THE METHOD OF TRANSMUTATIONS
by
REUBEN HERSH
D e p a r t m e n t of M a t h e m a t i c s
and Statistics
U n i v e r s i t y of New M e x i c o
A l b u q u e r q u e , New M e x i c o
87131
l.
INTRODUCTION
A standard m a t h e m a t i c a l
new problem,
strategy, when
faced with a
is to reduce it to a p r e v i o u s l y
lem, or at least to a simpler problem.
For example,
reduce a p r o b l e m with a s i n g u l a r c o e f f i c i e n t
regular c o e f f i c i e n t s ;
solved probto
to one with
to reduce a p r o b l e m c o n t a i n i n g a
small p a r a m e t e r to one i n d e p e n d e n t of the parameter;
to
transform a second-order
e q u a t i o n into a f i r s t - o r d e r
equation,
to t r a n s f o r m a G o u r s a t p r o b l e m
or vice versa;
into a C a u c h y problem,
or vice versa.
U s u a l l y it is not hard to v e r i f y
such a t r a n s f o r m a t i o n ,
the p r o p e r t i e s
once it has b e e n found.
lem is to find the right t r a n s f o r m a t i o n
lem.
tic m e t h o d to find such a t r a n s f o r m a t i o n .
the task of c o n s t r u c t i n g
no h a r d e r than the task of v e r i f y i n g
The prob-
into an old prob-
In this note we will show that there
often makes
of
is a s y s t e m a This m e t h o d
the t r a n s f o r m a t i o n
its p r o p e r t i e s .
will also d i s c u s s the c o n n e c t i o n of our m e t h o d with
We
HERSH
probability
theory.
265
It often happens that the transfor-
mation we seek can be expressed as the expected value of
a suitable random variable;
in a probabilistic
found.
indeed,
context that they have first been
We start out by listing
cludes as particular
examples.
we mention
tion of the Euler-Poisson-Darboux
Lions,
Our method in-
examples many formulas scattered
through the literature:
equation,
it has sometimes been
the "transmutation
equation to the wave
of Bragg and Dettman,
of S. Rosencrans,
mulas of A. Weinstein,
reduc-
operators" of Delsarte and
the "related equations"
"diffusion transform"
the classical
J. Donaldson,
and various
the
for-
W. Roth, M° Kac and
S. Kaplan.
We will not discuss
formulas,
the rigorous verification
or give precise conditions
for their validity,
for this has been done in the references we cite,
this score we have nothing new to say.
can be obtained by a single technique,
seen,
approach.
and on
Our purpose here
is to show that these seemingly scattered
uniform heuristic
of our
formulas all
which provides a
This technique,
as will be
is simply an operational version of the methods of
classical
transform theory including
form, Hankel transform,
the Fourier trans-
and Laplace transform.
The paper falls into four sections.
following this introduction,
of transmutation
these examples
formulas.
we collect five examples
In the third section, we use
to explain our general method for con-
structing transmutation
formulas.
In the last section
we comment briefly on the probabilistic
formulas,
In Section 2,
aspect of these
and also on the application of transmutation
formulas to problems on regular and singular perturbations.
266
2.
HERSH
EXAMPLES
EXAMPLE
i.
The
method of spherical means, and
Darbov~ equation
In the s e c o n d
13,
volume
it is shown,
using
differentiation,
that
of C o u r a n t - H i l b e r t ,
spherical
if
means
u(x,t)
and
and
Ch. VI, par.
fractional
v(x,t)
are c o n -
n e c t e d b y the f o r m u l a
v(x,t)
where
= ~i
u(x,t~) (i _
2) (n-3)/2 d~,
x =
(x ! , .... x ),
and if
n
dimensional wave equation,
utt
then
v
gu,
satisfies
u(x,0)
The Darboux
Av,
equation
example we solve
ut(x,0)
the n-
= 0
equation,
v(x,0)
is s i n g u l a r
a singular
satisfies
= f,
the D a r b o u x
n-i
vtt + -~--v t
u
= f,
at
vt(x,0)
t = 0,
equation
0.
so in this
in terms
of a r e g u -
lar e q u a t i o n .
EXAMPLE
and Lions
Let
r(t)
The transmutation operators of Delsarte
2.
[2,3]
D = d/dt.
and
operator
q(t)
a solution
of
L = D
are g i v e n
independent
tial o p e r a t o r
t i o n of
Let
with
of
t
2
+ r(t) D + q(t) ,
functions.
(usually
space-dependent
D2u + Au = 0
Lv + Av = 0
by
Let
a partial
be an
differen-
coefficients)°
is t r a n s f o r m e d
setting
A
where
v = Hu,
Then
into a s o l u where
the
HERSH
operator
H
= 0
v = Hu,
and
satisfies
HD 2
then,
=
267
LH.
In fact,
assuming
A
if
D 2u
+ Au
c o m m u t e s with
H,
we h a v e
Lv + Av = ( L + A ) H u
To c o n s t r u c t
the
= H(D2+A)u
"transmutation operator"
sarte and Lions seek a kernel
action of
H
is given by
h(s,t) ,
Hf =
tuting this r e p r e s e n t a t i o n of
HD 2 = LH,
= 0.
H,
Del-
so that the
l h ( s , t ) f(t)dt.
H
Substi-
into the e q u a t i o n
and i n t e g r a t i n g b y parts,
one easily d e r i v e s
a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n
for
h(s,t) :
h
= h
+ r(s)h
+ q(s) h,
with boundary conditions
tt
ss
s
d e t e r m i n e d b y the side c o n d i t i o n s of the two o p e r a t o r s
D2
and
L.
hyperbolic
Since
h.s,t),' satisfies
a second-order
e q u a t i o n in two i n d e p e n d e n t v a r i a b l e s ,
be e x p r e s s e d in terms of the R i e m a n n
equation.
f u n c t i o n of that
In particu].ar,
if we specialize
A = -A,
-
r(t)
n-i
t
,
it c a n
q(t)
-: 0,
then the R i e m a n n func~-ion is e x p r e s s i b l e e x p l i c i t l y
elementary
Example
terms,
and we r e c o v e r
in
the same formula as in
i.
EXAMPLE
3.
From a second-order to a first-order
equation
If
u(t)
is a v e c t o r - v a l u e d
abstract Cauchy problem
{utt
function,
satisfying
Au,
= f,
u(0)
then
v(t)
= l/ ~-~-/0
~ u(s) e -s 2/4 t
ds
the
ut(0) = 0 } ,
268
HERSH
satisfies
v(0)
the a b s t r a c t
= f}.
edly;
see
This f o r m u l a
problem
has been
{v t = Av,
rediscovered
repeat-
[5-9].
A = d2/dx 2 '
If
Cauchy
our f o r m u l a
u = ~1 (f(x+t)
then
reduces
to the c l a s s i c a l
+ f(x-t)) ,
Poisson
and
solution
of
the h e a t e q u a t i o n ,
v
= v
t
1
v
EXAMPLE
=
- -
,
xx
v(0)
= f,
-s2/4t
L
e
f(s+x)
ds.
From a Cauchy problem to a Goursat
4.
problem, and back
In the t h e s i s
u(t)
of W. J. R o t h
is a v e c t o r - v a l u e d
linear operator,
and
[4] it is s h o w n
function
{utt
Au,
and
A
u(0)
t h a t if
is a c l o s e d
f,
ut(0)
0},
then
2 f~/2
v(r,s)
satisfies
s > 0,
from
u(t)
= v(r,0)
r/~s sin O)dO
{v
= Av
if
r > 0,
rs
and m o r e o v e r w e r e c o v e r
= f}
u
b y the f o r m u l a
-
dt ~0
(By a m o d i f i c a t i o n
alize
u(2
7~ 0
the G o u r s a t p r o b l e m
v(0,s)
v
=
t sin
v
of D u h a m e l ' s
to the c a s e w h e r e
and a r b i t r a r y . )
0
v(r,0)
(tsn sn0)
2
'
formula,
and
2
dO
"
one c a n g e n e r -
v(0,s)
are unequal
HERSH
EXAMPLE
269
From a first-order equation to a family of
5.
higher-order equations
Suppose
u(t)
-~ < t < ~
A
is a v e c t o r - v a l u e d
and satisfies
is closed and
P(d/dt, d/dx)
{u t = Au,
f £ ~0(A~) .
is a hyperbolic differential
be t-dependent.)
P(d/dt,
u(0)
Let
gk(t,x)
-d/dX) gk = 0,
where
operator,
or
(Its coefficients m a y
be a fundamental
solution
all of whose Cauchy data
vanish except for the k'th;
let
Then it is easily verified
(see
v(t) =
= f},
Suppose m o r e o v e r that
is parabolic of positive genus.
of
function for
L
(d/dt) k g k (0,x) = 6(x) .
[i0])
that
U(S) gk (t,s) ds
is a solution of
{P(d/dt,A) v = 0 ,
v(0) = f,
v(0) = 0
if
j ~k}.
If
P(d/dt,A) = d / d t - A 2,
then one has
g0 =
2
-s /4t
(1/2 / ~ ) e
, almost as in Example 3 above.
If
1
P (d/dt, A)
d2/dt 2
A 2 , one has
_
q0 t, s) = ~ (@ (s+t) +
=
-
(s-t)) ,
i/2
Is
< t
0
fs
> t
%l(t,s) :
so that
{Vtt = A2v, V(0) = v 0 , v(0)
v(t)
=
[u0(s) g0(t,s)
= v I}
is solved by
+ Ul(S)gl(t,s) ]ds
--00
= ~
[u0(t) + U0(-t)
+
Ul(S) ds]
t
where
270
HERSH
du O
dt
= Au0 '
u0(0)
= v0
du 1
dt
- AUl '
ul(0)
= Vl"
3.
A S Y S T E M A T I C A P P R O A C H TO D I S C O V E R Y OF T R A N S M U T A T I O N
FORMULAS
In the references
above
cited for the five examples given
(and in many other such works)
rigorous proof that the f u n c t i o n
terms of
u(t)
there is p r o v i d e d
v(t) ,
expressed
by the given t r a n s m u t a t i o n
indeed satisfy the conditions
on
v.
formula,
in
does
What is often lack-
ing is a clue to explain how such formulas may be d i s c o v ered.
Or in some cases,
as in E x a m p l e i, a d e r i v a t i o n
given which is quite special,
leaving no clue how to pro-
ceed to relate a d i f f e r e n t pair of p r o b l e m s
and
is
for
u(t)
v(t).
Our p u r p o s e h e r e is to d e s c r i b e a u n i f o r m a p p r o a c h
discovering
formulas
such as those of Examples 1-5; the
task of v e r i f y i n g the formula,
is often s t r a i g h t f o r w a r d .
we h a v e nothing
In general,
to
once it is w r i t t e n down,
On this score,
in any case,
to add to the cited references.
a transmutation
formula can be r e g a r d e d
in the following light:
We have
solution
two p r o b l e m s
to the first problem,
function of
A,
u = u(t,A) .
Similarly,
tion
v
involving some o p e r a t o r
d e p e n d i n g on
depends on
A,
u(t) ,
t
A.
The
we regard as a
as a p a r a m e t e r :
in the second problem,
the solu-
as well as on a p a r a m e t e r
HERSH
s: v = v(s,A) .
sent
v
Then,
in t e r m s
of t h e
function
family
of
to t r a n s m u t e
of
u
the kernel
a distribution,
the given
couple
of o u r
examples.
In Example
i, l e t u s
number--i.e.,
for
A.
Then we
a representation
of t h e o n e - p a r a m e t e r
assume
can
a genuine
function
is u l t i m a t e l y
A.
regard
operate
to b e
L e t us w o r k
the operator
as
if i t w e r e
we have
express
u(t)
To f i n d
1
operator
we can
we
find
v -- to r e p r e -
u(t,l)dt
may be
and where
by
on w h i c h
: fh(s,t)
h(s,t)
placed
bol
in terms
to
u(t,-) :
v(s,l)
where
u
-- we m u s t
v(s,.)
functions
271
re-
through
A
a
as a s y m -
a complex
a functional
u(t)
or
calculus
symbolically,
= cos(t/~f.
a comparable
e x p r e s s i o n for
v,
we can reduce
n-i
the e q u a t i o n
v
+ -v
- Av = 0
to a f o r m a l " B e s s e l
tt
t
t
e q u a t i o n " of o r d e r
((n/2) - i)
by the "substitution"
w(z)
Therefore
= t ((n/2) -i)
v(t) ,
l-(n/2)
J
of t h e
sions
for
to t h e s e
method
/~.
we have
v = ct
where
z = t ((n/2 - i)
(z)
is as u s u a l
first kind.
u
(t(n/2)-i
and
v;
expressions.
is to u s e
/~) f
J(n/2)-i
these
the Bessel
So f a r w e h a v e
function
two
formal
the p r o b l e m
is to g i v e
The
essence
of
two
formal
the
of o r d e r
expres-
a meaning
transmutation
expressions
to r e l a t e
272
v
HERSH
to
u;
then
v(t)
is known if
The problem of expressing
u(t)
v(t)
is known.
in terms of
u(t)
is
evidently the problem of expressing a Bessel function in
terms of cosine; this, however,
is well-known; we have
the standard formula
1
J (z)
(see
(z/2) v
I / 2 F ( ~ +i)
=
I±
( l - s 2) ~---2 cos(zs) ds
J0
[Ii], formula 3.7) .
Now, if we replace
cos(st (n/2)-I / ~ )
z
by
cited in Example 1.
by
t(n/2)-i / ~
u(st (n/2)-l)
and
we get the formula
This approach to the Euler-Poisson-
Darboux equation is similar to that used by J. Donaldson
[12].
Example 4 is closely related to the first example.
As Roth
v(r,s),
into
[4] points out, his formula for transforming
the solution for an interior Goursat problem,
u(t),
the solution of a Cauchy problem, can be
obtained by formally "solving"
{utt = Au,
by
u(0) = f,
ut(0) }
u = cos(t/A) f
and "solving"
{v
rs
= Av,
by
To express
u
and
v(0,s)
= v(r,0)
= f}
v : J0 (2 r/~sA) f.
v
in terms of each other, again we
resort to classical identities from the theory of Bessel
H E RSH
functions.
clear
That
v
can be expressed
from the c o m p l e t e n e s s
essentially,
to asking
the cosine
to express
for a Hankel
the r e p r e s e n t a t i o n s
u
of
transform
v
cos ~.
in Example
u
is
functions;
in terms of
transform
given
in terms of
of the cosine
one is seeking
and conversely,
2 73
of
J0;
amounts
We obtain
4 above by using
the identities
f /2
2
= -~ J0
J0 (~)
d.
cos(~sin
f~/2
cos ~ - d ~ j 0
If in these
v,
formulas
according
we obtain
A
we replace
cos
by
"solutions"
the transmutations
symbolic
sin 8)de.
from
u
u
written
to
expressions
and
v
J0
by
above,
and back;
for functions
of
drop out of our formulas.
These
general
two examples may be enough
u(t,A)
special
u(s,l)
>~.
A
case,
v(t,l) ,
A,
I, and let
of Problem
is m u l t i p l i c a t i o n
Similarly,
it is enough
A = I.
special
the
let
v(t,A)
II.
be the solution
struct a t r a n s m u t a t i o n
where
clear
an operator
of Problem
of Problem
case where
complex number
u(s,A),
involving
be the solution
be the solution
Let
to make
pattern:
Given two p r o b l e m s
and
~ sin 8 J0(~
to the formal
the questionable
@)d@,
define
to represent
to consider
For if we can solve
I in the
by a real or
v(t,l).
v(t,A)
in terms of
this special
I
and if the c o m p l e x - v a l u e d
are related b y a kernel,
and
To con-
II
case
in this
functions
u(s,l)
274
HERSH
v(t,l)
where
F
is some curve, usually an interval on the real
axis, then,
for
= ]F g(s,t) u(s,l)ds
if
v(t,A),
u(s,A)
exists
for
s c F,
the candidate
the solution of Problem II, is evidently
F g(s,t) u(s,A)ds.
The integral
is a Bochner
is a classical
function.
integral
If
function or "distribution",
integral
if the kernel
g(s,t)
g(s,t)
is a generalized
as in Example
5 above,
the
is "symbolic" and may be interpreted b y a formal
integration by parts.
It should be emphasized
is applicable
that the transmutation method
even if neither Problem I nor Problem I!
is "well-posed".
The existence
of a transmutation operator
implies
that the class of admissible data for Problem II includes
the admissible
is a complete
posed--then
data for Problem I.
linear space--i.e.,
so is Problem II.
If this second class
if Problem I is well-
On the other hand, the
method retains its validity even if the data are highly
restricted.
For instance, b y choosing
A = -£,
we see
from Example 3 that the admissible data for the backward
heat equation include the admissible data for Cauchy's
problem for the Laplace equation.
We have in this instance
properly posed problem
a transmutation
of one im-
to a second improperly posed
problem.
Let us go on to show how our method yields the formula
of Example 3.
Problem I is
HE RSH
{U
Problem
=
tt
AU,
u(0)
:
f,
ut(0)
v(0)
= f}.
= 0}.
II is
{V t = Au,
Again we have
at l e a s t
Now,
2 75
if
u = cos
A = I
t / Z A f,
is a c o m p l e x
in the F o u r i e r
v = etAf,
and e v i d e n t l y
number.
transform
formula
2
--S
e
substitute
Example
I = /Z~,
e
cos
and we o b t a i n
sl ds
the f o r m u l a
of
3.
A slightly
Dettman
-
[6];
I, s e t t i n g
different
formula
they c h o o s e
u(0)
= 0,
As in the o t h e r
is g i v e n b y B r a g g
the d a t a d i f f e r e n t l y
ut(0)
examples,
and
in P r o b l e m
= f.
the v e r i f i c a t i o n
is
straightforward.
Example
For
5 is similar.
We n o w h a v e
the sake of s i m p l i c i t y ,
roots
T. (A) ;
3
assume
u(t,s)
P(T,A)
: etAf.
has
then
v(t,A)
=Lc.e
j 3
tT. (A)
3
f
where
k
T. c : 6
3
3
j,k"
Then
v(t,il)
: ~c.e
j 3
T. (il)
3
f = f e isl g(t,s) ds
simple
276
HERSH
where
g(t,s)
is the F o u r i e r
The e x i s t e n c e
lows
of
g(t,s)
f r o m the a s s u m e d
tuting
A
for
Finally,
Lions.
il,
transform
hyperbolicity
be
{Lv + 1 2 v : v
+ r(t)v
v(o,~)
u(t,i)
=
i,
= cos
h(t,s)
Since
tl,
formula.
of D e l s a r t e -
of
+ q(t)v
+ 12v = 0
(o,I)
= 0}.
of
=
= I,
u
t
(0,x)
= 0}.
fh(t,s) u(s,l)ds
is the F o u r i e r
cosine
it f o l l o w s
Lh - h
an o p e r a t o r
and s u b s t i -
and w e h a v e
L v + 1 2 v = 0,
Define
t
u(0,1)
v(t,l)
where
v
b e the s o l u t i o n
u(t,l)
fol-
t
{D2u + 12u = 0,
Then
P,
2, the p r o b l e m
the s o l u t i o n
tt
Let
of
we get the desired
v(t,~)
.
T.(il)
>....~~c.e
3
f.
3
distribution
as a S c h w a r t z
w e l o o k at E x a m p l e
Let
of
H
ss
transform
of
v(t,l).
that
= 0.
by
Hf = f h ( t , s ) f ( s ) d s .
Then,
from
tegrations
Lh = h
by parts
,
it f o l l o w s ,
ss
o n the left, t h a t
on u s i n g
two in-
LH = HD 2.
This
is the f o r m u l a b y w h i c h L i o n s
fine t h e i r
transmutation
b y our m e t h o d
case.
operator
we can recover
H;
and D e l s a r t e
de-
t h u s w e see t h a t
their procedure
as a s p e c i a l
HERSH
277
4.
P R O B A B I L I S T I C INTERPRETATIONS;
ON A SMALL P A R A M E T E R
In some of the principal
it is possible
where
T(t)
For instance,
mean
0
t,
as a Gaussian
its density
expectation
= E{u(T)}
time, d i s t r i b u -
probability
law, and
3, the appropriate
normal
T
random variable,
is
with
Such a random variable has as
2
1
-s /4t
St(s) = ~
e
, and so, by
elementary
formula
of a function
v(t)
v(t)
t.
function
the familiar
as
operator.
in Example
and variance
DEPENDING
of transmutations,
a random
to some appropriate
is the e x p e c t e d - v a l u e
distributed
examples
the formula
is, for each
ted according
E
to rewrite
EQUATIONS
= E[u(T)]
in p r o b a b i l i t y
for the
of a random variable,
=
u(s) St(s)ds.
--OO
In fact,
it was
that the formula
with a limit
An earlier
in this p r o b a b i l i s t i c
in Example
theorem
3 arose in
on random
example
representation
[7], in connection
evolutions.
is due to M. Kac
[13].
He found
that if
T =
where
N(s)
and if
u
is a P o i s s o n process
v(t)
ds,
with intensity
satisfies
{utt
and
(-l) N(s)
Au,
= E{u(T) },
u(0)
then
: f,
v
Ut(0)
satisfies
= 0}
a,
278
HERSH
{vtt + 2a v t = Av,
v(O)
= f,
v t(O)
=
This formula was g e n e r a l i z e d by K a p l a n
case w h e r e
a = a(t),
0}.
[14] to the
a given f u n c t i o n of
t;
Kaplan
points out that by use of the d e n s i t y function as a k e r nel,
v(t)
can be w r i t t e n in terms of
gral over the real axis,
u(t)
as an inte-
and in that form the K a c - K a p l a n
result is a special case of the D e l s a r t e
transmutation
method.
In the work on random e v o l u t i o n s
were e x t e n d e d to more general
ables.
[7], these results
operators
A systematic p r o b a b i l i s t i c
the K a c - K a p l a n and H e r s h - G r i e g o
and random vari-
approach w h i c h u n i f i e s
examples,
and gives
new and more general results by systematic
Lemma,
was given by S. Rosencrans,
diffusion transform
use of Ito's
in his work on the
[15].
From the v i e w p o i n t of the p r e s e n t paper,
listic r e p r e s e n t a t i o n s
where
the kernel
to
be a d e l t a function,
singular part.)
comprise
g(s,t)
tive with respect
s
the p r o b a b i -
just those transmutations
is, for each
t,
the d e r i v a -
of a finite m e a s u r e .
(g
could
if the p r o b a b i l i t y m e a s u r e has a
Any such kernel
g(s,t)
as the d e n s i t y of a s u i t a b l y c o n s t r u c t e d
TCt)
some
can be r e g a r d e d
random time
.
Thus R o s e n c r a n s was able
representation
to give a p r o b a b i l i s t i c
to the formulas of our Example
solution of the E u l e r - P o i s s o n - D a r b o u x
equation
i; the
is ob-
tained as the m e a n of s o l u t i o n s of the wave equation,
e v a l u a t e d at a c e r t a i n random time.
A probabilistic
use p r o b a b i l i s t i c
r e p r e s e n t a t i o n makes
limit theorems
it p o s s i b l e
to
(laws of large numbers,
HERSH
central
limit theorems)
for solutions
see
279
to prove
of such equations
asymptotic
as
estimates
sutt + au t = Au;
[18].
However,
for such purposes
tions
itself provides
quite
aside
an equation
the m e t h o d
a convenient
and powerful
from its p r o b a b i l i s t i c
connections.
in an operator
A
it m a y be possible
v
Thus the p r o b l e m
kernel
h(s,t) :
of sending
function
s
to zero is reduced
function
h
s
the results
v ;
are independent
operator
stitute
This p r o g r a m was carried
P.
A
wide class of p o l y n o m i a l s
P
the singular p e r t u r b a t i o n
which one happens
in
is solved by transmuting
vs(t,L)
out for a
[16]; in particular,
s÷0
to
u(t,L) ,
the
of
u
In
to sub-
problem
evtt + v t = Lv,
solution
to
instead of a
of the p a r t i c u l a r
into
is entirely
= f h s ( s , t ) u(t,A)dt.
the study of a real-valued
vector-valued
s,
to some function
in such a way that the s-dependence
vs(S,A)
Given
= 0,
to t r a n s m u t e
carried by the transmutation
approach,
and a small p a r a m e t e r
P £ (d/dt,A)v
u(t,A),
of transmuta-
tt
[17], the singularly
= Lu.
perturbed
singular
equations
280
HERSH
U
+ --U
: U
t
t
XX
tt
and
cu
were
studied
In the
method
tial
second
the
time
:
t
U
XX
c ÷ 0.
of t h e s e
loss
singularity
tt
--U
t > 0,
simultaneously
tions:
the
for
1
t
+
two
equations,
overcomes
of an i n i t i a l
a transmutation
two d i s t i n c t
condition
of t h e c o e f f i c i e n t
of
u
as
t
complica~ ÷ 0,
at the
and
ini-
t = 0.
This research was
G P - 3 4 1 8 8 A #i.
supported
in p a r t b y N S F
Grant
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