Journal of Inequalities in Pure and
Applied Mathematics
ON THE STABILITY OF A CLASS OF FUNCTIONAL EQUATIONS
BELAID BOUIKHALENE
Département de Mathématiques et Informatique
Faculté des Sciences BP 133,
14000 Kénitra, Morocco.
EMail: bbouikhalene@yahoo.fr
volume 4, issue 5, article 104,
2003.
Received 20 July, 2003;
accepted 24 October, 2003.
Communicated by: K. Nikodem
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
098-03
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Abstract
In this paper, we study the Baker’s superstability for the following functional
equation
XZ
f(xkϕ(y)k−1 )dωK (k) = |Φ|f(x)f(y), x, y ∈ G
(E (K))
ϕ∈Φ K
where G is a locally compact group, K is a compact subgroup of G, ωK is the
normalized Haar measure of K, Φ is a finite group of K-invariant morphisms of
G and f is a continuous complex-valued function on G satisfying the Kannappan type condition, for all x, y, z ∈ G
Z Z
Z Z
(*)
f(zkxk−1 hyh−1 )dωK (k)dωK (h)= f(zkyk−1 hxh−1 )dωK (k)dωK (h).
K K
K K
We treat examples and give some applications.
On the Stability of A Class of
Functional Equations
Belaid Bouikhalene
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2000 Mathematics Subject Classification: 39B72.
Key words: Functional equation, Stability, Superstability, Central function, Gelfand
pairs.
The author would like to greatly thank the referee for his helpful comments and remarks.
Contents
1
Introduction, Notations and Preliminaries . . . . . . . . . . . . . . . . 3
2
General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
References
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1.
Introduction, Notations and Preliminaries
Let G be a locally compact group. Let K be a compact subgroup of G. Let
ωK be the normalized Haar measure of K. A mapping ϕ : G 7−→ G is a morphism of G if ϕ is a homeomorphism of G onto itself which is either a grouphomorphism, i.e (ϕ(xy) = ϕ(x)ϕ(y), x, y ∈ G), or a group-antihomorphism,
i.e (ϕ(xy) = ϕ(y)ϕ(x), x, y ∈ G). We denote by M or(G) the group of morphisms of G and Φ a finite subgroup of M or(G) of a K-invariant morphisms
of G (i.e ϕ(K) ⊂ K). The number of elements of a finite group Φ will be designated by |Φ|. The Banach algebra of bounded measures on G with complex
values is denoted by M (G) and the Banach space of all complex measurable
and essentially bounded functions on G by L∞ (G). C(G) designates the Banach space of all continuous complex valued functions on G. We say that a
function f is a K-central function on G if
(1.1)
f (kx) = f (xk), x ∈ G, k ∈ K.
In the case where G = K, a function f is central if
(1.2)
f (xy) = f (yx) x, y ∈ G.
On the Stability of A Class of
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Belaid Bouikhalene
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See [2] for more information.
In this note, we are going to generalize the results obtained by J.A. Baker in
[8] and [9]. As applications, we discuss the following cases:
a) K ⊂ Z(G), (Z(G) is the center of G).
b) f (hxk) = f (x), h, k ∈ K, x ∈ G (i.e. f is bi-K-invariant (see [3] and
[6])).
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c) f (hxk) = χ(k)f (x)χ(h), x ∈ G, k, h ∈ K (χ is a unitary character of
K) (see [11]).
d) (G, K) is a Gelfand pair (see [3], [6] and [11]).
e) G = K (see [2]).
In the next section, we note some results for later use.
On the Stability of A Class of
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2.
General Properties
In what follows, we study general properties. Let G, K and Φ be given as above.
Proposition 2.1. For an arbitrary fixed τ ∈ Φ, the mapping
Φ −→ Φ,
ϕ −→ ϕ ◦ τ
is a bijection.
On the Stability of A Class of
Functional Equations
Proof. Follows from the fact that Φ is a finite group.
Proposition 2.2. Let ϕ ∈ Φ and f ∈ C(G), then we have:
R
R
i) K f (xkϕ(hy)k −1 )dωK (k) = K f (xkϕ(yh)k −1 )dωK (k), x, y ∈ G, h ∈
K.
ii) If f satisfy (*), the for all z, y, x ∈ G, we have
Z Z
K
f (zhϕ(ykxk −1 )h−1 )dωK (h)dωK (k)
K
Z Z
=
f (zhϕ(xkyk −1 )h−1 )dωK (h)dωK (k).
K
Proof.
K
i) Let ϕ ∈ Φ and let x, y ∈ G, h ∈ K, then we have
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Case 1: If ϕ is a group-homomorphism, we obtain, by replacing k by
kϕ(h)−1
Z
Z
−1
f (xkϕ(hy)k )dωK (k) =
f (xkϕ(h)ϕ(y)k −1 )dωK (k)
K
K
Z
=
f (xkϕ(y)ϕ(h)k −1 )dωK (k)
K
Z
=
f (xkϕ(yh)k −1 )dωK (k).
K
Case 2: if ϕ is a group-antihomomorphism, we have, by replacing k by
kϕ(h)
Z
Z
−1
f (xkϕ(hy)k )dωK (k) =
f (xkϕ(y)ϕ(h)k −1 )dωK (k)
K
K
Z
=
f (xkϕ(h)ϕ(y)k −1 )dωK (k)
K
Z
=
f (xkϕ(yh)k −1 )dωK (k).
On the Stability of A Class of
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Belaid Bouikhalene
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K
ii) Follows by simple computation.
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Proposition 2.3. For each τ ∈ Φ and x, y ∈ G, we have
XZ
XZ
−1
(2.1)
f (xkϕ(τ (y))k )dωK (k) =
f (xkψ(y)k −1 )dωK (k).
ϕ∈Φ
K
ψ∈Φ
K
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Proof. By applying Proposition 2.1, we get that when ϕ is iterated over Φ, the
morphism of the form ϕ ◦ τ annihilates all the elements of Φ.
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Belaid Bouikhalene
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3.
The Main Results
Theorem 3.1. Let G be a locally compact group; let K be a compact subgroup
of G with the normalized Haar measure ωK and let Φ given as above.
Let δ > 0 and let f ∈ C(G) such that f satisfies the condition (*) and the
functional inequality
X Z
(3.1)
f (xkϕ(y)k −1 )dωK (k) − |Φ|f (x)f (y) ≤ δ, x, y ∈ G.
K
ϕ∈Φ
Then one of the assertions is satisfied:
On the Stability of A Class of
Functional Equations
Belaid Bouikhalene
(a) If f is bounded, then
(3.2)
|f (x)| ≤
|Φ| +
p
|Φ|2 + 4δ|Φ|
.
2|Φ|
(b) If f is unbounded, then
i) f is K-central,
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ii) f ◦ τ = f , for all τ ∈ Φ,
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iii) K f (xkyk −1 )dωK (k) = K f (ykxk −1 )dωK (k), x, y ∈ G.
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a) Let X = sup |f |, then we get for all x ∈ G
J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003
|Φ||f (x)f (x)| ≤ |Φ|X + δ,
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from which we obtain that
|Φ|X 2 − |Φ|X − δ ≤ 0,
such that
X≤
|Φ| +
p
|Φ|2 + 4δ|Φ|
.
2|Φ|
b) i) Let x, y ∈ G, h ∈ K, then by using Proposition 2.2, we find
|Φ||f (x)||f (hy) − f (yh)|
= ||Φ|f (x)f (hy) − |Φ|f (x)f (yh)|
X Z
≤
f (xkϕ(hy)k −1 )dωK (k) − |Φ|f (x)f (hy)
ϕ∈Φ K
X Z
−1
+
f (xkϕ(yh)k )dωK (k) − |Φ|f (x)f (yh)
K
ϕ∈Φ
≤ 2δ.
Since f is unbounded it follows that f (yh) = f (hy), for all h ∈ K, y ∈ G.
ii) Let τ ∈ Φ, by using Proposition 2.3, we get for all x, y ∈ G
|Φ||f (x)||f ◦ τ (y) − f (y)|
= ||Φ|f (x)f (τ (y)) − |Φ|f (x)f (y)|
On the Stability of A Class of
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Belaid Bouikhalene
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X Z
−1
≤
f (xkϕ(τ (y))k )dωK (k) − |Φ| f (x)f (τ (y))
ϕ∈Φ K
X Z
+
f (xkψ(y)k −1 )dωK (k) − |Φ|f (x)f (y)
K
ψ∈Φ
≤ 2δ.
Since f is unbounded it follows that f ◦ τ = f , for all τ ∈ Φ.
iii) Let f be an unbounded solution of the functional inequality (3.1), such
that f satisfies the condition (*), then, for all x, y ∈ G, we obtain, by using
Part i) of Proposition 2.2:
Z
Z
−1
−1
|Φ||f (z)| f (xkyk )dωK (k) −
f (ykxk )dωK (k)
K
K
Z
= |Φ|
f (z)f (xkyk −1 )dωK (k)
K
Z
−1
− |Φ|
f (z)f (ykxk )dωK (k)
K
Z
Z
≤ Σϕ∈Φ
f (zhϕ(xkyk −1 )h−1 )dωK (h)dωK (k)
K
Z K
−1
− |Φ|
f (z)f (xkyk )dωK (k)
On the Stability of A Class of
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K
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Z
Z
+ Σϕ∈Φ
f (zhϕ(ykxk −1 )h−1 )dωK (h)dωK (k)
K
K
Z
−1
−|Φ|
f (z)f (ykxk )dωK (k)
K
≤ 2δ.
Since f is unbounded we get
Z
Z
−1
f (xkyk )dωK (k) =
f (ykxk −1 )dωK (k), x, y ∈ G.
K
K
On the Stability of A Class of
Functional Equations
Belaid Bouikhalene
The main result is the following theorem.
Theorem 3.2. Let δ > 0 and let f ∈ C(G) such that f satisfies the condition
(*) and the functional inequality
X Z
−1
(3.3)
f (xkϕ(y)k )dωK (k) − |Φ|f (x)f (y) ≤ δ, x, y ∈ G.
K
ϕ∈Φ
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Then either
|f (x)| ≤
(3.4)
|Φ| +
p
|Φ|2 + 4δ|Φ|
, x ∈ G,
2|Φ|
or
(E (K))
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XZ
ϕ∈Φ
K
f (xkϕ(y)k −1 )dωK (k) = |Φ|f (x)f (y), x, y ∈ G.
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Proof. The idea is inspired by the paper [1].
If f is bounded, by using Theorem 3.1, we obtain the first case of the theorem.
Now let f be an unbounded solution of the functional inequality (3.3), then
there exists a sequence (zn )n∈N in G such that f (zn ) 6= 0 and limn |f (zn )| =
+∞.
For the second case we will use the following lemma.
Lemma 3.3. Let f be an unbounded solution of the functional inequality (3.3)
satisfying the condition (*) and let (zn )n∈N be a sequence in G such that f (zn ) 6=
0 and limn |f (zn )| = +∞. It follows that the convergence of the sequences of
functions:
i)
On the Stability of A Class of
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Belaid Bouikhalene
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R
P
(3.5)
x 7−→
ϕ∈Φ
K
f (zn kϕ(x)k −1 )dωK (k)
, n ∈ N,
f (zn )
to the function
x 7−→ |Φ|f (x).
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ii)
P
(3.6) x 7−→
ϕ∈Φ
R
K
f (zn hϕ(xkϕ(τ (y))k −1 )h−1 )dωK (h)
,
f (zn )
n ∈ N, τ ∈ Φ, k ∈ K, y ∈ G
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to the function
x 7−→ |Φ|f (xkτ (y)k −1 ) τ ∈ Φ, k ∈ K, y ∈ G,
is uniform.
By inequality (3.1), we have
P
R
−1
f
(z
kϕ(y)k
)dω
(k)
δ
n
K
ϕ∈Φ K
− |Φ|f (y) ≤
,
|f (zn )|
f (zn )
then we have, by letting n 7−→ +∞, that
R
P
−1
ϕ∈Φ K f (zn kϕ(y)k )dωK (k)
lim
= |Φ|f (y),
n
f (zn )
On the Stability of A Class of
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Belaid Bouikhalene
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and
P
lim
n
ϕ∈Φ
R
f (zn hϕ (xkϕ (τ (y)) k −1 ) h−1 ) dωK (h)
K
= |Φ|f (xkτ (y)k −1 ).
f (zn )
Since by Proposition 2.3, we have
R
P
−1 −1
XZ
ϕ∈Φ K f (zn hϕ(x)kϕ(τ (y))k h )dωK (h)
dωK (k)
f
(z
n)
K
τ ∈Φ
R
P
−1 −1
XZ
ϕ∈Φ K f (zn hϕ(x)kψ(y)k h )dωK (h)
dωK (k),
=
f (zn )
ψ∈Φ K
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combining this and the fact that f satisfies the condition (*), we obtain
R
−1 −1
X Z P
ϕ∈Φ K f (zn hϕ(x)kϕ(τ (y))k h )dωK (h)
dωK (k)
f (zn )
τ ∈Φ K
R
P
−1
f
(z
kψ(y)k
)dω
(k)
δ
n
K
ψ∈Φ K
−|Φ|f (x)
.
≤
|f (zn )|
f (zn )
Since the convergence is uniform, we have
XZ
f (xkϕ(y)k −1 )dωK (k) − |Φ|2 f (x)f (y) ≤ 0,
|Φ|
K
On the Stability of A Class of
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Belaid Bouikhalene
ϕ∈Φ
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thus (E (K)) holds and the proof is complete.
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4.
Applications
If K ⊂ Z(G), we obtain the following corollary.
Corollary 4.1. Let δ > 0 and let f be a complex-valued function on G satisfying
the Kannappan condition (see [10])
f (zxy) = f (zyx), x, y ∈ G,
(*)
and the functional inequality
X
(4.1)
f (xϕ(y)) − |Φ|f (x)f (y) ≤ δ, x, y ∈ G.
On the Stability of A Class of
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Belaid Bouikhalene
ϕ∈Φ
Then either
(4.2)
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|f (x)| ≤
|Φ| +
p
|Φ|2 + 4δ|Φ|
, x ∈ G,
2|Φ|
or
(4.3)
X
f (xϕ(y)) = |Φ|f (x)f (y), x, y ∈ G.
ϕ∈Φ
If G is abelian, then the condition (*) holds and we have the following:
If Φ = {i} (resp. Φ = {i, σ}), where i(x) = x and σ(x) = −x, we find the
Baker’s stability see [8] (resp. [9]).
If f (kxh) = χ(k)f (x)χ(h), k, h ∈ K and x ∈ G, where χ is a character of
K (see [11]), then we have the following corollary.
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Corollary 4.2. Let δ > 0 and let f ∈ C(G) such that f (kxh) = χ(k)f (x)χ(h),
k, h ∈ K, x ∈ G,
Z Z
(*)
K
K
f (zkxhy)χ(k)χ(h)dωK (k)dωK (h)
Z Z
=
f (zkyhx)χ(k)χ(h)dωK (k)dωK (h)
K
K
and
(4.4)
X Z
f
(xkϕ(y))χ(k)dω
(k)
−
|Φ|f
(x)f
(y)
≤ δ, x, y ∈ G.
K
K
On the Stability of A Class of
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Belaid Bouikhalene
ϕ∈Φ
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Then either
|f (x)| ≤
(4.5)
|Φ| +
p
|Φ|2 + 4δ|Φ|
, x ∈ G,
2|Φ|
or
(4.6)
XZ
ϕ∈Φ
f (xkϕ(y))χ(k)dωK (k) = |Φ|f (x)f (y), x, y ∈ G.
K
Proposition 4.3. If the algebra χωK ? M (G) ? χωK is commutative then the
condition (*) holds.
Proof. Since f (kxh) = χ(k)f (x)χ(h), k, h ∈ K, x ∈ G, then we have χωK ?
f ? χωK = f . Suppose that the algebra χωK ? M (G) ? χωK is commutative,
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then we get:
Z Z
K
f (xkyk −1 hzh−1 )dωK (k)dωK (h)
K
Z Z
=
f (xkyhzh−1 k −1 )dωK (k)dωK (h)
K
K
= hδz ? χωK ? δy ? χωK ? δx , f i
= hδz ? χωK ? δy ? χωK ? δx , χωK ? f ? χωK i
= hχωK ? δz ? χωK ? δy ? χωK ? δx ? χωK , f i
= hχωK ? δz ? χωK ? δx ? χωK ? δy ? χωK , f i
Z Z
=
f (ykxk −1 hzh−1 )dωK (k)dωK (h).
K
K
On the Stability of A Class of
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Belaid Bouikhalene
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Let f be bi-K-invariant (i.e f (hxk) = f (x), h, k ∈ K, x ∈ G), then we
have:
Corollary 4.4. Let δ > 0 and let f ∈ C(G) be bi-K-invariant such that for all
x, y, z ∈ G,
Z Z
Z Z
(*)
f (zkxhy)dωK (k)dωK (h) =
f (zkyhx)dωK (k)dωK (h),
K
K
K
K
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and
(4.7)
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X Z
f (xkϕ(y))dωK (k) − |Φ|f (x)f (y) ≤ δ, x, y ∈ G.
K
ϕ∈Φ
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Then either
|f (x)| ≤
(4.8)
|Φ| +
p
|Φ|2 + 4δ|Φ|
, x ∈ G,
2|Φ|
or
(4.9)
XZ
ϕ∈Φ
f (xkϕ(y))dωK (k) = |Φ|f (x)f (y), x, y ∈ G.
K
Proposition 4.5. If the pair (G, K) is a Gelfand pair (i.e ωK ? M (G) ? ωK is
commutative), then the condition (*) holds.
Proof. We take χ = 1 (unit character of K) in Proposition 4.3 (see [3] and
[6]).
In the next corollary, we assume that G = K is a compact group.
Lemma 4.6. If f is central, then f satisfies the condition (*). Consequently, we
have
Z
Z
−1
(4.10)
f (xtyt )dt =
f (ytxt−1 )dt, x, y ∈ G.
G
G
Corollary 4.7. Let δ > 0 and let f be a complex measurable and essentially
bounded function on G such that
X Z
(4.11)
f (xtϕ(y)t−1 )dt − |Φ|f (x)f (y) ≤ δ, x, y ∈ G.
G
ϕ∈Φ
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Then
|f (x)| ≤
(4.12)
|Φ| +
p
|Φ|2 + 4δ|Φ|
, x ∈ G.
2|Φ|
Proof. Let f ∈ L∞ (G) be a solution of the inequality (4.11), then f is bounded,
if not, then f satisfies the second case of Theorem 3.2 which implies that f is
central (i.e the condition (*) holds) and f is a solution of the following functional equation
XZ
(4.13)
f (xtϕ(y)t−1 )dt = |Φ|f (x)f (y), x, y ∈ G.
ϕ∈Φ
G
In view of the proposition in [5], we have that f is continuous. Since G is
compact, then the proof is accomplished.
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Belaid Bouikhalene
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On the Stability of A Class of
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[7] E. HEWITT AND K.A. ROSS, Abstract Harmonic Analysis, Vol. I and II.,
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[10] Pl. KANNAPPAN, The functional equation f (xy) + f (xy −1 )
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Université de Nancy I, 1980.
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Functional Equations
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J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003
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