Some Remarks on Spectral Synthesis Sets
Relations between the difference spectra of unions and intersections are studied and their implications on some problems in spectral synthesis are observed. Вивчаються взаємозв'язки між різницєвими спектрами об'єднань та перетинів. Встановлено їхні наслідки для деяких проблем спектрального...
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| Cite this: | Some Remarks on Spectral Synthesis Sets / J. Joseph & T.K. Muraleedharan // Український математичний журнал. — 2015. — Т. 67, № 10. — С. 1434–1438. — Бібліогр.: 11 назв. — англ. |
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Joseph, J. Muraleedharan, T.K. 2020-02-16T20:36:16Z 2020-02-16T20:36:16Z 2015 Some Remarks on Spectral Synthesis Sets / J. Joseph & T.K. Muraleedharan // Український математичний журнал. — 2015. — Т. 67, № 10. — С. 1434–1438. — Бібліогр.: 11 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165866 517.9 Relations between the difference spectra of unions and intersections are studied and their implications on some problems in spectral synthesis are observed. Вивчаються взаємозв'язки між різницєвими спектрами об'єднань та перетинів. Встановлено їхні наслідки для деяких проблем спектрального синтезу. en Інститут математики НАН України Український математичний журнал Короткі повідомлення Some Remarks on Spectral Synthesis Sets Деякі зауваження щодо множин спектрального синтезу Article published earlier |
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Some Remarks on Spectral Synthesis Sets Joseph, J. Muraleedharan, T.K. Короткі повідомлення |
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Some Remarks on Spectral Synthesis Sets |
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Some Remarks on Spectral Synthesis Sets |
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some remarks on spectral synthesis sets |
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Joseph, J. Muraleedharan, T.K. |
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Joseph, J. Muraleedharan, T.K. |
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Деякі зауваження щодо множин спектрального синтезу |
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Relations between the difference spectra of unions and intersections are studied and their implications on some problems in spectral synthesis are observed.
Вивчаються взаємозв'язки між різницєвими спектрами об'єднань та перетинів. Встановлено їхні наслідки для деяких проблем спектрального синтезу.
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1027-3190 |
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https://nasplib.isofts.kiev.ua/handle/123456789/165866 |
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Some Remarks on Spectral Synthesis Sets / J. Joseph & T.K. Muraleedharan // Український математичний журнал. — 2015. — Т. 67, № 10. — С. 1434–1438. — Бібліогр.: 11 назв. — англ. |
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2025-11-26T00:12:39Z |
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К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.9
J. Joseph, T. K. Muraleedharan (St. Joseph’s College Devagiri, Kozhikode, India)
SOME REMARKS ON SPECTRAL SYNTHESIS SETS
ДЕЯКI ЗАУВАЖЕННЯ ЩОДО МНОЖИН СПЕКТРАЛЬНОГО СИНТЕЗУ
Relations between the difference spectra of unions and intersections are studied and their implications on some problems
in spectral synthesis are observed.
Вивчаються взаємозв’язки мiж рiзницевими спектрами об’єднань та перетинiв. Встановлено їхнi наслiдки для
деяких проблем спектрального синтезу.
1. Introduction. The question whether the union of two spectral synthesis sets is again a spectral
synthesis set and whether every spectral synthesis set is a Wiener – Ditkin set are outstanding open
problems in commutative harmonic analysis. In connection with the study of such problems in spectral
synthesis Stegeman [9] introduced the difference spectrum and later the idea was systematically
exploited by him in [10]. Difference spectrum is a union of perfect sets and is closed if the group
is metrizable. But, for non metrizable groups, Salinger and Stegeman [8] proved that the difference
spectrum need not be closed. This article studies certain relations between the difference spectrum
of the unions and intersections of closed sets, and the corresponding unions and intersections of the
difference spectra. Their implications on some problems in spectral synthesis are also studied.
If G is the character group of the locally compact Abelian group, then A(G), Fourier algebra, is
the algebra of Fourier transforms of functions in the convolution algebra L1(Γ). If f ∈ A(G), then
f = φ̂, for some φ ∈ L1(Γ). Define ‖f‖A(G) = ‖φ‖L1(Γ), then A(G) becomes a commutative regular
semisimple Banach algebra isometrically isomorphic to L1(Γ), which is identified as a predual of a
group von Neumann algebra [2]. For the notations we follow [4], [6] and [10].
Suppose x ∈ G, f, g ∈ A(G) and I, J are ideals in A(G). We use the notation f =x g if f
and g agree in some neighborhood of x, f ∈x I if f =x g for some g ∈ I, I ⊂x J if f ∈x J for
every f ∈ I and I =x J if I ⊂x J and J ⊂x I. For a closed subset E of G, I(E) = {f ∈ A(G) :
f = 0 on E}, j(E) = {f ∈ A(G) : suppf ∩ E = ∅} and J(E) = j(E). Then J(E) ⊂ I ⊂ I(E)
for every closed ideal I of A(G) with zero set E. E is called a set of synthesis or spectral synthesis
set if I(E) = J(E) and E is said to be Wiener – Ditkin set ( Calderon set or C-set ) if f ∈ fj(E)
for every f ∈ I(E). Evidently, every Wiener – Ditkin set is a spectral set. We recall the definition
of ∆(E), the difference spectrum of a closed set E : ∆(E) = {x : I(E) 6=x J(E)} so that ∆(E)
is a union of perfect subsets of ∂(E), the boundary of E. E is a set of spectral synthesis if
and only if ∆(E) = ∅ [10]. Also the n-difference spectrum is ∆n(E) =
⋃
f∈I(E){x ∈ G :
fn /∈x J(E)} so that E is an n-weak synthesis set if and only if ∆n(E) = ∅ and ∆m(E) 6= ∅
for all m < n [6]. The 1-difference spectrum is nothing but the difference spectrum ∆(E) and the
1-weak synthesis set is the synthesis set. We use the notation B(x, r) for the closed ball with center
x and radius r ≥ 0.
c© J. JOSEPH, T. K. MURALEEDHARAN, 2015
1434 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
SOME REMARKS ON SPECTRAL SYNTHESIS SETS 1435
2. Difference spectra of sets and its subsets. The entire set G itself is a synthesis set, so in
general, it is not true that a subset of a spectral synthesis set is again a spectral synthesis set, unless
G is discrete.
Theorem 1. If E ⊂ R, x ∈ E and V = E ∩ B(x, r), then ∆(V ) ⊂ ∆(E). Moreover if r > 0,
then there are points x1, x2, . . . in E which satisfy ∆(E) ⊂ ∪∆(Vi), where Vi = E ∩B(xi, r).
Proof. Let y ∈ ∆(V ) then y ∈ ∂E. If y ∈ ∂B(x, r), boundary of an interval, and since ∆(V ) is
a union of perfect sets there is a y′ ∈ ∆(V ) such that |x−y′| < |x−y|. Otherwise let y′ = y. Choose
a neighborhood U of y′ such that U ∩E ⊂ V. Pick k ∈ A(G) with k = 1 near y′ and supp k ⊂ U. So
there exists f ∈ I(V ) such that f /∈y′ J(V ). Then fk ∈ I(E), f =y′ fk and since J(V ) ⊃ J(E),
y′ ∈ ∆(E). Since y is the limit of such points y′ and ∆(E) is closed, the first part of the statement
follows.
For the second part, let x ∈ ∆(E) and V = E ∩B(x, r), choose f ∈ I(E) and f /∈x J(E), and
r1 =
r
2
. Let U be the open ball with center x and radius r1. Choose k ∈ A(G) with k = 1 near x
and supp k ⊂ U. Then f =x fk and J(E) =x J(V ) hence f /∈x J(V ). The same argument works
not only for x but for all the points in the r1 neighborhood of x and hence the proof.
The first part of the above theorem is not true for Rn if n > 2, for let E = {x ∈ Rn :
‖x‖ ≥ 1} ∪ {0}. At the same time, the second part of the above theorem is true for Rn, n > 1,
as well (essentially the same proof works). So if E ⊂ Rn then ∆(E) ⊂
⋃
xi∈E ∆(Vi), where
Vi = E ∩ B(xi, r), which, in particular, implies the known result that if each point of a set has a
relative synthesis neighborhood then the set itself is a synthesis set. For the n-difference spectrum,
however, we have the following result: ∆n(E) ⊂
⋃
xi∈E ∆n(Vi), so that if each Vi is a weak
synthesis set with characteristic less than or equal to n then E itself is a weak synthesis set with
characteristic less than or equal to n.
Using the above theorem it is easy to construct examples of non synthesis sets in R3. Let G1 and
G2 are locally compact Abelian groups and let α : G1 → G2 be injective, continuous homomorphism.
Assume that E1 ⊂ G1 is compact and E2 = αE1. Then it is proved in [1] that E1 is a synthesis
set if and only if E2 is a synthesis set. This result shows that any sphere of positive radius is a
non synthesis set in R3. Due to the symmetry of S2, the unit sphere in R3, ∆(S2) = S2. Now
S2 ∩B(x, r), r > 0, x ∈ S2, is a non synthesis set in R3. So the union of pieces of different spheres
is also a non-synthesis set in R3. Similarly many curves in R2 are examples of synthesis set in R2
since S1 is a synthesis set.
For E,F ⊂ G and x ∈ G, we recall that E =x F means E ∩V = F ∩V for some neighborhood
V of x. Now we use the following lemma to prove a theorem.
Lemma 1 [6]. Let E,F ⊂ G and x ∈ G. If E =x F, then (i) J(E) =x J(F ) and
(ii) ∆(E) =x ∆(F ).
Theorem 2. For a closed subset E of R and r > 0, there is a collection of closed sets Vi =
= E ∩ B(xi, r), i = 1, 2, 3, . . . , such that ∆(E) = ∪∆(Vi). Moreover for i 6= j, ∆(Vi) ∩∆(Vj) is
either empty or a singleton set.
Proof. Choose a closed interval F containing E. Let x1, x2, . . . be a sequence of real numbers in
F such that for i 6= j, B(xi, r)∩B(xj , r) is either empty or a singleton set and F = ∪B(xi, r). Then
E = E ∩ F = ∪(E ∩ B(xi, r)) = ∪Vi. Let
⋃
i,j,i6=j Vi ∩ Vj = {y1, y2, . . .}. So ∆(E) = ∆(∪Vi) ⊂
⊂ ∪∆(Vi) ∪ {y1, y2, . . .}. This follows from the fact that if y ∈ ∆(E) and y 6= yn, n = 1, 2, . . . ,
then y ∈ Vk for some k. Now choose U, a neighborhood of y such that U ∩ Vj = ∅ for j 6= k. Now
∪Vi =y Vk and y ∈ ∆(Vk) by Lemma 1.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
1436 J. JOSEPH, T. K. MURALEEDHARAN
Now if yn ∈ ∆(∪Vi), then yn ∈ ∆(Vj) for some j so that ∆(∪Vi) ⊂ ∪∆(Vi). To prove this,
let yn ∈ ∆(∪Vi). Then yn belongs to exactly one of the sets, say Vk or two such sets Vl and Vm. If
yn ∈ Vk for a unique k, then as above Vk =yn ∪Vi. Otherwise if yn ∈ Vl ∩ Vj and since ∆(∪Vi) is
a perfect set there is a sequence in ∆(∪Vi) which converges to yn. But such sequences are either in
Vl or in Vm. As in the first case, yn s are in ∆(Vl) or in ∆(Vm) so its limits also. Now the result
follows from the above theorem.
Choose a non synthesis set E in R, the existence of such sets follows from the Malliavin’s
theorem, so that ∆(E) 6= ∅. For any r > 0, there are sets V1, V2 . . . , as described above, which
satisfy the relation ∆(E) ⊂ ∪∆(Vi). So for some i, ∆(Vi) 6= ∅. Since translates of a non synthesis
set are again non synthesis sets, any interval of positive length contains a non synthesis set. Due to
the relation ∆(E1 × E2) ⊃ E1 ×∆(E2) ∪∆(E1)× E2 [6], it is easy to see that if either E1 or E2
is a non synthesis set, then E1 × E2 is a non synthesis set. So in Rn, n ≥ 1, any set of positive
Lebesgue measure contains a non synthesis set.
For a closed setE of R, let V1 = E∩B(x, r1) and V2 = E∩B(x, r2). Then ∆(V1∩V2) = ∆(V1)∩
∩∆(V2). Now the natural question is whether ∆ (
⋂∞
i=1 Vi) =
⋂∞
i=1 ∆(Vi), where V1 ⊃ V2 ⊃ . . . and
Vi = E ∩ B(x, ri). In R3, the corresponding result is not true: for, let E =
{
x ∈ R3 : ‖x‖ ≤ 2
}
\{
x ∈ R3 :
1
2
< ‖x‖ < 1
}
and Vn = E ∩ B
(
x, 1 +
1
n
)
. Then V1 ⊃ V2 ⊃ . . . and ∆(∩Vn) = S2,
but ∩∆(Vn) = ∅. Even in R this result is not true. Consider a non synthesis set E in R so
that ∅ 6= E′ = ∆(E), a union of perfect sets. Choose x ∈ E′ and for each positive integer, let
Vn = B
(
x,
1
n
)
∩E. Each Vn is a non synthesis set so that ∆(Vn) 6= ∅. As n→∞, ∆(∩Vn) = ∅
but ∩∆(Vn) = {x}.
Corollary 1. Let E be a closed subset of R. Then ∆ (
⋂∞
i=1 Vi) ⊂
⋂∞
i=1 ∆(Vi), where Vi =
= E ∩B(x, ri).
Proof. ∆(∩Vi) ⊂ ∆(Vi) for each i since ∩Vi ⊂ Vi.
LetE = E0,0 = [0, 1].Now see the following sequence of closed sets {Vi} satisfying the condition
V1 ⊃ V2 ⊃ . . . . For any two disjoint closed balls E(1,1), E(1,2) in E, let V1 = E(1,1) ∪ E(1,2). In
general, for n = 1, 2, 3, . . . , let Vn = E(n,1) ∪E(n,2) ∪ . . .∪E(n,2n), where E(n,2i−1), E(n,2i) are any
two disjoint closed balls in E(n−1,i), i = 1, 2, 3, . . . , 2n−1. Clearly, ∆ (
⋂n
i=1 Vi) =
⋂n
i=1 ∆(Vi). Is
it possible to conclude that ∆ (
⋂∞
i=1 Vi) ⊂
⋂∞
i=1 ∆(Vi)? If it is true, in particular, it shows that the
Cantor set is a set of spectral synthesis, a result proved by Herz [3].
3. Difference spectra of unions and intersections. A closed subset E of G is said to be
a Wiener – Ditkin set if whenever f ∈ I(E), f ∈ fj(E). For a subset E of G define Γ(E) =
= ∪f∈I(E){x ∈ G : f /∈x fj(E)} so that, E is a Wiener – Ditkin set if and only if Γ(E) = ∅.
Clearly ∆(E) ⊂ Γ(E), but the other inclusion, the C-set-S-set problem, is a major unsolved problem
in commutative harmonic analysis. A weaker form of this, ‘the weak C-set-S-set problem’ is discussed
recently in [7].
As we mentioned in the introduction, the question whether the union of two spectral synthesis
sets is again a spectral synthesis set is a well known, difficult open problem. For any two closed
sets E1 and E2, the relation ∆(E1 ∪ E2) ⊂ ∆(E1) ∪∆(E2) or a positive answer to the C-set-S-set
problem gives an easy solution to the union problem. Our next aim is to find a relation between the
difference spectrum of unions and unions of difference spectra. For closed subsets E1 and E2 of
G, we introduce two sets E1,2 and E2,1 as follows. Let E1,2 = {x ∈ ∂E1 ∩ ∂E2 ∩ ∂(E1 ∪ E2) :
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
SOME REMARKS ON SPECTRAL SYNTHESIS SETS 1437
x is a limit point of E1 \ E2} and E2,1 = {x ∈ ∂E1 ∩ ∂E2 ∩ ∂(E1 ∪ E2) : x is a limit point of
E2 \ E1}.
Lemma 2. Let E1 and E2 be closed sets in G. Then ∆(E1 ∪E2) ⊂ ∆(E1)∪∆(E2)∪Γ(E1,2 ∩
∩ E2,1).
Proof. Assume that x does not belong to the right-hand side of the desired inclusion and
x ∈ ∂(E1 ∪E2). Then I(E1 ∪E2)j(E1,2 ∩E2,1) ⊂x J(E1 ∪E2). To prove this, if x ∈ (E1,2 ∩E2,1)
then j(E1,2 ∩ E2,1) ∈x j(E1 ∪ E2) and if x ∈ (E1,2 ∩ E2,1)c ∩ E1, I(E1 ∪ E2) ⊂ I(E1) =x
=x J(E1) =x J(E1∪E2). Similar proofs hold for x ∈ (E1,2∩E2,1)c∩E2. Now let f ∈ I(E1∪E2).
Then f ∈x fj(E1,2 ∩ E2,1) ⊂x I(E1 ∪ E2)J(E1,2 ∩ E2,1) ⊂x J(E1 ∪E2). So x /∈ ∆(E1 ∪E2) and
the result follows.
Theorem 3. If E1, E2 are synthesis sets and if E1,2∩E2,1 is a Wiener – Ditkin set, then E1∪E2
is a synthesis set.
Proof. Since ∆(E1) = ∆(E2) = Γ(E1,2 ∩ E2,1) = ∅, the result follows.
In the above relation one can replace E1,2 ∩E2,1 with any set F that lies in between E1,2 ∩E2,1
and E1 ∪ E2. So if E1 and E2 are synthesis sets and F is a Wiener – Ditkin set, then E1 ∪ E2 is a
synthesis set. Whether Γ(E1,2 ∩ E2,1) can be replaced with ∆(E1,2 ∩ E2,1) in the above lemma is
another interesting problem (see also [10]).
For results about intersections, we define two sets as follows. For two closed sets E1, E2 in
G, let E′1,2 = {x ∈ ∂E1 ∩ ∂E2 : x is a limit point of E1 \ E2} and E′2,1 = {x ∈ ∂E1 ∩ ∂E2 :
x is a limit point of E2 \ E1}.
Lemma 3. If E1, E2 are closed sets in G, then ∆(E1∩E2) ⊂ ∆(E1)∪∆(E2)∪Γ(E′1,2∩E′2,1).
Proof. Assume that x /∈ ∆(E1)∪∆(E2)∪Γ(E′1,2∩E′2,1) and x ∈ ∂(E1∩E2) ⊂ ∂(E1)∩∂(E2).
Then I(E1 ∩ E2)j(E′1,2 ∩ E′2,1) ⊂x J(E1 ∩ E2). To prove this let x ∈ E′1,2 ∩ E′2,1. Then j(E′1,2 ∩
∩ E′2,1) ⊂x j(E1 ∩ E2). If x ∈ (E′1,2 ∩ E′2,1)c ∩ E1, then I(E1 ∩ E2) =x I(E1) =x J(E1) ⊂x
⊂x J(E1 ∩ E2). If f ∈ I(E1 ∩ E2) ⊂ I(E′1,2 ∩ E′2,1), then f ∈x fj(E′1,2 ∩ E′2,1) ∈x J(E1 ∩ E2).
Consequently x /∈ ∆(E1 ∩ E2) and the theorem follows.
Now we have the following result for the intersection of two synthesis sets, which is an improve-
ment of a result in [4] and the proof follows immediately from Lemma 3.
Theorem 4. If E1, E2 are synthesis sets and if E′1,2∩E′2,1 is a Wiener – Ditkin set, then E1∩E2
is a synthesis set.
Consider R3 and let E1 = B((0, 0, 0), 2) and E2 = B((3, 0, 0), 2). Then E1,2 ∩ E2,1 is a circle
which contains perfect sets. Similar conclusion holds for n > 3 also. In [11], Varopoulose proved
that every compact Abelian group G contains two sets of spectral synthesis whose intersection is not
of synthesis. Let E1 and E2 be non synthesis sets in a compact group G such that E1 ∩E2 is not of
synthesis. So, R×E1 and R×E2 are synthesis sets and R×E1 ∩R×E2 = R× (E1 ∩E2) is not
a synthesis set. Now by the Structure theorem, the compact Abelian group in Varopoulose result can
be replaced by any non discrete locally compact Abelian group (see [5]). Recall that any set which
does not contain a perfect set is a Wiener – Ditkin set. So there are synthesis sets E1 and E2 in R
such that E′1,2 ∩E′2,1 contains perfect sets. But in R, can the set E1,2 ∩E2,1, a still smaller set, ever
contain a perfect set? If the answer is no, it shows that the union of two synthesis sets is again a
synthesis set in R. In R2 also the details of such sets are not known.
The idea of weak Wiener – Ditkin set (weak C-set) has been introduced and studied in [7]. A
closed subset E of G is called an n-weak Wiener – Ditkin set if fn ∈ fj(E) for all f ∈ I(E). Thus
E is an n-weak Wiener – Ditkin set if and only if Γn(E) = ∅ and Γm(E) 6= ∅ for all m < n, where
Γn(E) =
⋃
f∈I(E){x : fn /∈x fj(E)}. Note that the 1-weak Wiener – Ditkin set is nothing but the
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
1438 J. JOSEPH, T. K. MURALEEDHARAN
Wiener – Ditkin set and Γ1(E) = Γ(E). Now for the n-difference spectrum we have the following
result.
Theorem 5. Let E1 and E2 be closed sets in G. Then ∆2n−1(E1 ∪E2) ⊂ ∆n(E1)∪∆n(E2)∪
∪ Γn(E1 ∩ E2).
Proof. Assume that x does not belong to the right-hand side of the inclusion and x ∈ ∂(E1∪E2).
Then In(E1 ∪ E2)j(E1 ∩ E2) ⊂x J(E1 ∪ E2) as in Lemma 2. Let f ∈ I(E1 ∪ E2), then fn ∈x
∈x fj(E1 ∩ E2). So f2n−1 ∈ fnj(E1 ∩ E2) ⊂x In(E1 ∪ E2)j(E1 ∩ E2) ⊂x J(E1 ∪ E2). So x
does not belong to the left side of the inclusion and the result follows.
Similar arguments give the following improvement of 3.3 in [6].
Theorem 6. Let E1, E2 be closed subsets of G. Then
i) ∆2n−1(E1 ∩ E2) ⊂ ∆n(E1) ∪∆n(E2) ∪ Γn(E′1,2 ∩ E′2,1),
ii) ∆2n−1(E1 ∪ E2) ⊂ ∆(E1) ∪∆(E2) ∪ Γn(E1,2 ∩ E2,1),
iii) ∆2n−1(E1) ∪∆2n−1(E2) ⊂ ∆n(E1 ∪ E2) ∪ Γn(E1,2 ∩ E2,1).
Acknowledgement. The authors thank UGC, Govt. of India for the financial support.
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Received 04.02.13,
after revision — 04.08.15
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
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