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Comptes Rendus
Mathématique
Alex Bergman
A Sharp Entropy Condition for the Density of Angular Derivatives
Volume 363 (2025), p. 29-33
Online since: 6 March 2025
https://doi.org/10.5802/crmath.695
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 29-33
https://doi.org/10.5802/crmath.695
Research article / Article de recherche
Operator theory / Théorie des opérateurs
A Sharp Entropy Condition for the Density of
Angular Derivatives
Une condition d’entropie forte pour la densité des
dérivées angulaires
Alex Bergman a
a Matematikcentrum, Sölvegatan 18A, 22100 Lund, Sweden
E-mail: alex.bergman@math.lu.se
Abstract. Let f be a holomorphic self-map of the unit disc. We show that if log(1− | f (z)|) is integrable on
a sub-arc of the unit circle, I , then the set of points where the function f has finite Carathéodory angular
derivative on I is a countable union of Beurling–Carleson sets of finite entropy. Conversely, given a countable
union of Beurling–Carleson sets, E , we construct a holomorphic self-map of the unit disc, f , such that the
set of points where the f function has finite Carathéodory angular derivative is equal to E and log(1−| f (z)|)
is integrable on the unit circle. Our main technical tools are the Aleksandrov disintegration Theorem and a
characterization of countable unions of Beurling–Carleson sets due to Makarov and Nikolski.
Résumé. Soit f une auto-application holomorphe du disque unitaire. Nous montrons que si log(1− | f (z)|)
est intégrable sur un sous-arc du cercle unitaire, I , alors l’ensemble des dérivées angulaires de Carathéodory
de f sur I est une union dénombrable d’ensembles de Beurling–Carleson d’entropie finie. Inversement,
étant donné une union dénombrable d’ensembles de Beurling–Carleson, E , nous construisons une auto-
application holomorphe du disque unitaire, telle que son ensemble de dérivées angulaires de Carathéodory
est égal à E et que log(1−| f (z)|) est intégrable sur le cercle unitaire. Nos principaux outils techniques sont le
théorème de désintégration d’Aleksandrov et une caractérisation des unions dénombrables d’ensembles de
Beurling–Carleson due à Makarov et Nikolski.
Keywords. Angular derivative, holomorphic map, Beurling–Carleson sets.
Mots-clés. Dérivée angulaire, carte holomorphe, ensembles de Beurling–Carleson.
2020 Mathematics Subject Classification. 30J99, 30C35, 30H10, 46E22.
Manuscript received 22 September 2024, revised 17 October 2024, accepted 21 October 2024.
1. Introduction
In this note D is the open unit disc in the complex plane, T is the unit circle, and m is the
normalized Lebesgue measure on T. A holomorphic self-map of the unit disc, f : D→ D, is said
to have angular derivative at λ ∈ T in the sense of Carathéodory if the following two conditions
hold :
(i) f has non-tangential limit at λ and f (λ) = limr→1− f (rλ) satisfies | f (λ)| = 1.
(ii) The derivative, f ′, has finite non-tangential limit at λ.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
30 Alex Bergman
The set of points where the function f has finite Carathéodory angular derivative will be denoted
C ( f ). Carathéodory angular derivatives play an important role in several areas of analysis. In
particular, they are important in perturbation theory of self-adjoint and unitary operators [7],
conformal mapping [2], and model and de Branges–Rovnyak spaces [6]. The above list of topics
and references is far from exhaustive. We shall be concerned with holomorphic self-maps of the
unit disc which are locally non-extreme in the following sense∫
I
log
(
1−| f (z)|)dm(z) >−∞, (1)
where I ⊂ T is a relatively open arc. If (1) holds then | f | < 1 almost everywhere on I and hence
C ( f )∩ I has measure 0. Our main result is that C ( f )∩ I is in fact much smaller than a generic set
of measure 0. When I =T condition (1) is equivalent to f being a non-extreme point of the unit
ball of H∞, here H∞ is the set of bounded holomorphic functions on the unit disc. This condition
plays a crucial role in the theory of de Branges–Rovnyak spaces, see [6].
We now introduce the relevant entropy condition. A closed set E ⊂T of measure zero, |E | = 0,
is called a Beurling–Carleson set of finite entropy if∑
I
|I | log
(
1
|I |
)
<∞, (2)
where the sum is taken over the maximal complementary arcs of E , here and in the sequel
|E | = m(E). Beurling–Carleson sets have appeared in connection with many problems in complex
analysis. To be brief we only mention that they are the zero sets of analytic functions in the disc
which extend to smooth functions on the circle, [8].
Our main result is the following characterization of C ( f )∩ I for locally non-extreme holomor-
phic self-maps of the unit disc.
Theorem 1. Let f : D→D be holomorphic and I ⊂T an open arc. If∫
I
log
(
1−| f |)dm >−∞.
Then C ( f )∩ I is a countable union of Beurling–Carleson sets. Conversely, if E ⊂ T is a countable
union of Beurling–Carleson sets there exists an analytic function f : D→D satisfying∫
T
log
(
1−| f |)dm >−∞,
and C ( f ) = E.
The sum in (2) increases the more spread out the points of E are. Thus the Beurling–Carleson
condition is telling us that the Carathéodory angular derivatives of f must have a certain amount
of clumping.
Our proof of Theorem 1 is based on the Aleksandrov disintegration Theorem and a characteri-
zation of countable unions of Beurling–Carleson sets by Makarov and Nikolski. We discuss these
results in the next section.
2. Preliminaries
For each α ∈T let µα be the measure given by the Herglotz integral formula
1−| f (z)|2
|α− f (z)|2 =
∫
1−|z|2
|ζ− z|2 dµα(ζ).
The measure µα is of finite mass and positive. The measures {µα}α∈T are called the Aleksandrov–
Clark measures of f . Information about Aleksandrov–Clark measures can be found in, for
Alex Bergman 31
example, the survey article [4] and the notes [5]. Let (µα)a denote the absolutely continuous
part of the measure µα. Standard properties of Poisson integrals imply that
d(µα)a(z) = 1−| f (z)|
2
|α− f (z)|2 dm(z).
The Aleksandrov–Clark measures satisfy the following disintegration Theorem due to Aleksan-
drov, [1].
Theorem 2. Let h be a continuous function on the unit circle. Then∫
T
h(z)dm(z) =
∫
T
∫
T
h(z)dµα(z)dm(α).
In fact, Aleksandrov showed that this can be extended to L1(T) functions in a suitable way.
The Carathéodory angular derivatives of f can be characterized using the Aleksandrov–Clark
measures. We state the following realization of this, which is Lemma 3.4 in [5]. It can also be
deduced by combining (VI-7), (VI-9), and (VI-10) from [6].
Lemma 3. Let f be a holomorphic self-map of the unit disc with Aleksandrov–Clark measure µα
and I ⊂ T a relatively open arc. Suppose µα is absolutely continuous with respect to Lebesgue
measure on I . Then f has a Carathéodory angular derivative at λ ∈ I if and only if∫
I
dµα(z)
|z −λ|2 <∞.
Remark 4. It was pointed out by the referee that it is enough to assume that µα has no point
masses on I in the previous Lemma. This is contained in Lemma 3.4 in [5].
Let H 2 denote the Hardy space of the unit disc and A∞ be the class of holomorphic func-
tions on the unit disc with smooth extension to the unit circle. As mentioned in the intro-
duction Beurling–Carleson sets are the boundary zero sets A∞. Indeed, Beurling–Carleson sets
are also the strong boundary zero sets of such functions, in the sense that if ϕ ∈ A∞, then{
λ ∈T : (z −λ)−1ϕ ∈ A∞} is a Beurling–Carleson set. It is a theorem of Makarov and Nikolski that
countable unions of Beurling–Carleson sets are the strong boundary zero sets for H 2, [3].
Theorem 5. Let ϕ ∈ H 2. Then the set {
λ ∈T : ϕ
z −λ ∈ H
2
}
is a countable union of Beurling–Carleson sets. Conversely, let E ⊂ T be a countable union of
Beurling–Carleson sets. Then there exists an outer function ϕ ∈ H 2, such that
E =
{
λ ∈T : ϕ
z −λ ∈ H
2
}
.
We remark that if ϕ ∈ H 2, then the sets
En =
{
λ ∈T : ∥(z −λ)−1ϕ∥2 ≤ n
}
,
are Beurling–Carleson sets.
3. Proof of the Main Result
We now turn to the proof of Theorem 1. We begin with a lemma.
Lemma 6. Let f and I be as in Theorem 1. Then for Lebesgue-almost every α ∈T the restriction of
µα to I is absolutely continuous.
32 Alex Bergman
Proof. Recall that the absolutely continuous part of µα is given by
d(µα)a = 1−| f |
2
|1−α f |2 dm.
Thus for each α ∈T
µα(I ) ≥ (µα)a(I ) =
∫
I
1−| f (z)|2
|1−α f (z)|2 dm(z).
Integrating with respect to α and using Fubini’s Theorem gives∫
T
∫
I
dµα(z)dm(α) ≥
∫
I
∫
T
1−| f (z)|2
|1−α f (z)|2 dm(α)dm(z).
Using the Aleksandrov disintegration Theorem on the left-hand side and that | f (z)| < 1 almost
everywhere on I on the right-hand side gives
|I | =
∫
T
∫
I
dµα(z)dm(α) ≥
∫
I
∫
T
1−| f (z)|2
|1−α f (z)|2 dm(α)dm(z) = |I |.
Thus ∫
T
(
µα(I )− (µα)a(I )
)
dm(α) = 0.
Since µα(I ) − (µα)a(I ) ≥ 0 for all α it follows that µα(I ) = (µα)a(I ) for almost every α ∈ T as
claimed. □
Remark 7. From the proof of the lemma we see that the conclusion remains valid if we merely
assume that | f (z)| < 1 almost everywhere on I .
We are now ready for the proof of Theorem 1.
Proof of Theorem 1. Recall that f is a holomorphic self-map on the unit disc. Suppose first that
f is locally non-extreme on the open interval I ,∫
I
log
(
1−| f (z)|)dm(z) >−∞.
We must show that C ( f )∩ I is a countable union of Beurling–Carleson sets. By Lemma 6 we may
assume without loss of generality that µ=µ1 is absolutely continuous on I . Indeed, if it is not we
may replace f by α f with a suitable α ∈T. Under this assumption f has a Carathéodory angular
derivative at λ ∈T if and only if ∫
T
1
|z −λ|2 dµ(z) <∞,
by Lemma 3. For λ ∈ I this is equivalent to local summability∫
I
1
|z −λ|2
1−| f (z)|2
|1− f (z)|2 dm(z) =
∫
I
1
|z −λ|2 dµ(z) <∞.
Since 1−| f | is log-integrable on I there exists a unique outer function, F ∈ H 2, such that
|F (z)|2 =

1−| f (z)|2
|1− f (z)|2 (z ∈ I ),
1 (z ∈T\ I ).
Then ∫
T
|F (z)|2
|z −λ|2 dm(z) =
∫
T\I
dm(z)
|z −λ|2 +
∫
I
dµ(z)
|z −λ|2 .
Hence, {
λ ∈T : F
z −λ ∈ H
2
}
=C ( f )∩ I .
Now by Theorem 5 the left-hand side is a countable union of Beurling–Carleson sets.
Alex Bergman 33
Suppose now that E ⊂ T is a countable union of Beurling–Carleson sets. We will construct a
non-extreme holomorphic self-map of the unit disc, f , such that C ( f ) = E . By Theorem 5 there
exists an outer function ϕ ∈ H 2 of norm 1 such that
E = {λ ∈T : (z −λ)−1ϕ ∈ H 2}.
Let f : D→D, f (0) = 0, be the unique holomorphic function such that
1+ f (z)
1− f (z) =
∫
T
ζ+ z
ζ− z |ϕ|
2dm(ζ).
It follows from Lemma 3 that f has Carathéodory angular derivative at λ ∈T if and only if∫
T
|ϕ(z)|2
|z −λ|2 dm(z) <∞.
Thus, by construction, C ( f ) = E . It remains to check that f is non-extreme. Indeed, by standard
properties of Poisson integrals,
1−| f (z)|2
|1− f (z)|2 = |ϕ|
2, for almost every z ∈T.
Since 1− f (z) and ϕ both belong to H 2 and hence are log integrable it follows that∫
T
log
(
1−| f (z)|)dm(z) >−∞,
as claimed. □
Declaration of interests
The author does not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and has declared no affiliations other than their research
organizations.
References
[1] A. B. Aleksandrov, “Multiplicity of boundary values of inner functions”, Izv. Akad. Nauk
Armyan. SSR Ser. Mat. 22 (1987), no. 5, pp. 490–503.
[2] J. B. Garnett and D. E. Marshall, Harmonic measure, New Mathematical Monographs, Cam-
bridge University Press, 2005, pp. xvi+571.
[3] N. G. Makarov, “The unitary point spectrum of almost unitary operators”, Zap. Nauchn.
Semin. Leningr. Otd. Mat. Inst. Steklova 126 (1983), pp. 143–149.
[4] A. Poltoratski and D. E. Sarason, “Aleksandrov-Clark measures”, in Recent advances in
operator-related function theory, Contemporary Mathematics, American Mathematical So-
ciety, 2006, pp. 1–14.
[5] E. Saksman, “An elementary introduction to Clark measures”, in Topics in complex analysis
and operator theory, Universidad de Málaga, 2007, pp. 85–136.
[6] D. E. Sarason, Sub-Hardy Hilbert spaces in the unit disk, The University of Arkansas Lecture
Notes in the Mathematical Sciences, John Wiley & Sons, 1994, pp. xvi+95.
[7] B. Simon, “Spectral analysis of rank one perturbations and applications”, in Mathematical
quantum theory II. Schrödinger operators (Vancouver, BC, 1993), CRM Proceedings & Lec-
ture Notes, American Mathematical Society, 1995, pp. 109–149.
[8] B. A. Taylor and D. L. Williams, “Ideals in Rings of Analytic Functions with Smooth Boundary
Values”, Can. J. Math. 22 (1970), no. 6, pp. 1266–1283.


A Sharp Entropy Condition for the Density of Angular Derivatives

Djalil Chafaï

Max Fathi

On cutoff via rigidity for high dimensional curved diffusions

Anne-Laure Basdevant

Olivier Hénard

Édouard Maurel-Segala

Arvind Singh

On cases where Litt’s game is fair

Manuel Ladra

Andrés Pérez-Rodríguez

Comptes Rendus
Mathématique
Manuel Ladra and Andrés Pérez-Rodríguez
Regular evolution algebras are closed under subalgebras
Volume 363 (2025), p. 1461-1465
Online since: 28 November 2025
https://doi.org/10.5802/crmath.804
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 1461-1465
https://doi.org/10.5802/crmath.804
Research article
Algebra
Regular evolution algebras are closed under
subalgebras
Manuel Ladra ,a and Andrés Pérez-Rodríguez ,a
a Department of Mathematics & CITMAga, Universidade de Santiago de Compostela,
15782 Santiago de Compostela, Spain
E-mails: manuel.ladra@usc.es, andresperez.rodriguez@usc.es
Abstract. The main goal of this note is to show that subalgebras of regular evolution algebras are themselves
evolution algebras. This allows us to assume, without loss of generality, that every subalgebra in the regular
setting has a basis consisting of vectors with disjoint supports. Finally, we use this result to characterise the
existence of codimension-one subalgebras in regular evolution algebras.
Keywords. Evolution algebras, regular evolution algebras, subalgebras.
2020 Mathematics Subject Classification. 17D92, 17A60.
Funding. This work was partially supported by Agencia Estatal de Investigación (Spain), grants PID2020-
115155GB-I00 and PID2024-155502NB-I00 (European FEDER support included, UE) and by Xunta de Galicia
through the Competitive Reference Groups (GRC), ED431C 2023/31. The second author was also supported
by FPU21/05685 scholarship, Ministerio de Educación y Formación Profesional (Spain).
Manuscript received 10 March 2025, revised 20 October 2025, accepted 21 October 2025.
Given an arbitrary fieldK, an evolution algebra overK is aK-algebra E which admits a distin-
guished basis B = {e1, . . . ,en , . . . }, called a natural basis, satisfying the condition that ei e j = 0 for
all i ̸= j . In this note, we focus on finite-dimensional evolution algebras, meaning that B is a finite
set. For a given natural basis B = {e1, . . . ,en} in E , the scalars ai j ∈K satisfying e2i =
∑n
j=1 ai j e j are
called the structure constants of E relative to B . The matrix MB (E ) = (ai j )ni , j=1 is said to be the
structure matrix of E relative to B . Evolution algebras were introduced by J. P. Tian and P. Vo-
jtěchovský as a mathematical framework for modelling inheritance mechanisms that do not fol-
low classical Mendelian genetics. Their commutative but non-associative multiplication reflects
self-replication patterns observed in specific non-Mendelian genetic models (see [10,11]). It is
worth noting that linear algebra has provided powerful tools for modelling and analysing genetic
systems since they were first studied by Etherington [8].
Since evolution algebras are not defined by identities, one of the main challenges in working
with them is that they are neither closed under subalgebras (see [2, Example 2.3]) nor under ideals
(see [2, Example 2.7]). However, unlike the case of subalgebras, the structure of ideals has been
extensively studied (see [3,4]). In particular, [1, Proposition 4.2] establishes that if an evolution
algebra E is regular (E = E 2, or equivalently, MB (E ) is non-singular) then every nonzero ideal is
basic, that is, every ideal admits a natural basis consisting of vectors from the natural basis.
Regular evolution algebras have been thoroughly investigated and possess particularly desir-
able properties. For instance, they have a unique natural basis (see [6, Theorem 4.4]), their auto-
morphism groups are finite (see [6, Theorem 4.8]), and they are universally finite (see [5]). More-
over, their algebras of derivations have also been described [7, Theorem 4.1]. Indeed, this short
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1462 Manuel Ladra and Andrés Pérez-Rodríguez
paper aims to explore subalgebras in the regular setting. First, note that the one-dimensional
subalgebras are given by the non-trivial solutions of a non-linear polynomial system of equa-
tions.
Lemma 1. Let E be a regular evolution algebra with natural basis B = {e1, . . . ,en} and structure
matrix MB (E ). Then, the one-dimensional subspace span
{
α1e1 + ·· ·+αnen
}
is a proper nonzero
subalgebra of E if and only if the vector (α1, . . . ,αn) is proportional to a non-trivial solution ofx
2
1
...
x2n
= (MB (E )t )−1
x1...
xn
 . (1)
Proof. A subspace span
{
α1e1 +·· ·+αnen
}
is a subalgebra if and only if there exists a scalar k ̸= 0
such that ( n∑
i=1
αi ei
)2
=
n∑
i=1
α2i (ai 1e1 +·· ·+ai nen) =
n∑
j=1
( n∑
i=1
ai jα
2
i
)
e j = k
( n∑
i=1
αi ei
)
.
For this condition to hold, it is necessary that
∑n
j=1 a j iα
2
j = kαi for all i = 1, . . . ,n. Moreover, since
it can be assumed that k = 1 (by setting α′i = αik ), we obtain (1). □
In particular, our main theorem establishes that regular evolution algebras are closed under
subalgebras, a property that provides a more precise and structured approach to understanding
subalgebras and significantly simplifies their study in higher dimensions.
Theorem 2. Let E be a regular evolution algebra over any field K. Then, every subalgebra of E
admits a natural basis.
Proof. Let E be a regular evolution algebra with natural basis {e1, . . . ,en}. Consider a proper
subalgebra U = span{ui : 1 ≤ i ≤ m}, where each ui = ∑nj=1µi j e j with µi j ∈ K, of dimension
m < n. Without loss of generality, we assume that the matrix (µi j )m,ni , j=1 is in reduced row echelon
form. Furthermore, by appropriately reordering the natural basis, we can assume that the leading
coefficients form the identity matrix, i.e.
u1 = e1 + v1, u2 = e2 + v2, . . . , um = em + vm ,
where v1, . . . , vm ∈ span
{
em+1, . . . ,en
}
.
For the sake of contradiction, assume that U does not admit a natural basis. Then, there
exist distinct indices k, l ∈ {1, . . . ,m} such that uk ul ̸= 0. Moreover, as E is regular, we have
rank
{
e21, . . . ,e
2
n
} = n, which implies that rank{u1, . . . ,um} = rank{u21, . . . ,u2m} = m. Then, as
{u21, . . . ,u
2
m} is also a basis of the subalgebra U , there exist scalars α1, . . . ,αm ∈K such that
α1u
2
1 +·· ·+αmu2m +uk ul = 0.
Expanding each term,
α1(e
2
1 + v21)+·· ·+αm(e2m + v2m)+uk ul = 0.
Since uk ul ∈ span
{
e2m+1, . . . ,e
2
n
}
, the linear independence of {e21, . . . ,e
2
n} forces α1 = ·· · = αm = 0,
contradicting that uk ul ̸= 0. □
Remark 3. The converse of the previous result does not hold in general. As shown in [9,
Corollary 3.4], all subalgebras of a nilpotent evolution algebra with the maximum index of
nilpotency are basic ideals. For instance, the unique proper nonzero subalgebras of the evolution
algebra E with natural basis {e1,e2,e3} and product given by e21 = e2, e22 = e3 and e23 = 0 are
span{e3} and span{e2,e3}.
Recall that, given an evolution algebra E with natural basis B = {e1, . . . ,en} and an element
u =∑ni=1µi ei ∈ E , we define its support as suppB (u) := {i :µi ̸= 0}.
Manuel Ladra and Andrés Pérez-Rodríguez 1463
Corollary 4. Let E be a regular evolution algebra over any field K with natural basis B =
{e1, . . . ,en}. Then, every subalgebra U of E admits a basis {u1, . . . ,um} where the supports of its
elements are disjoint.
Proof. The result follows from Theorem 2 and from the fact that if we have two elements u
and v in a regular evolution algebra, then uv = 0 if and only if their supports are disjoint (see [6,
Remark 4.3]). □
This result significantly simplifies the study of subalgebras in the regular case. For example,
given a regular evolution algebra E with natural basis B = {e1, . . . ,en}, any subalgebra of codimen-
sion 1 can be written as
span
{
ei , v : i ̸= p, q
}
, (2)
where v ∈ span{ep ,eq } for some distinct indices p, q ∈ {1, . . . ,n}. Without loss of generality, in this
work we will assume that p < q . Using this description, we finally establish precise conditions for
the existence of subalgebras of codimension 1 in regular evolution algebras.
Proposition 5. Let E be a regular evolution algebra over any fieldK of dimension greater than two
with natural basis B = {e1, . . . ,en} and structure matrix MB (E ) = (ai j )ni , j=1.
Suppose that E has a subalgebra of codimension 1. In this case, there necessarily exist distinct
indices p, q ∈ {1, . . . ,n} such that
rank Mp,q ≤ 1,
where Mp,q denotes the submatrix of MB (E ) formed by the p-th and q-th columns after removing
the p-th and q-th rows.
Conversely, if rank Mp,q ≤ 1, the existence of subalgebras of codimension 1 can be characterised
as follows.
(a) If rank Mp,q = 1, then at least one row of Mp,q is non-zero; let (α,β) be such a row. In this
case, E has a subalgebra of codimension 1 if and only if
α2βapp +β3aqp =α3apq +αβ2aqq , (3)
and the corresponding subalgebra is given by (2) with v =αep +βeq .
(b) If rank Mp,q = 0, then the subalgebras of codimension 1 are (2) where v = ep +λeq for any
λ ∈K∗ satisfying the equation
aqpλ
3 −aqqλ2 +appλ−apq = 0, (4)
and additionally, span
{
ei : i ̸= q
}
when apq = 0 and span
{
ei : i ̸= p
}
when aqp = 0.
Proof. Let πp,q be the linear projection of E onto span{ep ,eq } along span
{
ei : i ̸= p, q
}
. Observe
that (2) with v =αep +βeq defines a subalgebra of E if and only if the following conditions hold:
πp,q (e
2
i ) = ai p ep +ai q eq ∈ span
{
αep +βeq
}
, for all i ̸= p, q , and (5)
πp,q
(
(αep +βeq )2
)= (α2app +β2aqp )ep + (α2apq +β2aqq )eq ∈ span{αep +βeq}. (6)
From (5), it follows that rank Mp,q ≤ 1 is a necessary condition. Conversely, now assume that
rank Mp,q ≤ 1 and study the following two cases separately.
(a) If rank Mp,q = 1, then (α,β) must correspond with a non-zero row of Mp,q for (2) to be a
subalgebra of E . Now, we consider the following scenarios for (6) to be satisfied:
(i) if α= 0 and β ̸= 0, then (6) holds if and only if aqp = 0;
(ii) if β= 0 and α ̸= 0, then (6) holds if and only if apq = 0;
(iii) if α,β ̸= 0, then (6) holds if and only if there exists a scalar k ∈ K∗ such that
α2app +β2aqp = kα and α2apq +β2aqq = kβ, or equivalently, if (3) holds.
1464 Manuel Ladra and Andrés Pérez-Rodríguez
Notice that if α= 0 and β ̸= 0 (resp. β= 0 and α ̸= 0), then (3) holds if and only if aqp = 0
(resp. apq = 0). Thus, we conclude that, in this case, (2) is a subalgebra if and only if (3).
(b) If rank Mp,q = 0, then it is evident that span
{
ei : i ̸= q
}
and span
{
ei : i ̸= p
}
are subalgebras
if and only if apq = 0 and aqp = 0, respectively. Moreover, (2) with v =αep +βeq for some
α,β ̸= 0, or equivalently, with v = ep +λeq for λ ̸= 0, is a subalgebra if and only if there
exists a scalar k ∈K∗ such that app+λ2aqp = k and apq +λ2aqq = kλ, which is equivalent
to λ being a nonzero solution of (4).
Thus, the result follows. □
Remark 6. Although every nonzero proper subalgebra of a regular evolution algebra of dimen-
sion 2 trivially admits a natural basis, as they are of dimension 1, the computations in Proposi-
tion 5 also play a key role in fully characterising all subalgebras in this case. Specifically, given
a regular evolution algebra E over any field K with natural basis {e1,e2} and structure matrix
MB (E ) = (ai j )2i , j=1, the subalgebras are given by span{e1+λe2} for any λ ∈K∗ satisfying the equa-
tion
a21λ
3 −a22λ2 +a11λ−a12 = 0;
and additionally, span{e1} when a12 = 0 and span{e2} when a21 = 0.
Corollary 7. Let E be a regular evolution algebra over any field K of dimension greater than two,
with natural basis B = {e1, . . . ,en} and structure matrix MB (E ) = (ai j )ni , j=1. If the subspace (2) is a
subalgebra for some v ∈ span{ep ,eq }, then the structure constants satisfy
a2i p ai q app +a3i q aqp = a3i p apq +ai p a2i q aqq (7)
for all i ̸= p, q. Particularly, if E is three-dimensional and K is algebraically closed or K= R, then
the converse also holds.
Proof. The necessity follows straightforwardly from Proposition 5.
For sufficiency in dimension 3, note first that for any distinct indices p, q ∈ {1,2,3} we always
have rank Mp,q ≤ 1. If rank Mp,q = 1, the claim follows immediately from the hypothesis (7)
together with condition (3). On the other hand, if rank Mp,q = 0, since K is algebraically closed
orK=R, then equation (4) with apq , aqp ̸= 0 always admits a solution inK. Therefore, the result
follows. □
We conclude this note with two examples which show how the hypothesis for equivalence
cannot be relaxed. Example 8 illustrates the importance of the choice of the field, while Example 9
highlights the necessity of the evolution algebra to have dimension 3.
Example 8. Let E be the regular evolution algebra over Q with natural basis {e1,e2,e3} and
structure matrix given by
MB (E ) =
1 0 01 −1 1
2 1 0
 .
First, note that M2,3 = (0,0) and that the structure constants verify the relation
a212a13a22 +a313a32 = 0 = a312a23 +a12a213a33.
However, we have that a23, a32 ̸= 0, and the polynomial a32x3 − a33x2 + a22x − a23 = x3 − x − 1
is irreducible over Q. Consequently, according to Proposition 5, there are no subalgebras of
dimension 2 of the form span{e1, v} with v ∈ span{e2,e3}. Moreover, since
a221a23a11 +a323a31 ̸= a321a13 +a21a223a33 and a231a32a11 +a332a21 ̸= a331a12 +a31a232a22,
E does not have subalgebras of dimension 2.
Manuel Ladra and Andrés Pérez-Rodríguez 1465
Example 9. Let E be a regular evolution algebra over any fieldKwith natural basis {e1,e2,e3,e4}
and the two last columns of the structure matrix MB (E ) given by
a13 a14
a23 a24
a33 a34
a43 a44
=

1 2
1 −1
−3 2
1 0
 .
Notice that a2i 3ai 4a33 +a3i 4a43 = a3i 3a34 +ai 3a2i 4a44 for i = 1,2. However,
rank M3,4 = rank
(
1 2
1 −1
)
= 2,
what yields, by Proposition 5, that there are no subalgebras of dimension 3 of the form
span{e1,e2, v} with v ∈ span{e3,e4}.
Acknowledgments
We thank the referee for their helpful and accurate comments and suggestions, which con-
tributed to the improvement of this paper.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] N. Boudi, Y. Cabrera Casado and M. Siles Molina, “Natural families in evolution algebras”,
Publ. Mat., Barc. 66 (2022), no. 1, pp. 159–181.
[2] Y. Cabrera Casado, M. Siles Molina and M. V. Velasco, “Evolution algebras of arbitrary
dimension and their decompositions”, Linear Algebra Appl. 495 (2016), pp. 122–162.
[3] Y. C. Casado, M. Kanuni and M. Siles Molina, “Basic ideals in evolution algebras”, Linear
Algebra Appl. 570 (2019), pp. 148–180.
[4] Y. C. Casado, D. Martín Barquero, C. Martín González and A. Tocino, “Connecting ideals in
evolution algebras with hereditary subsets of its associated graph”, Collect. Math. 76 (2025),
no. 2, pp. 357–372.
[5] C. Costoya, P. Ligouras, A. Tocino and A. Viruel, “Regular evolution algebras are universally
finite”, Proc. Am. Math. Soc. 150 (2022), no. 3, pp. 919–925.
[6] A. Elduque and A. Labra, “Evolution algebras and graphs”, J. Algebra Appl. 14 (2015), no. 7,
article no. 1550103 (10 pages).
[7] A. Elduque and A. Labra, “Evolution algebras, automorphisms, and graphs”, Linear Multi-
linear Algebra 69 (2021), no. 2, pp. 331–342.
[8] I. M. H. Etherington, “Genetic algebras”, Proc. R. Soc. Edinb., Sect. A, Math. 59 (1939),
pp. 242–258.
[9] M. Ladra, P. Páez-Guillán and A. Pérez-Rodríguez, “On the subalgebra lattice of solvable
evolution algebras”, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 119 (2025),
no. 3, article no. 83 (18 pages).
[10] J. P. Tian, Evolution algebras and their applications, Lecture Notes in Mathematics,
Springer, 2008.
[11] J. P. Tian and P. Vojtěchovský, “Mathematical concepts of evolution algebras in non-
Mendelian genetics”, Quasigroups Relat. Syst. 14 (2006), no. 1, pp. 111–122.


Regular evolution algebras are closed under subalgebras

Florent Noisette

Uniqueness for the Camassa–Holm equation with non-homogeneous boundary conditions

Benjamin Gammage

Justin Hilburn

Betti Tate’s thesis and the trace of perverse schobers

Alain Bretto

Alain Faisant

François Hennecart

Comptes Rendus
Mathématique
Alain Bretto, Alain Faisant and François Hennecart
About Berge–Füredi’s conjecture on the chromatic index of hypergraphs
Volume 363 (2025), p. 323-328
Online since: 8 April 2025
https://doi.org/10.5802/crmath.739
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
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www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 323-328
https://doi.org/10.5802/crmath.739
Research article / Article de recherche
Combinatorics / Combinatoire
About Berge–Füredi’s conjecture on the
chromatic index of hypergraphs
Sur la conjecture de Berge–Füredi sur l’indice
chromatique des hypergraphes
Alain Bretto a, Alain Faisant b and François Hennecart ∗,b
a Université Caen Normandie, GREYC CNRS-UMR 6072, Caen, France
b Université Jean Monnet, ICJ UMR5208, CNRS, École Centrale de Lyon, INSA Lyon, Université Claude
Bernard Lyon 1, 42023 Saint-Étienne, France
E-mails: alain.bretto@info.unicaen.fr, faisant@univ-st-etienne.fr, francois.hennecart@univ-st-etienne.fr
Abstract. We prove that the chromatic index of a hypergraph H satisfies the Berge–Füredi conjectured bound
q(H ≤ ∆([H2])+ 1 under certain hypotheses on the antirank ar(H ) or on the maximum degree ∆(H ).
This provides sharp information in connection with the Erdős–Faber–Lovász conjecture which deals with
the coloring of a family of cliques that intersect pairwise in at most one vertex.
Résumé. Nous montrons que l’indice chromatique d’un hypergraphe satisfait la borne conjecturée indépen-
damment par Berge et Füredi, q(H ≤∆([H2])+1 sous certaines conditions portant sur son antirang ar(H )
ou son degré maximum ∆(H ). Ces résultats fournissent des informations sur la conjecture de Erdős, Faber
and Lovász qui traite de la coloration d’une famille de cliques se coupant deux à deux en au plus un sommet.
Keywords. Hypergraphs, hyperedge coloring, chromatic index.
Mots-clés. Hypergraphes, coloration des hyperarêtes, indice chromatique.
2020 Mathematics Subject Classification. 05C15.
Manuscript received 12 March 2024, revised 10 March 2025, accepted 11 March 2025.
1. Introduction
Berge–Füredi’s conjecture (cf. [1,2,6] and Meyniel, unpublished communication) extends to
linear hypergraphs Vizing’s theorem on the chromatic index of graphs (see [3, Theorem 7.2.2]). It
asserts that the chromatic index of any linear hypergraph H = (V ,E) without loop satisfies
q(H ) ≤∆([H ]2)+1. (1)
This bound was first conjectured around 1986. Since then, to our knowledge, only few advances
have been made. It has been recently proved [8] that (1) holds whenever ∆
(
[H ]2
) ≤ 5. In the
present paper we obtain new conditional results where the antirank ar(H ) and the maximum
degree ∆(H ) are involved together with the maximum degree ∆
(
[H ]2
)
of the 2-section of H .
If H is linear then∆
(
[H ]2
)≤ |V |−1. Hence bound (1) implies Erdős–Faber–Lovász conjecture
asserting that q(H ) ≤ |V | for any linear hypergraph H .
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
324 Alain Bretto, Alain Faisant and François Hennecart
2. Notation and statements
Let H = (V ,E) be a hypergraph. For x ∈V and e ∈ E , we let
degH (x) =
∣∣∣{a ∈ E : x ∈ a}∣∣∣, dH (e) = ∣∣∣{a ∈ E ∖ {e} : a ∩e ̸=∅}∣∣∣,
be the degree of vertex x and the degree of hyperedge e in H , respectively. Then, we define
respectively the rank, the antirank, the maximum degree and the minimum degree of H by
r(H ) = max
e∈E
|e|, ar(H ) = min
e∈E
|e|, ∆(H ) = max
x∈V
degH (x), δ(H ) = minx∈V degH (x).
A loop in a hypergraph H is a hyperedge e such that |e| = 1.
The 2-section of H = (V ,E) is the loopless multigraph [H ]2 = (V , A) with k many edges
between x, y ∈V , x ̸= y , if there are k hyperedges in H containing both x and y . We have
∆
(
[H ]2
)≥ (ar(H )−1)∆(H ). (2)
The chromatic index of H , denoted by q(H ), is the least integer q such that there exists a map
c : E → {1,2, . . . , q} satisfying
∀ e,e ′ ∈ E , (c(e) = c(e ′) =⇒ e ∩e ′ =∅ or e = e ′).
The map c is called a hyperedge-coloring of H and the image of E through c forms the set of
colors.
A hypergraph H = (V ,E) is said linear if for any pair of distinct hyperedges e,e ′ ∈ E , we have
|e ∩e ′| ≤ 1. In this case [H ]2 reduces to a simple graph.
Recently it has been established that if H is a linear hypergraph satisfying
ar(H ) ≥
√
|V |, (3)
then q(H ) ≤ |V | (cf. [7]). This upper bound has been improved in [4] where it is shown that under
the same hypotheses on H , then (1) actually holds. In this note we show that condition (3) can
be weakened by proving the following sharper result.
Theorem 1. Let H = (V ,E) be a loopless hypergraph such that ar(H ) ≥
√
∆
(
[H ]2
)+1. Then
q(H ) ≤∆([H ]2)+1.
Note that we do not assume in the above statement that H is linear. We will propose an
approach supported by the notion of critical hypergraph (see also Section 6.1 for additional
comments).
We will prove the following key result which has its own interest. H is said k-uniform (resp.
d-regular) if ∀ e ∈ E , |e| = k (resp. ∀ x ∈V , degH (x) = d).
Theorem 2. Let k ≥ 2 and H = (V ,E) be a k-uniform and (k+1)-regular linear hypergraph. Then
q(H ) ≤∆([H ]2)+1.
The following corollary will be derived from Theorem 1.
Corollary 3. Let H = (V ,E) be a loopless hypergraph such that
∆(H ) ≤
√
∆
(
[H ]2
)+1+1. (4)
Then q(H ) ≤∆([H ]2)+1.
Alain Bretto, Alain Faisant and François Hennecart 325
3. Proof of Theorem 1
In [4] we exploited the notion of critical hyperedge. A hyperedge e is said critical if q(H ∖ e) =
q(H ) − 1. Hypergraph H is a critical hypergraph if all its hyperedges are critical. Since the
addition of e to H ∖ e needs a new color, e must intersect every color of H ∖ e. So we have
the following.
Lemma 4 ([4, Lemma 3.2]). Let H = (V ,E) be a hypergraph without loop and e ∈ E be a critical
hyperedge. Then q(H )−1 ≤ dH (e).
We may assume that H is connected and critical since ar(H ′) ≥ ar(H ) and ∆([H ′]2) ≤
∆
(
[H ]2
)
for any partial hypergraph H ′ of H . Let e ∈ E be of minimum cardinality k := ar(H ).
We have
dH (e) =
∑
a∈E∖{e}
a∩e ̸=∅
1 ≤ ∑
a∈E∖{e}
|a ∩e| = ∑
x∈V
∑
a∈E∖{e}
a∋x
1 = ∑
x∈e
(
degH (x)−1
)
, (5)
and equality occurs when H is linear. Then by Lemma 4 we get
q(H )−1 = q(H ∖e) ≤ dH (e) ≤ k
(
∆(H )−1)= (k −1)∆(H )− (k −∆(H )),
so
q(H ) ≤∆([H ]2)+1− (k −∆(H )). (6)
Our hypothesis implies ∆
(
[H ]2
)≤ k2 −1. We distinguish two cases.
• If ∆([H ]2) < k2 −1 we get by (2) that ∆(H ) < k +1 thus ∆(H ) ≤ k, yielding the required
upper bound for q(H ) by (6).
• Assume that ∆([H ]2)= k2 −1. We thus have to prove that q(H ) ≤ k2. If there exists e ∈ E
such that |e| = k and dH (e) ≤ k2 −1 then q(H ) ≤ dH (e)+1 ≤ k2 by Lemma 4 and we are
done. Otherwise
dH (e) ≥ k2 for any hyperedge e of cardinality k. (7)
We shall see that H is a very special hypergraph.
– Let e ∈ E be of cardinality k. By (2) we have ∆(H ) ≤ k +1, thus by (7) and (5) we get
k2 ≤ dH (e) ≤
∑
x∈e
(
degH (x)−1
)≤ k(∆(H )−1)≤ k(k +1−1) = k2.
We infer
dH (e) = k2 and degH (x)−1 = k for any x ∈ e. (8)
– Let a ∈ E such that a ∩e ̸=∅. Let x ∈ a ∩e. We have
k2 −1 =∆([H ]2)≥ deg[H ]2 (x) = ∑
b∈E
b∋x
(|b|−1)
= |a|−1+ ∑
b∈E∖{a}
b∋x
(|b|−1)
≥ |a|−1+ (degH (x)−1)(k −1)
= |a|−1+k(k −1)
≥ k2 −1,
giving |a| = k.
– Since H is connected we infer that |a| = k for any a ∈ E . Hence H is k-uniform and
(k + 1)-regular. By (5) and (8), we have |a ∩ e| = 1 whenever a ∩ e ̸= ∅. Thus H is
linear. The proof will be completed when Theorem 2 will be established.
326 Alain Bretto, Alain Faisant and François Hennecart
4. Proof of Theorem 2
By assumption we have
k|e| = (k +1)|V |, ∆([H ]2)= k2 −1 and ∀ e ∈ E , dH (e) = k2. (9)
Since H is linear, its line-graph L(H ) is the simple graph whose vertices are hyperedges of H
and such that there is an edge between e,e ′ ∈ E , e ̸= e ′, if e ∩ e ′ ̸=∅. We have degL(H )(e) = dH (e)
for any e ∈ E .
By (9), each vertex e of L(H ) has degree degL(H )(e) = k2. Hence L(H ) is k2-regular. Since
q(H ) coincides with χ
(
L(H )
)
, namely the chromatic number of the simple graph L(H ), we infer
from Brooks’ theorem (see [3, Theorem 7.1.9, p. 238]) that either
q(H ) ≤∆(L(H ))= k2 =∆([H ]2)+1
or L(H ) is complete or an odd-length cycle. If L(H ) were a complete graph, we would have
|e| = k2 +1 implying by (9) that (k +1) divides k(k2 +1), which gives a contradiction since k ≥ 2.
If L(H ) were a cycle, we would have dH (e) = degL(H )(e) = 2 for any e ∈ E , a contradiction. This
ends the proof of Theorem 2.
5. Proof of Corollary 3
Let H ′ = (V ,E ′) be a critical partial hypergraph of H such that q(H ′) = q(H ).
• If ar(H ′) ≥
√
∆
(
[H ′]2
)+1 we infer from Theorem 1 that
q(H ) = q(H ′) ≤∆([H ′]2)+1 ≤∆([H ]2)+1.
• If ar(H ′) <
√
∆
(
[H ′]2
)+1 then we take e ∈ E ′ such that |e| = ar(H ′). By Lemma 4 and
using (5) with H ′ in place of H , we get
q(H )−1 = q(H ′)−1 ≤ dH ′ (e)
≤ ∑
x∈e
(
degH ′ (x)−1
)
≤ |e|(∆(H ′)−1)
= ar(H ′)(∆(H ′)−1)
<
√
∆
(
[H ′]2
)+1(∆(H ′)−1)
≤
√
∆
(
[H ]2
)+1(∆(H )−1)
≤∆([H ]2)+1
by (4). Hence q(H ) ≤∆([H ]2)+1 as wanted.
6. Remarks and questions
6.1.
We should notice that when ar(H ) >
√
∆
(
[H ]2
)+1 then the desired bound q(H ) ≤∆([H ]2)+1
can be equally deduced from the greedy algorithm for coloring hyperedges (see for instance [5,
p. 586]) which yields
q(H ) ≤ max
r(H )≥k≥ar(H )
k
⌊
∆
(
[H ]2
)
k −1 −1
⌋
+1.
Here we use a different approach which allows us to solve in addition the equality case ar(H ) =√
∆
(
[H ]2
)+1.
Alain Bretto, Alain Faisant and François Hennecart 327
6.2.
When the maximum degree ∆(H ) satisfies ∆(H ) >
√
∆
(
[H ]2
)+1+ 1, Corollary 3 is no longer
usable. However at the cost of an additional condition on the rank r(H ) = maxe∈E |e|, we can
notice that (1) remains valid for linear hypergraphs H . Let δ2 :=
√
∆
(
[H ]2
)
and assume that
r(H ) = δ2 −u, u > 0,
∆(H ) = δ2 +1+ v, v > 0.
Then
q(H ) =χ(L(H ))≤∆(L(H ))+1 ≤ r(H )(∆(H )−1)+1 = (δ2 −u)(δ2 + v)+1.
Thus (1) holds whenever (v −u)δ2 ≤ uv , that is 1u ≤ 1v + 1δ2 .
6.3.
If H is supposed linear, in the extremal case ∆
(
[H ]2
) = |V | − 1, ar(H ) = |V |1/2 in Theorem 1
the authors obtained in [4] a much better upper bound for q(H ), namely q(H ) ≤ 12 (∆
(
[H ]2
)+
1+
√
∆
(
[H ]2
)+1)+ 1. How can the upper bound for q(H ) be improved under the sole condi-
tion ar(H ) =
√
∆
(
[H ]2
)+1?
6.4.
Theorem 2 deals with k-uniform and d-regular linear hypergraphs when d = k +1 and provides
the required bound (1). For general k,d satisfying k ≤ d − 2, the line-graph L(H ) is k(d − 1)-
regular and Brooks’ theorem provides the (non sufficiently sharp) inequality q(H ) = χ(L(H )) ≤
∆
(
L(H )
)= k(d −1) while ∆([H ]2)= d(k −1).
6.5.
More generally let H be a k-uniform linear hypergraph with k ≥ 2 and set δ2 :=
√
∆
(
[H ]2
)
.
Then (1) holds if at least one of the following conditions is satisfied:
(1) k = 2: Vizing’s theorem (see [3, Theorem 7.2.2, p. 249]);
(2) k ≥
√
∆
(
[H ]2
)+1: Theorem 1;
(3) ∆
(
[H ]2
)= k2 which implies ∆(H ) ≤√∆([H ]2)+1: Corollary 3;
(4) 3 ≤ k < δ2 and 1δ2−k ≤
1
∆(H )−δ2−1 +
1
δ2
: see Section 6.2.
In order to solve Berge–Füredi’s conjecture for all linear uniform hypergraphs, it thus remains to
consider the case where
3 ≤ k < δ2 <∆(H )−1 and 1
δ2 −k
> 1
∆(H )−δ2 −1
+ 1
δ2
.
We may likewise ask the following question: is Berge–Füredi’s conjecture in all its generality
implied by its validity for linear uniform hypergraphs?
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
328 Alain Bretto, Alain Faisant and François Hennecart
References
[1] C. Berge, “On the chromatic index of a linear hypergraph and the Chvátal Conjecture”, Ann.
N.Y. Acad. Sci. 555 (1989), pp. 40–44.
[2] C. Berge, “Motivations and history of some of my conjectures”, Discrete Math. 165-166
(1997), pp. 61–70.
[3] A. Bretto, A. Faisant and F. Hennecart, Elements of Graph Theory: From Basic Concepts to
Modern Developments, EMS Textbooks in Mathematics, European Mathematical Society,
2022, pp. xv+486.
[4] A. Bretto, A. Faisant and F. Hennecart, “The Berge–Füredi Conjecture on the chromatic index
of hypergraphs with large hyperedges”, 2024. Online at https://arxiv.org/abs/2403.06850.
[5] T. Dvořák, “Chromatic index of hypergraphs and Shannon’s Theorem”, Eur. J. Comb. 21
(2000), pp. 585–591.
[6] Z. Füredi, “The chromatic index of simple hypergraphs”, Graphs Comb. 2 (1986), no. 1,
pp. 89–92.
[7] Z. Wang, “The Erdős–Faber–Lovász Conjecture for gap-restricted hypergraphs”, Engineering
16 (2024), pp. 47–59.
[8] G.-H. Zhang and B. Skinner, “Chromatic index of simple hypergraphs”, Discrete Math. 343
(2020), no. 12, article no. 112087 (7 pages).


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