t h o d s
p r o b l e m
=
E u l e r - L a g r a n g e
t h e n
of
|x|3"signa\
i n f i m u m
p r o p e r t y
C o m p u t a t i o n
p o s s i b l e
=
is
p r o p e r t y
m i n i m i z e r
~
6
(2.3)
e q u a t i o n
u*(x)
p
dx
satisfies
=
s o l u t i o n
eul]
2
J
u ^
+
a b s o l u t e
w h e r e
b y
> [14]
2
— : r )
fpp(X)U,p)
p r o b l e m
E u l e r - L a g r a n g e
=
r e p u l s i o n
— l , v ( l )
inf
T h e
t h e
s
( u
s h o w n
=
+ e p
=
w a s
w h i c h
f o r m
[ - 1 , 1 ] .
a r e
x
for
t h e
2 . 2
W h a t
4
of t h e a t
w i t h
J_
a s
a n d
g i v e n
0, t h e r e
- f o o
in
is
( x
>
s m o o t h
A 3
=
e
t h o s e
( - l , l )
s o l u t i o n
p h e n o m e n o n
A s
p
W ^
f ( x , u , p )
sufficiently
i.e.
/ ( # , u , p )
H o w e v e r ,
p h e n o m e n o n
i n t e g r a n d s , a n
i n t e g r a n d
c o n v e x .
L a v r e n t i e v
S u c h
t h e
B a l l
m i n i m i z e r s
d e t e c t i n g
s u c h
s i n g u l a r
m i n i -
m i n i m i z i n g
f ( x , u , u
x
) d x
J a i n
A i
w h e r e A u
a , first
f r o m
/3
a r e
{ u e
g i v e n
m e t h o d its
=
W ^ f a b )
=
a , u ( b )
=
/?},
c o n s t a n t s .
p r o p o s e d
d e r i v a t i v e .
: u(a)
T h u s
b y
B a l l
g i v e n
& e
K n o w l e s >
0
w e
[12] c o n s i s t s
in
d e c o u p l i n g
m i n i m i z e
rb I ( u , v )
=
/
f ( x , u , v ) d x
J a a m o n g
p i e c e w i s e
affine
f u n c t i o n s
u
i n
A \
o n
a
u n i f o r m
m e s h
of
size
h ,
C o m p u t a t i o n
a n d
f u n c t i o n s
v
t h e
c o n s t r a i n t
t h a t
a r e
of
p i e c e w i s e
m i n i m i z e r s
c o n s t a n t
o n
5
t h e
s a m e
g r i d ,
s u b j e c t
t o
b
tp(ux
- v ) d x
for all
all
0
is
v
e
p i , P 2
s u i t a b l e t e e i n g
s u i t a b l e
R ,
w h e r e
£
R .
g r o w t h
t h a t
t o
m i n i m i z e r s
,
n
U
/ i , e - > 0 ,
effect
M a n i a affine b u t i n
f ( x , u , p ) / M ( ^
c a l l y
a s
M
e n s u r i n g
in
A \
h
v?(P2))
T h e n
b e
u n d e r g u a r a n -
s h o w n
f u n c t i o n
after
ip(p)
+
p a r t i c u l a r
it
p o s s i b l y
inf
is
t h a t
t h e
I ( u )
=
t h a t
7 ,
c o n -
e x t r a c t i o n
of
a
J ( w * ) .
=
t h e
/ M
w h i c h
m o n o t o n i a s
|p|
~»
0 0 ,
i n t e g r a l
) d x
x
a
p h e n o m e n o n ,
t h a t
e x a m p l e , for
i n s t e a d ,
s u c c e e d s
f o r m
M ,
( x , u , u
A l t h o u g h
a s
f a c t
(2.1) a m o n g
i n t e g r a n d
m i l d
M
r u l e
[12]; i n
x .
t r u n c a t e d
f
i n
of
t r u n c a t i o n
|p|
s u i t a b l e
i n
s t u d i e d
s p e c i a l
z
u
is
t h e
00.
a l s o
t r u n c a t e d
h a s
f u n c t i o n s ,
l a r g e
it
t h i s
m a y
in
t h e
t h e s i n c e
t h e n
m e t h o d
n o t c a s e
b e
is
a
c a n
w o r k s
e a s y
w h e n
i n t e g r a l it
w e
t o
I
m i n i m i z e
t h e o r e t i c a l l y , find
a n
a t t a i n s
(2.3), t h e r e l o c a l
first
a
a b s o l u t e m i n i m u m
is t h e
m i n i m i z e r
it
d a n g e r
of
IM
for
of all
M ) .
i n t e r e s t i n g
u s e
for
t r a p e z o i d a l
w h e n
L a v r e n t i e v
m i n i m i z e r
sufficiently
o n
(i)
C((p(pi)
=
a t t a i n e d )
=
w h e n e v e r
d r a w b a c k s IM
u , p )
- > i n
A \
r e p l a c e d =
^ , P )
5
o n e
g i v e n of
t h e
b o u n d a r y
t r a c t i o n .
d e r i v a t i o n
l e a d s
y
specified
a p p l i e d
r e q u i r e m e n t
t h i s
( 3 . 2 )
m i n i m i z a t i o n
t o
c o n s e q u e n c e
(3.3)
a r e a
m e a n i n g f u l ,
a n o t h e r
=
t
c o n d i t i o n
w h e r e
p a r t i c u l a r
g u a r a n t e e
(see,
m i n i m i z a t i o n
G a r r o n i
c o m p l e t e
c o m p l e x i t y
i n t e g r a t e d .
N o n l i n e a r
p o s i t i v e
d e t
T o
s p a c e
of
&
p h e n o m e n o n .
c o r r e s p o n d i n g
ft.
m o d e l
h y p o t h e s i s
c o m p u t a t i o n
f a m i l i a r
c o n d i t i o n
m a p p i n g .
o n
b e
a m
p h e n o m e n a
t h e
t h e
I
M a s o
b o u n d e d
a t
f u n c t i o n
f u n c t i o n
i n t r o d u c t i o n ,