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INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY
Class. Quantum Grav. 20 (2003) 4775–4784 PII: S0264-9381(03)63063-7
Determining the shape of the universe using discrete
sources
GIGomero
Instituto de F´
ısica Te´
orica, Universidade Estadual Paulista, Rua Pamplona, 145 S˜
ao Paulo,
SP 01405-900, Brazil
E-mail: german@ift.unesp.br
Received 6 May 2003
Published 3 October 2003
Online at stacks.iop.org/CQG/20/4775
Abstract
Suppose we have identified three clusters of galaxies as being topological copies
of the same object. How does this information constrain the possible models for
the shape of our universe? It is shown here that, if our universe has flat spatial
sections, these multiple images can be accommodated within any of the six
classes of compact orientable three-dimensional flat space forms. Moreover,
the discovery of two more triples of multiple images in the neighbourhood of
the first one would allow the determination of the topology of the universe, and
in most cases the determination of its size.
PAC S numbers: 04.20.Gz, 98.65.Cw, 98.80.Es, 98.80.Jk
1. Introduction
The last two decades have seen a continuously increasing interest in the study of cosmological
models with multiply connected spatial sections (see [1] and references therein). Since
observational cosmology is becoming an increasingly high precision science, it would be of
great interest to develop methods of systematically constructing specific candidates for the
shape of our universe in order to analyse whether these models are consistent with observational
data.
Since one of the simplest predictions of cosmological models with multiply connected
spatial sections is the existence of multiple images of discrete cosmic objects, such as clusters of
galaxies1,the following question immediately arises: suppose we have identified three clusters
of galaxies as being different topological copies of the same object, how does this information
constrain the possible models for the shape of our universe? The initial motivation for this
work was the suggestion of Roukema and Edge that the x-ray clusters RXJ 1347.5−1145 and
CL 09104+4109 may be topological images of the Coma cluster [4]. Even if these particular
1Provided that the scale of compactification is small enough (see [2, 3]).
0264-9381/03/224775+10$30.00 © 2003 IOP Publishing Ltd Printed in the UK 4775
4776 GIGomero
clusters do not turn out to be topological copies of the same object, the suggestion of Roukema
and Edge raises an interesting challenge: what if one day a clever astrophysicist discovers
three topological copies of the same object?
It is shown here that these (would be) multiple images could be accommodated within
any of the six classes of compact orientable three-dimensional flat space forms. Moreover,
and this is the main result of this paper, the discovery of two more triples of multiple images in
the neighbourhood of the first one would be enough to determine the topology of the universe,
and in most cases even its size. Thus, two interesting problems arise now: (i) does our
present knowledge of the physics of clusters of galaxies (or alternatively, of quasars) allow
the identification of a triple of multiple images if they actually exist? and (ii) given that such
an identification has been achieved, how easily can other triples of topological copies near the
first one be identified? The present paper does not deal with these two problems; however, it
should be remembered that a recent method proposed by Bernui and me in [5] (see also [6])
could be used to test, in a purely geometrical way, the hypothesis that any two given clusters
of galaxies are topological copies.
The model building procedure is explained in the following section, while section 3
presents some numerical examples illustrating specific candidates for the shape of our universe,
under the presumed validity of the Roukema–Edge hypothesis. Section 4discusses the main
result of this paper: how the topology of space could be determined with the observation of
just two more triples of images; and how, in most cases, one could even determine the size of
our universe. Finally, section 5consists of discussions on the results presented in this paper
and suggestions for further research.
2. Model building
Suppose that three topological copies of the same cluster of galaxies have been identified. Let
C0be the nearest copy to us, C1and C2the other two copies, d1and d2the distances from C0
to C1and C2respectively, and θthe angle between the geodesic segments C0C1and C0C2.
Roukema and Edge [4] have suggested an example of this configuration, the Coma cluster
being C0and the clusters RXJ 1347.5–1145 and CL 09104+4109 being C1and C2(or vice
versa). The distances between these clusters and Coma are 970 and 960h−1Mpc, respectively
(for 0=1and =0), and the angle between them, with the Coma cluster at the vertex, is
≈88◦.Under the assumption that this multiplicity of images was due to two translations of
equal length and in orthogonal directions, they constructed FL cosmological models whose
compact flat spatial sections of constant time were (i) 3-torii, (ii) manifolds of class G2or
(iii) manifolds of class G4,all of them with square cross sections, and scale along the third
direction larger than the depth of the catalogue of x-ray clusters used in the analysis.
Let us consider the possibility that at least one of the clusters Ciis an image of C0by
ascrew motion, and do not assume that the distances from C0to C1and C2are equal, or
that they form a right angle (with C0at the vertex). It is shown in this section that one can
accommodate this generic configuration of clusters within any of the six classes of compact
orientable three-dimensional flat space forms, thus providing a plethora of models for the
shape of our universe consistent with the (would be) observational fact that these clusters are
in fact the same cluster. Moreover, one could also consider the possibility that one of the
clusters Ciis an image of C0by a glide reflection, thus giving rise to non-orientable manifolds
as models for the shape of space. However, these cases will not be considered here since
they do not give qualitatively different results and the corresponding calculations can be done
whenever needed.
Determining the shape of the universe using discrete sources 4777
Tab le 1 . Diffeomorphism classes of compact orientable three-dimensional Euclidean space forms.
The first row contains Wolf’s notation for each class, and the second gives the generators of the
corresponding covering groups.
Class G1G2G3G4G5G6
(A1,a)
Generators a, b, c (A1,a),b, c (B, a),b, c (C, a),b, c (D, a),b, c (A2,b+c)
(A2,b−c)
The diffeomorphic and isometric classifications of three-dimensional Euclidean space
forms given by Wolf in [7], were described in detail by Gomero and Rebouc¸as in [3]. The
generators of the six diffeomorphic compact orientable classes are given in table 1,where an
isometry in Euclidean 3-space is denoted by (A, a),where ais a vector and Ais an orthogonal
transformation, and the action is given by
(A, a) :p→ Ap +a, (1)
for any point p.The orientation preserving orthogonal transformations that appear in the
classification of the Euclidean space forms take the matrix forms
A1=
10 0
0−10
00−1
,A
2=
−10 0
010
00−1
,A
3=
−100
0−10
001
,
(2)
B=
10 0
00−1
01−1
,C=
10 0
00−1
01 0
and D=
10 0
00−1
01 1
,
in the basis formed by the set {a, b, c}of linearly independent vectors that appear in table 1.
We will fit the set of multiple images {C0,C
1,C
2}within manifolds of classes G2–G6,since
the class G1(the 3-torus) is trivial.
Let us first deal with the classes G2–G5.The generators for the corresponding covering
groups are α=(A, a), β =(I, b) and γ=(I, c),with A=A1,B,C and Dfor the classes
G2,G3,G4and G5respectively, and Iis the identity transformation. For these classes we
will consider the following non-trivial configuration: denoting the position of C0by p, C1is
located at α(p) and C2at β(p).The configuration in which C2is located at γ(p)is equivalent
to the former, while the configuration in which C1and C2are images of C0by pure translations
(strictly possible only in G2,and a convenient approximation in G4if θ≈90◦,and the distances
of C1and C2from C0are almost equal, as is the case in the Roukema–Edge hypothesis) is
equivalent to that of a torus.
Forspace forms of the classes G2–G5,thefollowing facts are easily derivable from the
generators of their corresponding covering groups (see [3] for details):
1. The vector ais orthogonal to both band c.
2. The angle between band cis a free parameter for the class G2,while its value is fixed at
120◦,90◦and 60◦for the classes G3,G4and G5,respectively.
3. Denoting by |a|the length of the vector a,and similarly for any other vector, one has that
|b|=|c|for the classes G3–G5,while both lengths are independent free parameters in the
class G2.Moreover, in all classes G2–G5,|a|is an independent free parameter.
4. Denoting the canonical unitary basis vectors in Euclidean space by {ˆ
ı, ˆ
, ˆ
k},one can
always write a=|a|ˆ
ı,b =|b|ˆ
and c=|c|cos ϕˆ
+|c|sin ϕˆ
k,forthebasis {a, b, c},
where ϕis the angle between band c,asestablished in item 2.
4778 GIGomero
Tab le 2 . The second column gives the position of C1for each class of the manifolds considered
in the first column. The third column gives the distance between C0and C1and the last one the
cosine of the angle between the segments C0C1and C0C2,cos(α, β).
Class α(p) δα(p) δα(p) cos(α, β)
G2(x +|a|,−y,−z) |a|2+4(y2+z2)−2y
G3(x +|a|,−1
2y−√3
2z, √3
2y−1
2z) |a|2+3(y 2+z2)−√3
2(√3y+z)
G4(x +|a|,−z, y) |a|2+2(y 2+z2)−(y +z)
G5(x +|a|,1
2y−√3
2z, √3
2y+1
2z) |a|2+(y2+z2)−1
2(y +√3z)
Tab le 3 . Partial solutions for the positions of C0for flat manifolds of the classes G2–G5.
Class yz
G2−1
2d1cos θ±1
2√d2
1sin2θ−|a|2
G3±√3
6√d2
1sin2θ−|a|2−1
2d1cos θ∓1
2√d2
1sin2θ−|a|2−√3
6d1cos θ
G4±1
2√d2
1sin2θ−|a|2−1
2d1cos θ∓1
2√d2
1sin2θ−|a|2−1
2d1cos θ
G5±√3
2√d2
1sin2θ−|a|2−1
2d1cos θ∓1
2√d2
1sin2θ−|a|2−√3
2d1cos θ
Writing p=(x, y ,z) for the components of the position of C0in the basis {ˆ
ı, ˆ
, ˆ
k},2
one can easily work out the expressions for the components of the position of C1,α(p),the
distance function δα(p) and the cosine of the angle between C0C1and C0C2,cos(α, β ).The
resulting expressions are shown in table 2.
Forthe configuration we are dealing with, one trivially has d2=δβ(p) =|b|,since βis a
pure translation. More interestingly, from δα(p) =d1and cos(α, β) =cos θ,one can partially
solve the equations for the components of the position of C0.The resulting expressions are
shown in table 3.Observe that for each class we have two solutions in terms of the free
parameter |a|.Fortheclasses G3–G5the two solutions are those for which d1cos θis given in
the fourth column in table 2.
Tworemarks are in order here. First, it is convenient to write the components of
the position of C0in terms of the parameter |a|,because this parameter can be easily
determined once two more triples of multiple images, say {D0,D
1,D
2}and {E0,E
1,E
2},
in the neighbourhood of {C0,C
1,C
2}have been identified, as shown in section 4.3Once this
has been done, the positions of C0,D
0and E0can be used to predict multiple images of them
due to the inverse isometry α−1,thus yielding a definitive observational test for the hypothesis
of the multiply connectedness of our universe. Secondly, note that the x-coordinate is not
constrained by this configuration of topological images. This freedom of the x-coordinate
is a consequence of homogeneity of manifolds of classes G2–G5along the x-axis. This
partial homogeneity is due to the fact that the orthogonal transformations involved in the
corresponding covering groups have the x-axis as their axis of rotation.
We now fit the multiple images {C0,C
1,C
2}within manifolds of class G6.The generators
for the covering group of a manifold of this class are α=(A1,a),β=(A2,b +c) and
µ=(A2,b −c).Thevectors {a, b, c}are mutually orthogonal but their lengths are free
2Note that the origin of a coordinate system is implicitly determined by the axes of rotation of the orthogonal
transformations in (2), and can be taken as the centre of the fundamental polyhedron for the corresponding manifold.
Moreover, this origin does not necessarily coincide with the position of our galaxy.
3Actually, much more can be done than that. If the topology of the universe turns out to be of any of the classes
G2–G6,thetriples {D0,D
1,D
2}and {E0,E
1,E
2}would be enough to decide which topology our universe has, and
except in the case of G2and a configuration in G6,itwould be possible to specify completely the parameters of the
manifold that models the spatial sections of the spacetime.
Determining the shape of the universe using discrete sources 4779
parameters. For manifolds of class G6we have two possible configurations, both of them with
C0located at p,
1. C1located at α(p) and C2at β(p),and
2. C1located at β(p) and C2at µ(p).
The case in which C1is at α(p) and C2at µ(p) is equivalent to the first configuration.
The expressions for the distances δα(p),δβ(p) and δµ(p),and angles cos(α, β) and
cos(β, µ) are
δα(p) =|a|2+4(y 2+z2)
δβ(p) =|b|2+4x2+(2z−|c|)2(3)
δµ(p) =|b|2+4x2+(2z+|c|)2
and
cos(α, β) =4z2−2(|a|x+|b|y+|c|z)
δα(p)δβ(p) (4)
cos(β, µ) =4x2+4z2+|b|2−|c|2
δβ(p)δµ(p) .
Forthe first configuration one has δα(p) =d1,δ
β(p) =d2and cos(α, β ) =cos θ,thus
yielding the equations
y2+z2=1
4d2
1−|a|2
4x2+(2z−|c|)2=d2
2−|b|2(5)
4z2−2(|a|x+|b|y+|c|z) =d1d2cos θ.
This is a system of three quadratic equations with six unknowns, the three coordinates (x, y ,z)
of the point pand the three coordinates (|a|,|b|,|c|)in the parameter space of the G6manifold
(see [3]). An algebraic solution of these equations for (x , y,z) in terms of (|a|,|b|,|c|),or
vice versa, would in general yield higher degree (decoupled) equations for each variable, and
thus is not so illuminating. Particular solutions can be obtained by (i) assuming specific values
for the parameters (|a|,|b|,|c|),and then calculating numerically the position of C0,or(ii)
assuming a particular position for C0,and then calculating the parameters (|a|,|b|,|c|).This
second method does not follow the strategy of determining the parameters of the manifold
using two more triples of clusters of galaxies (see section 4), thus it will not be pursued here.
The following section presents examples of application of the first method.
Finally, let us examine the second configuration which is simpler. One has δβ(p) =
d1,δ
µ(p) =d2and cos(β, µ) =cos θ,thus yielding the equations
4x2+(2z−|c|)2=d2
1−|b|2
4x2+(2z+|c|)2=d2
2−|b|2(6)
4x2+4z2+|b|2−|c|2=d1d2cos θ.
These equations can be partially solved giving
z=1
8|c|d2
2−d2
1
|c|=1
2d2
1+d2
2−2d1d2cos θ(7)
x2+z2=1
16 d2
1+d2
2+2d1d2cos θ−1
4|b|2.
4780 GIGomero
Tab le 4 . Examples of models within classes G3–G5.
Class |a|(h−1Mpc)y(h
−1Mpc)z(h
−1Mpc)
G31156 −21.4 −39.2 −32.9 −2.0
1142 22.2 82.8 108.5 73.5
G41156 −14.9 −45.8 −45.8 −14.9
1142 60.7 −121.3 −121.3 60.7
G51156 −3.5 −57.1 −67.9 −37.0
1142 127.3 −187.9 143.5 38.5
In this case the y-coordinate is not constrained by the configuration of topological images,
since the only orthogonal transformation involved in the calculations has the y-axis as its axis
of rotation.
3. Numerical examples
Let us now apply the results obtained in the previous section to the proposed multiple images of
Roukema and Edge [4], in a FL universe whose matter components are pressureless dust and a
cosmological constant. The models presented below are small universes with compactification
scales much smaller than the horizon radius, so they may seem to be in conflict with constraints
on the topology coming from observations of the CMBR. However, it must be recalled that
all current constraints for flat universes hold exclusively for models with (i) toroidal spatial
sections [8, 9], or (ii) any flat (compact and orientable) spatial section, but in cosmological
models without a dark energy component, and moreover, with the observer located on the
axis of rotation of a screw motion of the corresponding covering group [10]. As has been
shown by Inoue [9], the addition of a cosmological constant term makes the constraints less
stringent, whereas the effect of considering the observer out of an axis of rotation is totally
unknown. Since the models presented below consider both a cosmological constant term and
the observer off an axis of rotation, they cannot be considered as being ruled out by current
observational data.
The models constructed here consider C1as being the cluster RXJ 1347.5–1145 and
C2the cluster CL 09104+4109. Then for the values m0=0.3and 0=0.7, one has
d1=1158h−1Mpc,d
2=1142h−1Mpc and θ=87◦.Other examples can be built by simply
reversing these identifications, i.e. by considering C1as being the cluster CL 09104+4109 and
C2the cluster RXJ 1347.5–1145. As before, let us first examine the classes G2–G5.Onehas
|b|=1142h−1Mpc, and because of the expression √d2
1sin2θ−|a|2in table 3one also has
the constraint
|a|1156.4h−1Mpc.(8)
The models within class G2are special because they have a fixed value of y,sayy=
−30.3h−1Mpc; however the z-coordinate depends on the parameter |a|,and is remarkably
sensitive to this value as can be seen from the following two examples:
(1) First, consider the case when |a|is slightly lower than the maximum value allowed by
(8), say |a|=1156h−1Mpc. Then z=±15.5h−1Mpc.
(2) Second, consider the symmetric case when |a|=|b|=1142h−1Mpc. In this case one
has z=±91.0h−1Mpc.
Note that the classes G3–G5do not yield models with a fixed value of y;instead, both y
and zdepend on the parameter |a|.Intable 4we show the values of yand zcalculated from
Determining the shape of the universe using discrete sources 4781
table 3for |a|=1156 and 1142h−1Mpc. In this table the first column for each coordinate
corresponds to the first solution of table 3,and the second column to the second solution.
Now we deal with models within class G6.For the first configuration one obtains from
equations (5)
(2x+|a|)2+(2y+|b|)2=d2
1+d2
2−2d1d2cos θ−|c|2,
which implies that
|c|2d2
1+d2
2−2d1d2cos θ.
Furthermore, from the first and third equations in (5)one also has
|a|d1and |b|d2.
Afamily of simple examples is obtained by taking |a|=d1.Infact, in this case one has
y=z=0,x=−1
2d2cos θ, |b|2+|c|2=d2
2sin2θ.
Thus, taking |b|=|c|,one model of a universe with spatial sections of class G6that fits the
first configuration with the Roukema–Edge hypothesis is
|a|=1158h−1Mpc and |b|=|c|=1140.4h−1Mpc,
with Coma located at
x=−29.9h−1Mpc and y=z=0.
On the other hand, for the second configuration one has
|c|=791.6h−1Mpc,z=−5.8h−1Mpc,|b|834.1h−1Mpc,(9)
the last inequality being obtained from the last equation in (7). It is illustrative to give two
specific examples as was done with the G2models.
1. First, consider the case when |b|is slightly lower than its maximum value allowed by (9),
say |b|=834h−1Mpc, then one has x=±7h−1Mpc.
2. Second, consider the symmetric case when |b|=|c|=791.6h−1Mpc. In this case
x=±131.5h−1Mpc.
4. The case of three triples of images
In this section it is shown that the discovery of two additional triples of clusters of galaxies close
to {C0,C
1,C
2}would allow the determination of the topology of the universe, and in most
cases the determination of its size. Let us denote by {D0,D
1,D
2}and {E0,E
1,E
2}these two
additional triples of topological images. Mathematically, to characterize the closeness relation
between two triples {Ci}and {Di},itsuffices to say that the lengths of the geodesic segments
CiDi(i =0,1,2)be the same and smaller than the injectivity radius. Observationally, it is
enough that C0and D0are two nearby clusters of galaxies, while the distances between Ci
and Di(i =1,2)are equal (within the observational error bounds) to the distance between
C0and D0.4
By parallel transporting the triangle C1D1E1along the geodesic segment C0C1,one
obtains two triangles with a common vertex, namely the triangle of nearby clusters C0D0E0,
and that of transported clusters of C1D1E1.Itisjust a matter of elementary analytic geometry
to determine the unique rotation that takes one triangle to the other. Note however that one
can also easily find the unique reflection that takes one triangle to the other, if it exists. If the
4Strictly speaking, this closeness relation is not a necessary condition, but observationally it would be simpler to
look for other triples of images in the neighbourhood of the first one.
4782 GIGomero
angle of rotation is different from π,2π/3, π/2orπ/3, then the isometry that takes C0to
C1is not a screw motion, but a reflection, and the universe would be spatially non-orientable.
In contrast, if the angle of rotation is π,2π/3,π/2orπ/3, then one can think this is not
by coincidence, so the universe would be spatially orientable. In such a case, if the angle of
rotation is different from π,ituniquely determines to which class the topology of the universe
belongs, namely G3,G4or G5,respectively5.
Let us restrict our analysis to the orientable case in order to be specific. The determination
of the rotation taking C0D0E0to the parallel transportation of C1D1E1also provides the
direction of the axis of rotation of the screw motion linking C0with C1.Iftheuniverse has
atopology of class G3,G4or G5,thetranslation vector is parallel to this axis, so elementary
geometry can be used to determine the parameter |a|and the position of the axis. Moreover,
the isometry linking C0with C2has to be a translation, and a parallel transport of the triangle
C2D2E2to C0D0E0would confirm it. A remarkable fact is that, if the topology of the
universe has been identified to be of class G3,G4or G5,thevector cis automatically fixed, and
observational searches can be performed to find the topological images of C0,D
0and E0due
to the isometries γand γ−1for validation of the model.
Let us now consider the case when the angle of rotation taking C0D0E0to the parallel
transport of C1D1E1is π.Inthis case the topology of the universe has to be of class G2
or G6.One can decide between these two possibilities by parallel transporting C2D2E2to
C0D0E0.Ifthe angle of rotation between these triangles is null, then the isometry linking
C0to C2is a translation, and the universe has topology of class G2.Onthe other hand, if
the angle of rotation is π,the universe has topology of class G6.Intheformer case one can
proceed as before and determine the length |a|and the position of the axis of the screw motion.
However, since for the class G2,thevector cis a free parameter, its modulus and direction
remain undetermined.
If the topology of the universe turns out to be of class G6,the multiple images can be fitted
within the two inequivalent configurations described in section 2.One can decide between
both configurations by just looking at the directions of the axes of rotation, for if they are
orthogonal the first configuration would be the correct one, while if they are parallel the correct
one is the second. Using elementary geometry one can completely determine the three axes
of rotation and translation (thus determining the global shape of space) if the multiple images
fit with the first configuration. On the other hand, with the second configuration one can
determine the vectors band c,and thus the direction of a,butitis impossible to determine the
length |a|,ascould have been anticipated from equation (6). However, in the latter case, one
can design effective search procedures to look for multiple images due to the isometries αand
α−1,thus providing at least robust constraints for the parameter |a|(see [5]).
5. Discussion and further remarks
The work presented in this paper originated with the following problem in the context of
cosmological models with flat spatial sections: suppose we have identified three clusters of
galaxies as being different topological images of the same object. How do these multiple
images constrain the possible models for the shape of our universe? A natural extension
of this work would be the study of this problem in the context of universes with non-flat
spatial sections, specifically those with positive curvature (since multiply connected spaces of
negative curvature are very unlikely to have a detectable topology [2]).
5Note however that if there exists a reflection taking one triangle to the other, in order to settle definitely the
orientability of space, it would be necessary to identify a fourth triple of multiple images.
Determining the shape of the universe using discrete sources 4783
It has been shown here that one can accommodate any of the six classes of compact
orientable three-dimensional flat space forms to fit with any configuration of three topological
images of a cosmic object. It can be seen from the construction of the models that one could
also easily fit any of the non-orientable flat manifolds. Moreover, the main result in this paper
is that the identification of two more triples of multiple images of clusters of galaxies, in the
neighbourhood of the first one, is enough to completely determine the topology of space, as
well as its size in most cases.
Even if the primary goal of this paper is not to construct specific candidates for the shape
of our universe, but to present a systematic procedure for building such models, it turns out
that the illustrative examples constructed by using the Roukema–Edge hypothesis are not in
contradiction with current observational data.
In view of these results, it seems of primary importance to state and test hypotheses such
as that of Roukema and Edge, i.e. that the clusters RXJ 1347.5–1145 and CL 09104+4109 are
topological images of the Coma cluster, since the identification of a very small quantity of
multiple images is, as has been shown here, enough to determine (or almost determine) the
global shape of the universe. The problem of testing this kind of hypothesis can be solved by
the local noise correlation (LNC) method proposed in [5]. The problem of generating such
kinds of hypotheses seems to be much harder, although currently efforts are being made to
find multiple images of our galaxy [11], clusters of galaxies [12] and radio-loud AGNs [13].
In conclusion, let us stress that only those models have been considered here in
which the topological images are related by the generators of the covering groups of the
corresponding manifolds. This need not be the case, for one could also consider other
isometries (compositions of the generators) as being responsible for the multiple images.
Thus the list of possible models presented here is not exhaustive.
Acknowledgments
Iwould like to thank CLAF/CNPq and FAPESP (contract 02/12328-6) for the grants under
which this work was carried out, and the CBPF for kind hospitality. I also thank Bruno Mota,
Marcelo Rebouc¸as and Armando Bernui for critical reading of previous versions of this work
and for their valuable suggestions.
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