algebra, and we let $S_{G,ext}(M)$ denote its space of ultrafilters (on which
$G(M)$ acts).

######  Lemma 6.3.

$S_{G,ext}(M)$ is isomorphic, as a $G(M)$-flow, to $S_{G,M}(N)$.

If all types over $M$ are definable, then of course externally definable
subsets of $G(M)$ coincide with definable subsets, and everything fits
together.

######  Definition 6.4.

(i) A map $f$ from $G(M)$ to a compact space $X$ is said to be definable if it
factors through a continuous function from $S_{G}(M)$ to $X$. Likewise $f$ is
to be externally definable if it factors through a continuous function from
$S_{G,ext}(M)$ to $X$.
(ii) An action of $G(M)$ on a compact set $X$ (by homeomorphisms) is said to
be definable if for each $x\in X$, the map taking $g\in G(M)$ to $gx\in X$ is
a definable map from $G$ to $X$. Likewise for the notion of the action being
externally definable.

Let me remark that the notion of a “definable action” of a definable group on
a compact space is supposed to be some kind of analogue of a (jointly)
continuous action of a topological group on a compact space. But maybe we did
not yet find the correct analogue. Anyway with these notions we obtain some
kind of canonicity or universality of the action of $G(M)$ on the various type
spaces:

######  Proposition 6.5.

(i) The action of $G(M)$ on $S_{G,ext}(M)$ is externally definable, and
moreover with $x_{0}$ the identity of $G(M)$, $(S_{G,ext}(M),x_{0})$ is the
universal externally definable $G(M)$-ambit.
(ii) Likewise, if all types over $M$ are definable, $(S_{G}(M),x_{0})$ is the
universal definable $G(M)$-ambit.

Hence, for various reasons, if we consider the group $G(M)$, and $X=S_{G}(M)$,
then the Ellis group $u*{\cal M}$ and quotient $(u*{\cal M})/H$ attached to
$X$ (i.e. constructed from the semigroup $E(X)=S_{G,ext}(M)$) can be
considered fairly canonical, and there is no harm calling them the definable
Ellis group and definable generalized Bohr compactification (although using
“externally definable” in place of definable, would also make sense).

###  6.1 Newelski conjecture

In the paper [49] already mentioned above, Newelski asked some questions at
the end, including about what he calls weak generic types (which I have not
defined) and clarifying relationships between groups such as $G/G^{00}$ and
Ellis groups. Such as, are they equal assuming a bounded number of weak
generics? This led to some people such as myself dubbing as “Newelski’s
conjecture” that in certain tame contexts, such as $NIP$ or $o$-minimal, given
a group $G$ definable over a structure $M$, then $G/G_{M}^{00}$ equals the
Ellis group $u^{*}{\cal M}$ described above. Newelski noted already that there
is a natural surjective homomorphism from this Ellis group to $G/G^{00}_{M}$.
So the question was whether this must be an isomorphism.

The first result on the question was negative, in the sense of giving a
counterexample for groups in $o$-minimal structures without any further
hypotheses. See [13]. See also [51] for a generalization to
$SL(2,\mathbb{Q}_{p})$.

######  Proposition 6.6.

Let $M$ be the structure $(\mathbb{R},+,\times)$ and $G=SL_{2}$, so
$G(M)=SL_{2}(\mathbb{R})$. In this case all types over $M$ are definable. Then
the Ellis group is nontrivial (namely the $2$ element group), whereas
$G/G^{00}$ is trivial.

So the rather loose “Newelski conjecture” was tightened to ask whether
$G/G^{00}$ equals the Ellis group, when $T$ is $NIP$ and $G$ is definably
amenable. (Actually the assumptions are related to Newelski’s condition of a
bounded number of weak generic types). Anyway this was proved in [8].

######  Proposition 6.7.

Assume $T$ is $NIP$ and $G$ is definably amenable. Then the natural
homomorphism from the Ellis group to $G/G^{00}$ is an isomorphism.

The paper [8] also includes a classification of the invariant Keisler measures
on definably amenable groups in $NIP$ theories.

A possible refinement of the Newelski conjecture, which takes account of the
counterexample, is that when $T$ is $NIP$ and $G$ is a definable group then
the Ellis group $u*{\cal M}$ of $G(M)$ coincides with the definable
generalized Bohr compactification $(u*{\cal M})/H$ which just means that the
$\tau$-topology on $u*{\cal M}$ is already Hausdorff. This is true in the
definably amenable case, and the isomorphism of the Ellis group with
$G/G^{00}_{M}$ is actually an isomorphism of topological groups where the
Ellis group is equipped with its $\tau$-topology (although it is not part of
the statement in [8]), which follows from the discussion in the next
subsection.

###  6.2 $G/G_{M}^{000}$

As mentioned earlier the group $G/G_{M}^{000}$ belongs to model theory,
although it is not a compact, Hausdorff group, and in fact its status as a
mathematical object has always been a bit problematic. For $g\in
G=G({\bar{M}})$ the class of $g$ modulo $G_{M}^{000}$ depends only on
$tp(g/M)$, so $G/G_{M}^{000}$ is a “quotient” of $S_{G}(M)$ by what turns out
to be an $F_{\sigma}$ equivalence relation (on the compact space $S_{G}(M)$).
When $T$ and $M$ are countable, we are in the environment of Borel equivalence
relations on Polish spaces, and it turned out to be profitable to study the
complexity of the relevant equivalence relation. The general conjecture was
“smooth” implies that $G_{M}^{000}=G_{M}^{00}$ and so the quotient is compact,
Hausdorff. (The Lascar group analogue was mentioned earlier.)

However, already looking at examples, one had the feeling that more structure
could be given to $G/G_{M}^{000}$, in fact a structure belonging to classical
noncommutative geometry. This turns out to be true using the topological
dynamics machinery. See [36].

######  Proposition 6.8.

The natural map $f$ from $S_{G,ext}(M)$ to $G/G_{M}^{000}$ induces a
surjective homomorphism $\bar{f}$ from $(u*{\cal M})/H$ (the definable
generalized Bohr compactification) to $G/G_{M}^{000}$.

We also have a surjective homomorphism $\pi$ say from $G/G_{M}^{000}$ to the
definable Bohr compactification $G/G_{M}^{00}$, and let $K$ be the kernel of
$\pi$ which identifies with $G_{M}^{00}/G_{M}^{000}$.

######  Proposition 6.9.

${\bar{f}}^{-1}(K)$ is a closed subgroup of $(u*{\cal M}/H)$, and the kernel
of the surjective homomorphism from ${\bar{f}}^{-1}(K)$ to $K$ given by
${\bar{f}}$ is a dense subgroup. Hence $K$ has canonically the structure of
the quotient of a compact Hausdorff group by a dense subgroup.

###  6.3 Amenability and automorphism groups

The study of automorphism groups (as topological groups) from the point of
view of topological dynamics has been very profitable, and in [32] the
connection was made with structural Ramsey theory (for automorphism groups of
countable structures). When the theory $Th(M)$ of the countable structure $M$
is $\omega$-categorical, then $Aut(M)$, as a topological group, is an
invariant of $Th(M)$ (in fact $Th(M)$ can be recovered from $Aut(M)$), so the
topological dynamics of $Aut(M)$ is something also model-theoretic.

In [37] we proved that if $G$ is a definable group which is definably
amenable, then $G_{M}^{000}=G_{M}^{00}$, using elaborations of the stabilizer
theorem for approximate subgroups (and [45]). This was adapted in the same
paper to the automorphism group context to show that if $T$ is
$\omega$-categorical and the automorphism group of the (unique) countable
model of $T$ is amenable, then $T$ is $G$-compact, i.e.
$Gal_{L}(T)=Gal_{KP}(T)$. In [23] we introduced the notion of an amenable
first order theory $T$ (which is considerably weaker than $Aut(M)$ being
amenable as a topological group when $T$ is $\omega$-categorical) and proved:

######  Proposition 6.10.

If $T$ is amenable as a first order theory, then $T$ is $G$-compact.

##  7 The free group

An important question of Tarski was whether all free groups $F_{n}$ on $n$
generators for $n\geq 2$ are elementarily equivalent. This was solved by two
independent groups of researchers, Kharlampovich-Myasnikov and Sela, in a
series of papers. (See [33], [67] for example, also for more references.) So
we have a complete theory $T_{fg}$ in the language of groups (with symbols for
the group operation, inversion, and the identity element) which is the common
theory of the $F_{n}$ ($n\geq 2$).

Subsequently Sela [68] proved that $T_{fg}$ is stable. This result is accepted
by logicians and model theorists, although there has as yet been no
independent proof (by logicians). One would like an account based on the
combinatorics of words in the alphabet or language mentioned above. Possibly
this is too naive a set-up, compared with the algebraic-topological (geometric
group-theoretic) framework of Sela. Sela’s solutions, both to Tarski’s problem
and stability involved proving a relative quantifier elimination theorem down
to Boolean combinations of $\forall\exists$ formulas.

Nevertheless, we can choose to take the stability of $T_{fg}$ for granted, and
see where it leads us, in terms of problems and questions. $T_{fg}$ is a new
stable group (like a “new leaf” in the film with Walter Matthau and Elaine
May, or like a “new galaxy” as in no film that I know), with very different
properties from the already known stable groups. The subject “algebraic
geometry over the free group” is about the structure of solutions of equations
in free groups, but in fact maybe should be about the category $Def(T_{fg})$.

Sela’s result that the natural inclusions of the $F_{n}$ for $n\geq 2$ are
elementary, implies that the free groups on infinitely many generators are
models of $T_{fg}$. It is natural to ask whether (parts of) the proof of
stability of $T_{fg}$ can be simplified by looking directly at the models
$F_{\kappa}$ for $\kappa$ infinite. There are a couple of examples, noted by
Poizat in his early but fundamental paper on stable groups [64]. The set $I$
of free generators of $F_{\omega}$ is an indiscernible set in $F_{\omega}$,
and for any definable (in fact invariant over a finite set) subset $X$ of
$F_{\omega}$, $X$ intersects $I$ in a finite or cofinite set. It follows that:

######  Fact 7.1.

The free group is connected (no proper definable subgroup of finite index) and
moreover the type of some/any element of $I$ over $\emptyset$ is the unique
generic type of $T_{fg}$ over $\emptyset$, which we call $p_{0}$.

In the free group elements have unique square roots. If $a$ realizes $p_{0}$
then $a$ is not a square but $a$ is algebraic over $a^{2}$, from which Poizat
concludes:

######  Fact 7.2.

$T_{fg}$ is not superstable.

We did not define regular types in this article. Regular types are
generalizations of types of Morley rank $1$ or $U$-rank $1$. Nevertheless the
same argument shows that the $p_{0}$ is not regular. Some more technical
results about $p_{0}$ were proved in [61], and [74], including that $p_{0}$
has infinite weight, even witnessed in $F_{\omega}$: for $a$ realizing $p_{0}$
there is some infinite $f$-independent sequence $(b_{i}:i<\omega)$ such that
$a$ forks with $b_{i}$ for all $i$.

We still know very little about the category $T_{fg}$, although Sklinos and
collaborators have made some progress (but using heavily the set-up and
context of Sela, where not everything is accessible to model-theorists). Quite
possibly a firmer understanding of why $T_{fg}$ is stable would make a
difference. As a related case where our knowledge is basically complete, let
us consider the theory of $F_{1}$, otherwise known as $T=Th(\mathbb{Z},+)$.
Now $T$ is not totally transcendental (as there is an infinite descending
chain $2^{n}\mathbb{Z}$ of subgroups), but it is superstable, even of rank
$1$, and it is the archetypical example of a superstable, non $t.t$ theory,
but where both categories $Def(T)$ and $Mod(T)$ are well-understood. One of
the main things is that as an abelian group (with no additional structure),
$(\mathbb{Z}.+)$ is a $1$-based group, so the definable sets (in arbitrary
models) are Boolean combinations of cosets of $acl(\emptyset)$-definable
subgroups of $\mathbb{Z}^{n}$. These can be easily computed. And also any
model of $T$ is of the form $G\oplus\mathbb{Q}^{(\kappa)}$ where $G$ is an
elementary substructure of the profinite completion ${\hat{\mathbb{Z}}}$ of
$\mathbb{Z}$. In particular (assuming $CH$), the $\aleph_{1}$-saturated models
of $T$ can be described as ${\hat{\mathbb{Z}}}\oplus\mathbb{Q}^{\kappa}$ (for
$\kappa$ a cardinal $\geq\aleph_{1}$).

So this begs the question of whether there is a classification of the
$\aleph_{1}$-saturated models of $T_{fg}$. Now a nonstructure property for
$\aleph_{1}$-saturated models of a stable (possibe unsuperstable) theory $T$
is that $T$ has the so-called $DOP$ (dimensional order property of Shelah).
See [63] for both the definitions and the proof that $T_{fg}$ has the $DOP$.
So there is NO reasonable structure theorem for $\aleph_{1}$-saturated models
of $T_{fg}$. This may explain why it has so far been difficult to describe
“nonstandard” models of $T_{fg}$ (where nonstandard means not free), in
particular saturated models.

Let now $F$ denote an arbitrary model of $T_{fg}$, such as a “standard model”
($F_{n}$ for $n\geq 2$). Some obvious definable subgroups of $F$ are the
centralizers of elements $a$, $C_{F}(a)$. Some important observations are that
these definable subgroups, with the induced structure from $F$, are just
models of $Th(\mathbb{Z},+)$ (maybe with constants added). So these are
examples of $1$-based $U$-rank $1$ groups definable/interpretable in $T_{fg}$.

Here are some questions.

######  Question 7.3.

(i) What are the $U$-rank $1$ formulas/ types, in $T_{fg}$. Are they only the
centralizers described earlier?
(ii) What are the superstable formulas (definable sets) in $T_{fg}$?
(iii) What are the definable/interpretable fields in $T_{fg}$?
(iv) What are the definable/interpretable groups in $T_{fg}$?
(v) What are the regular types in $T_{fg}$?

For example in [5] it is proved that there are no definable infinite fields.

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equivalent to $\tau>(\gamma-1)\mathbb{I}_{\\{\gamma>1\\}}$ by choosing $q_{1}$
as close to its upper limit $\min\\{1,\gamma\\}$ as possible. Therefore,
$\alpha_{n}=C\epsilon_{n}n^{-\tau}<C\epsilon_{n}n^{-(\gamma-1)\mathbb{I}_{\\{\gamma>1\\}}}$,
which is equivalent to Assumption E1.

Proposition 5.1 implies that

$\bm{P}\left(\sum_{i=1}^{n}\widetilde{w}_{i}X_{i}>\overline{F}^{-1}(C\epsilon_{n})\right)\sim
C\epsilon_{n}\left(\sum_{i=1}^{\infty}\widetilde{w}_{i}^{\gamma}\right)$

holds as $\overline{F}^{-1}(C\epsilon_{n})\to\infty$, or equivalently, as
$n\to\infty$. In addition, from the definition of $\widetilde{w}_{i}$ in
equation (38),
$\sum_{i=1}^{n}\widetilde{w}_{i}^{\gamma}=n^{-\tau}=(C\epsilon_{n})^{-1}\alpha_{n}$,
which leads to

$\frac{C\epsilon_{n}\sum_{i=1}^{n}\widetilde{w}_{i}^{\gamma}}{\alpha_{n}}=1$

for any positive integer $n$. By taking the limit as $n\to\infty$ on both
sides, we have
$C\epsilon_{n}\sum_{i=1}^{\infty}\widetilde{w}_{i}^{\gamma}\sim\alpha_{n}$.
Hence $\bm{P}(S_{1,F,w}(\bm{U}_{n})>t_{\alpha,n})\sim\alpha_{n}.$ With
$a_{n,\gamma}=\left(\sum_{i=1}^{n}w_{i}^{\gamma}\right)^{-\frac{1}{\gamma}}$
as defined in (17), from the first line of (37), we can see that the event
$S_{1,F,w}(\bm{U}_{n})>t_{\alpha,n}$ is asymptotically equivalent to
$\overline{F}(a_{n,\gamma}S_{1,F,w}(\bm{U}_{n}))<\alpha_{n}$ for large enough
$n$. Therefore,

$\bm{P}\left(\overline{F}(a_{n,\gamma}S_{1,F,w}(\bm{U}_{n}))<\alpha_{n}\right)\sim\alpha_{n}$

as $n\to\infty$. This completes the proof of this theorem under Assumption B3
or B4 with a nonrandom weight Assumption A1 or A2 when $F$ satisfies a balance
condition with $p_{F}=1$ and $q_{F}=0$.

Now let us further assume that $X_{i}$ are nonnegative. Extending the above
result to UTAI, or Assumption B1, is straightforward based on Theorem 3.1 of
Chen and Yuen (2009) and by mimicking the argument in the proof of Theorem 2
of Bae and Ko (2017). Note that Theorem 3.1 of Chen and Yuen (2009) is written
for equal weights and possibly nonidentical distributions for $X_{i}$ in
$\mathcal{C}$. It is easy to see that if the distribution of $X_{i}$ is in
$\mathcal{C}$, for any constant $c>0$, the distribution of $cX_{i}$ is also in
$\mathcal{C}$. Therefore, their result also holds for Assumption A2.

Since Assumption B2 implies B1, the asymptotic tail-precise property of
$T_{1,F,w}(\bm{U}_{n})$ also holds for Assumption B2. ∎

###### Proof of Theorem 5.5.

Let $\alpha_{n}=C\epsilon_{n}n^{-\tau}$ for some $\epsilon_{n}\to 0$ as
$n\to\infty$ and positive real numbers $C$ and $\tau$ that do not depend on
$n$. The tail probability of the sum-combination statistic with the lower
threshold $t_{\alpha,n}$ that satisfies
$\overline{F}(t_{\alpha,n})=\alpha(\sum_{i=1}^{n}\bm{E}w_{i}^{\gamma})^{-1}$
can be written as

$\bm{P}(S_{1,F,w}(\bm{U}_{n})>t_{\alpha,n})=\bm{P}\left(\sum_{i=1}^{n}\frac{w_{i}}{n^{\frac{\tau}{\gamma}}(\sum_{j=1}^{n}\bm{E}w_{j}^{\gamma})^{\frac{1}{\gamma}}}X_{i}>\overline{F}^{-1}(C\epsilon_{n})\right).$

Let

$\widetilde{w}_{i}=\frac{w_{i}}{n^{\frac{\tau}{\gamma}}(\sum_{j=1}^{n}\bm{E}w_{j}^{\gamma})^{\frac{1}{\gamma}}}.$

Consider $0<\gamma<1$. By Lemma .5, $\sum_{i=1}^{n}\bm{E}w_{i}^{\gamma}\geq
1$, and therefore, $\bm{P}(0<\widetilde{w}_{i}<1)=1$. Lemma .2 implies that
$\bm{E}\widetilde{w}_{i}^{q_{1}}\vee\bm{E}\widetilde{w}_{i}^{q_{2}}=\bm{E}\widetilde{w}_{i}^{q_{1}}$
for any $0<q_{1}<q_{2}$. For any positive real value $q_{1}<\gamma<1$,

$\sum_{i=1}^{n}\bm{E}\widetilde{w}_{i}^{q_{1}}=\frac{\sum_{i=1}^{n}\bm{E}w_{i}^{q_{1}}}{n^{\frac{\tau
q_{1}}{\gamma}}(\sum_{j=1}^{n}\bm{E}w_{j}^{\gamma})^{\frac{q_{1}}{\gamma}}}\leq\frac{n^{1-q_{1}}}{n^{\frac{\tau
q_{1}}{\gamma}}}=n^{1-q_{1}-\frac{\tau q_{1}}{\gamma}}$

is uniformly bounded for all $n$ as long as
$\tau\geq\frac{\gamma}{q_{1}}-\gamma$.

Now consider $\gamma\geq 1$. By taking the exponent $q_{2}>\gamma\geq 1$,
expectation, and the exponent $1/q_{2}$ on both sides, and taking the sum over
$i=1,\ldots,n$,

$\sum_{i=1}^{n}\left(\bm{E}\widetilde{w}_{i}^{q_{2}}\right)^{\frac{1}{q_{2}}}=\frac{\sum_{i=1}^{n}\left(\bm{E}w_{i}^{q_{2}}\right)^{\frac{1}{q_{2}}}}{n^{\frac{\tau}{\gamma}}(\sum_{j=1}^{n}\bm{E}w_{j}^{\gamma})^{\frac{1}{\gamma}}}\leq\frac{n^{1-\frac{1}{q_{2}}}}{n^{\frac{\tau}{\gamma}}n^{\frac{1-\gamma}{\gamma}}}=n^{1-\frac{1}{q_{2}}-\frac{\tau}{\gamma}-\frac{1-\gamma}{\gamma}}=n^{2-\frac{1}{q_{2}}-\frac{\tau}{\gamma}-\frac{1}{\gamma}},$

where the inequality is due to Lemma .5. Therefore,
$\sum_{i=1}^{n}\left\\{\left(\bm{E}\widetilde{w}_{i}^{q_{1}}\right)^{\frac{1}{q_{1}}}\vee\left(\bm{E}\widetilde{w}_{i}^{q_{2}}\right)^{\frac{1}{q_{2}}}\right\\}\leq\sum_{i=1}^{n}\left(\bm{E}\widetilde{w}_{i}^{q_{2}}\right)^{\frac{1}{q_{2}}}$
is uniformly bounded for all $n$ if $\tau\geq 2\gamma-\frac{\gamma}{q_{2}}-1$
when $\gamma\geq 1$.

Note that when $\gamma=1$, the range of $2\gamma-\frac{\gamma}{q_{2}}-1$ is
roughly $(0,1)$, allowing up to $\alpha_{n}=o(n^{-0})=o(1)$. Similarly,
$\alpha_{n}$ up to $o(n^{-(1-\gamma)})$ is allowed when $\gamma<1$ and
$o(n^{-2(\gamma-1)})$ when $\gamma>1$, implying Assumption E3.

The rest of the proof is similar to that in the Proof of Theorem 5.4, using
Proposition 5.2. ∎

###### Proof of Theorem 5.6.

Consider the same $\alpha_{n}$, $t_{\alpha,n}$, and $\widetilde{w}_{i}$ as in
the Proof of Theorem 5.5. Consider $0<\gamma<1$. For any positive real value
$\delta\in(0,\min\\{\gamma,1-\gamma\\})$, by Lemma .5,

$\sum_{i=1}^{n}\bm{E}\widetilde{w}_{i}^{\gamma-\delta}=\frac{\sum_{i=1}^{n}\bm{E}w_{i}^{\gamma-\delta}}{n^{\frac{\tau(\gamma-\delta)}{\gamma}}(\sum_{j=1}^{n}\bm{E}w_{j}^{\gamma})^{\frac{\gamma-\delta}{\gamma}}}\leq\frac{n^{1-(\gamma-\delta)}}{n^{\frac{\tau(\gamma-\delta)}{\gamma}}}=n^{1-(\gamma-\delta)-\frac{\tau(\gamma-\delta)}{\gamma}}$

is uniformly bounded for all $n$ as long as
$\tau\geq\frac{\gamma}{\gamma-\delta}-\gamma$. Note that
$\delta\in(0,\min\\{(\gamma,1-\gamma\\})$ is equivalent to
$\gamma-\delta\in(\max\\{0,2\gamma-1\\},\gamma)$.

Now consider $\gamma\geq 1$. For some $\delta\in(0,\gamma)$, by Hölder’s
inequality,

$\begin{array}[]{ll}\sum_{i=1}^{n}\left(\bm{E}w_{i}^{\gamma-\delta}\right)^{\frac{1}{\gamma+\delta}}&\leq\left[\sum_{i=1}^{n}\left\\{\left(\bm{E}w_{i}^{\gamma-\delta}\right)^{\frac{1}{\gamma+\delta}}\right\\}^{{\gamma+\delta}}\right]^{\frac{1}{\gamma+\delta}}\left(\sum_{i=1}^{n}1^{\frac{\gamma+\delta}{\gamma+\delta-1}}\right)^{\frac{\gamma+\delta-1}{\gamma+\delta}}\\\
&=\left(\sum_{i=1}^{n}\bm{E}w_{i}^{\gamma-\delta}\right)^{\frac{1}{\gamma+\delta}}n^{\frac{\gamma+\delta-1}{\gamma+\delta}}.\end{array}$
(39)

We first consider $\gamma>1$. Notice that the smaller $\delta$ is, the smaller
the upper bound of (39) is. In particular, since $\gamma>1$, it is possible to
choose a positive constant $\delta$ such that $\gamma-\delta>1$. For such
$\delta$, since $\gamma+\delta>\gamma-\delta>1$, Lemma .2 implies that
$\bm{E}w_{i}^{\gamma+\delta}<\bm{E}w_{i}^{\gamma-\delta}<1$. Further by Lemma
.1, since $\gamma+\delta>1$,
$\left(\bm{E}w_{i}^{\gamma-\delta}\right)^{\frac{1}{\gamma+\delta}}<\left(\bm{E}w_{i}^{\gamma+\delta}\right)^{\frac{1}{\gamma+\delta}}$.
Thus, the upper bound of (39) is, using Lemma .5,

$\sum_{i=1}^{n}\left(\bm{E}\widetilde{w}_{i}^{\gamma-\delta}\right)^{\frac{1}{\gamma+\delta}}=\frac{\sum_{i=1}^{n}\left(\bm{E}w_{i}^{\gamma-\delta}\right)^{\frac{1}{\gamma+\delta}}}{n^{\frac{\tau(\gamma-\delta)}{\gamma(\gamma+\delta)}}(\sum_{j=1}^{n}\bm{E}w_{j}^{\gamma})^{\frac{\gamma-\delta}{\gamma(\gamma+\delta)}}}\leq\frac{n^{\frac{\gamma+\delta-1}{\gamma+\delta}}}{n^{\frac{\tau(\gamma-\delta)}{\gamma(\gamma+\delta)}}n^{\frac{(1-\gamma)(\gamma-\delta)}{\gamma(\gamma+\delta)}}}=n^{\frac{2\gamma^{2}-2\gamma+\delta-\tau(\gamma-\delta)}{\gamma(\gamma+\delta)}},$

which is uniformly bounded for any $n$ if
$\tau\geq\frac{2\gamma^{2}-2\gamma+\delta}{\gamma-\delta}$.

Now consider $\gamma=1$. In this case, $0<\gamma-\delta<1$ for any positive
constant $\delta<\gamma$. Using Lemma .5, the upper bound of (39) is
$n^{\frac{1-\gamma+\delta}{\gamma+\delta}}n^{\frac{\gamma+\delta-1}{\gamma+\delta}}=n^{\frac{2\delta}{\gamma+\delta}}=n^{\frac{2\delta}{1+\delta}}$.
Noting that $\sum_{j=1}^{n}\bm{E}w_{j}^{\gamma}=1$ when $\gamma=1$, we have

$\sum_{i=1}^{n}\left(\bm{E}\widetilde{w}_{i}^{\gamma-\delta}\right)^{\frac{1}{\gamma+\delta}}=\frac{\sum_{i=1}^{n}\left(\bm{E}w_{i}^{\gamma-\delta}\right)^{\frac{1}{\gamma+\delta}}}{n^{\frac{\tau(\gamma-\delta)}{\gamma(\gamma+\delta)}}(\sum_{j=1}^{n}\bm{E}w_{j}^{\gamma})^{\frac{\gamma-\delta}{\gamma(\gamma+\delta)}}}\leq\frac{n^{\frac{2\delta}{1+\delta}}}{n^{\frac{\tau(1-\delta)}{1+\delta}}}=n^{\frac{2\delta-\tau(1-\delta)}{1+\delta}},$

which is uniformly bounded for any $n$ if $\tau\geq\frac{2\delta}{1-\delta}$.

For the $\gamma<1$ case, $\frac{\gamma}{q_{1}}-\gamma$ is decreasing in
$q_{1}$, and therefore, $\alpha_{n}$ up to $o(n^{-(1-\gamma)})$ is allowed.
For the other two cases, $\gamma=1$ and $\gamma>1$, $\frac{2q_{2}}{1-q_{2}}$
and $\frac{2\gamma^{2}-2\gamma+q_{3}}{\gamma-q_{3}}$ are increasing in $q_{2}$
and $q_{3}$, respectively. This can be easily seen by noticing that the first
derivative of both functions is always positive since $q_{2}$ and $q_{3}$ are
not equal to 1. Therefore,$\alpha_{n}$ up to $o(n^{-\frac{0}{1}})=o(1)$ is
allowed when $\gamma=1$ and $o(n^{-2(\gamma-1)})$ is allowed when $\gamma>1$.

The rest of the proof is similar to that in the Proof of Theorem 5.4, using
Proposition 5.3. ∎

###### Proof of Theorem 5.8.

We first note that, since $\overline{F}$ is continuous, and therefore,
strictly decreasing, $c_{n}=\overline{F}^{-1}(n^{-1})\uparrow\infty$ as
$n\uparrow\infty$. Since $\overline{F}(c_{n})=1/n$ and
$\overline{F}(nt_{\alpha,n})=\alpha/n$,

$\overline{F}(nt_{\alpha,n})=\alpha\overline{F}(c_{n})\sim\overline{F}(\alpha^{-\frac{1}{\gamma}}c_{n})$

as $c_{n}\uparrow\infty$, where the last approximation is due to the
asymptotic scale invariance of a regularly varying distribution. Therefore,
$nt_{\alpha,n}\sim\alpha^{-\frac{1}{\gamma}}c_{n}$, or equivalently,

$c_{n}^{-1}\sim n^{-1}t_{\alpha,n}^{-1}\alpha^{-\frac{1}{\gamma}}$

as $n\uparrow\infty$, and further,

$\bm{P}\left(S_{3,F,w}(\bm{U}_{n})>t_{\alpha,n}\right)=\bm{P}\left(n^{-1}\bigvee_{i=1}^{n}X_{i}>t_{\alpha,n}\right)\sim\bm{P}\left(c_{n}^{-1}\bigvee_{i=1}^{n}X_{i}>\alpha^{-\frac{1}{\gamma}}\right).$

From (26),

$\bm{P}\left(c_{n}^{-1}\bigvee_{i=1}^{n}X_{i}<\alpha^{-\frac{1}{\gamma}}\right)\sim\exp\left(-(\alpha^{-\frac{1}{\gamma}})^{-\gamma}\right)=\exp(-\alpha)\sim
1-\alpha$

as long as $\alpha=\alpha_{n}\downarrow 0$ as $n\uparrow\infty$, which
finishes the proof. ∎

###### Proof of Theorem 5.9.

Since $X_{i}=\overline{F}^{-1}(U_{i})$ are independent,
$\bm{P}(\bigvee_{i=1}^{n}w_{i}X_{i}<t)=\prod_{i=1}^{n}\bm{P}(X_{i}<tw_{i}^{-1})$.
Taking log on both sides, we have

$\log\left(\bm{P}\left(\bigvee_{i=1}^{n}w_{i}X_{i}<t\right)\right)=\sum_{i=1}^{n}\log\bm{P}\left(X_{i}<tw_{i}^{-1}\right)=\sum_{i=1}^{n}\log
F(tw_{i}^{-1}).$

Note that $tw_{i}^{-1}\geq t\to\infty$ as $t\to\infty$ for all $i=1,\ldots,n$,
regardless of the value of $n$. Using the relationship $\log x\uparrow(x-1)$
as $x\uparrow 1$, we have $\log F(tw_{i}^{-1})\sim
F(tw_{i}^{-1})-1=-\overline{F}(tw_{i}^{-1})$ as $t\to\infty$ for all
$i=1,\ldots,n$, no matter what $n$ is. Therefore,

$\bm{P}\left(\bigvee_{i=1}^{n}w_{i}X_{i}>t\right)\sim
1-\exp\left(-\sum_{i=1}^{n}\overline{F}(tw_{i}^{-1})\right)$

as $t\to\infty$ for any $n$. In addition, $\min_{i}tw_{i}^{-1}\to\infty$
implies $\overline{F}(tw_{i}^{-1})\to 0$ as $t\to\infty$, and hence, the
relationship $1-\exp(-x)\sim x$ as $x\to 0$ implies

$\bm{P}\left(\bigvee_{i=1}^{n}w_{i}X_{i}>t\right)\sim\sum_{i=1}^{n}\overline{F}(tw_{i}^{-1})$

as $t\to\infty$ for any $n$. Therefore,

$\bm{P}\left(\bigvee_{i=1}^{\infty}w_{i}X_{i}>t\right)\sim\sum_{i=1}^{\infty}\overline{F}(tw_{i}^{-1}),$

which is equivalent to equation (27). ∎

###### Proof of Theorem 6.1.

From the GCLT in equations (29) and (30), the tail probability of the sum-
combining function can be approximated as, when $U_{i}$ are iid and $n$ is
large enough,

$\begin{array}[]{lll}\bm{P}(T_{1,F_{\gamma},w}(\bm{U}_{n})<\alpha_{n})&=&\bm{P}(\overline{F}(a_{n,\gamma}S_{1,F_{\gamma},w}(\bm{U}_{n}))<\alpha_{n})=\bm{P}\left(n\left(\sum_{i=1}^{n}U_{i}^{-\frac{1}{\gamma}}\right)^{-\gamma}<\alpha_{n}\right)\\\
&=&\bm{P}\left(\sum_{i=1}^{n}U_{i}^{-\frac{1}{\gamma}}>\alpha_{n}^{-\frac{1}{\gamma}}n^{\frac{1}{\gamma}}\right)\\\
&\sim&\left\\{\begin{array}[]{ll}\bm{P}\left(S(1,1,1,0)>\frac{\alpha_{n}^{-1}-\log
n}{\pi/2}\right),&\gamma=1\\\
\bm{P}\left(S(\gamma,1,1,\tan(\pi\gamma/2))>\alpha_{n}^{-\frac{1}{\gamma}}\left\\{\frac{2}{\pi}\Gamma(\gamma)\sin\left(\frac{\pi\gamma}{2}\right)\right\\}^{\frac{1}{\gamma}}\right),&\gamma\in(0,1),\end{array}\right.\end{array}$

where $S(\gamma,\beta,\sigma,\mu)$ indicates a random variable that follows
$S(\gamma,\beta,\sigma,\mu;0)$. Consider $\gamma\in(0,1)$. The proof is
straightforward using the tail approximation of a stable distribution (Nolan,
2020, Theorem 1.2) as long as $\alpha_{n}=o(1)$.

Now we consider $\gamma=1$. Since $\alpha_{n}=o(1/\log n)$, we have
$\alpha_{n}^{-1}-\log n\to\infty$ as $n\to\infty$. Therefore, following
Theorem 1.2 of Nolan (2020),
$\bm{P}\left(S(1,1,1,0)>\frac{\alpha_{n}^{-1}-\log
n}{\pi/2}\right)\sim(\alpha_{n}^{-1}-\log n)^{-1}$ as $n\to\infty$. ∎


# HE0359-3959: an extremely radiating quasar

M. L. Martínez-Aldama 1,∗, A. Del Olmo 1, P. Marziani 2, C. A. Negrete 3,
D. Dultzin 4, M. A. Martínez-Carballo 1

1Instituto de Astrofísica de Andalucía, IAA-CSIC, Granada, Spain
2INAF, Osservatorio Astronomico di Padova, Italy
3 CONACYT Research Fellow, Instituto de Astronomía, UNAM, Mexico
4Instituto de Astronomía, UNAM, Mexico
∗Correspondence<EMAIL_ADDRESS>
###### Abstract

We present a multiwavelength spectral study of the quasar HE0359-3959, which
has been identified as an extreme radiating source at intermediate redshift
($z$=1.5209). Along the spectral range, the different ionic species give
information about the substructures in the broad line region. The presence of
a powerful outflow with an extreme blueshifted velocity of $\sim$–6000$\pm$500
km s-1 is shown in the Civ$\lambda$1549 emission line. A prominent blueshifted
component is also associated with the 1900Å blend, resembling the one observed
in Civ$\lambda$1549\. We detect a strong contribution of very the
low–ionization lines, Feii and Near-Infrared Ca ii triplet.We find that the
physical conditions for the low, intermediate and high–ionization emission
lines are different, which indicate that the emission lines are emitted in
different zones of the broad line region. The asymmetries shown by the
profiles reveal different forces over emitter zones. The high–ionization
region is strongly dominated by radiation forces, which also affect the low
and intermediate–ionization emitter region, commonly governed by virial
motions. These results support the idea that highly radiating sources host a
slim disk.

Keywords: quasars: emission lines, quasars: outflows, quasars: individuals
HE0359-3959. quasars: supermassive black holes, galaxy evolution: feedback

##  1 Extreme Population A sources along the 4DE1 Main Sequence

The 4D Eigenvector 1 (4DE1) parameter space offers a formalism to distinguish
and classify type 1 Active Galactic Nuclei (AGN) considering their spectral
properties (Sulentic et al., 2000a, b). The Full Width at Half Maximum (FWHM)
of H$\beta$ broad component (H$\beta_{\rm{BC}}$), the strength of optical Feii
blend at 4570Å described by the ratio $\mathrm{R_{\rm FeII}}$=
I(Feii)/I(H$\beta_{\rm BC}$), the velocity shift of the Civ$\lambda$1549
profile, and soft X-ray photon index ($\Gamma\mathrm{{}_{soft}}$), provide
four observationally independent dimensions of the Eigenvector 1. In the 4DE1
optical plane, the type 1 AGN occupy a well defined sequence, driven mainly by
the Eddington ratio, L/LEdd. Along this sequence we observe a variation of the
physical parameters and orientation. Then, 4DE1 could be revealing an
evolution sequence for type 1 AGN (Sulentic et al., 2000a; Marziani et al.,
2010; Zamfir et al., 2010). For more information about the 4DE1 and update of
results, see Marziani et al. 2017 in this volume.

Using the 4DE1 we identify two populations with different spectral features: A
and B. Population A has a FWHM(H$\beta$BC)$\leq$4000 km s-1. It shows large
blue asymmetries in the high-ionization lines like Civ$\lambda$1549, and it is
majority populated by radio quiet sources. In contrast, population B shows a
FWHM(H$\beta$BC)$>$4000 km s-1 and it is mostly composed of radio-loud sources
(Sulentic et al., 2002; Zamfir et al., 2010). Each population can be divided
into small bins with $\Delta$FWHM(H$\beta$BC)=4000 km s-1 and
$\Delta$$\mathrm{R_{\rm FeII}}$=0.5, defining subpopulations shown in the
Figure 1. In this paper we focus in the subpopulation A3 and A4
($\mathrm{R_{\rm FeII}}$$>$1), which have been identified as highly radiating
sources (xA, Marziani $\&$ Sulentic, 2014). These kind of sources show high
Eddington ratios (L/LEdd$>$0.2) probably produced by a slim disk, which is
geometrically and optically thick and it could be formed in an advection-
dominated accretion flow (Abramowicz et al., 1988; Abramowicz $\&$ Straub ,
2014).

We have found selection criteria to identify the xA sources based on the 4DE1
formalism. In the optical region they show a $\mathrm{R_{\rm FeII}}$$>$1 (high
intensity of Feii) and in the UV range
Aliii$\lambda$1860/Siiii]$\lambda$1892$\geq$0.5 and
Ciii]$\lambda$1909/Siiii]$\lambda$1892$\leq$1.0 (Marziani $\&$ Sulentic,
2014). Also, they show strong blueshifted components associated with the high
ionization lines, for example in Civ$\lambda$1549 emission line, indicating
the presence of outflows. More details about the xA sources behavior can be
found in Martínez-Aldama et al. of this volume.

###  1.1 HE0359-3959: an extreme xA source

In our extreme luminosity Hamburg-ESO sample (Marziani et al., 2009; Sulentic
et al., 2017), we have identified four cases of highly radiating quasars that
show an extreme behavior, i.e., a high Eddington ratio and a strong blue
asymmetry (c($\nicefrac{{1}}{{2}}$)$<$–4000 km s-1; centroid at half
intensity) in the Civ$\lambda$1549 profile (Sulentic et al., 2017). The most
extreme case corresponds to the quasars HE0359-3959, with $z$=1.5209,
log(Lbol)=47.6 erg s-1 and a $\mathrm{R_{\rm FeII}}$=1.12. It is cataloged as
an A3 source (see Figure 1).

The aim of this paper is to analyze the spectral behavior of an extreme xA
source, the quasar HE0359–3959. We performed multicomponent fits in a wide
spectral range: UV, optical and Near–Infrared (Section 2); which gives us
information about the dynamics and the physical conditions of the broad line
region (BLR) (Section 3). In Section 4, we summarize the main results of our
work.

Figure 1: 4DE1 Optical Plane reproduced from Marziani $\&$ Sulentic (2014).
Grey points correspond to the sample of 470 bright low-$z$ QSOs from Zamfir et
al. (2010). The plane is divided in bins according to Sulentic et al. (2002).
Extreme accretor population A sources (xA) are located in A3 and A4 bins. The
black dot indicates the position of 1 Zw 1, the prototype of low-$z$ xA
sources. And, the red dot marks the location of HE0359-3959, an extreme xA
source with high-$z$.

##  2 Observations, data reduction and multicomponent fitting

###  2.1 Observations and data reduction

Ultraviolet (UV), optical and Near–Infrared spectra were observed with the
Very Large Telescope (VLT-ESO). Optical and Near–Infrared spectra were
obtained with the Infrared Spectrometer And Array Camera (ISAAC;
decommissioned in 2013) using a slit of 0.6′′. The near–infrared spectrum was
observed in 2010 in the K band with a total exposure time of 1120 seconds. The
optical spectrum was observed in 2004 in the J band with a total exposure time
of 3600 seconds. For the ultraviolet spectrum we used the Focal Reducer and
low dispersion Spectrograph (FORS1) a slit of 1.0′′ with a total exposure time
of 1440 seconds. It was observed in 2008. The data reduction was done using
the iraf package. The procedures followed are explained in Marziani et al.
(2009), Martínez-Aldama et al. (2015) and Sulentic et al. (2017).

###  2.2 Multicomponent fits

We perform multicomponent fits using specfit, an iraf routine (Kriss, 1994) to
get the information of the most important emission lines. In each spectral
range we fit a local continuum. The FWHM of all the broad components (BC) for
H$\beta$, Aliii$\lambda$1860, Siiii]$\lambda$1892, Civ$\lambda$1549 and
Siiv$\lambda$1397 was taken equal. In the Figure 2, we present the
multicomponents fits after continuum subtraction, for the Civ$\lambda$1549 and
Ca ii triplet range. The rest of the fits will be shown in an upcoming paper.

##  3 Results

###  3.1 Multiwavelength analysis

Low–ionization lines (LIL) have an ionization potential (IP) $\leq$20 eV. The
H$\beta$ line is the prototype of LIL. In population A3 and A4 sources
H$\beta$ has associated a blueshifted component (Bachev et al., 2004). In the
case of HE0359-3959, the blueshifted component has a contribution to the total
flux of $\sim$9$\%$, and shows a centroid a half intensity of
c($\nicefrac{{1}}{{2}}$)$\approx$–500$\pm$70 km s-1.

The Feii (IP$\sim$16 eV) has an important contribution in the optical and
near–infrared regions. To reproduce it we used the templates modeled by
Marziani et al. (2009) and García-Rissmann et al. (2012) for the optical and
near–infrared ranges, respectively. Several works have found (Joly, 1989;
Persson, 1988; Ferland $\&$ Persson, 1989; Dultzin et al., 1999; Martínez-
Aldama et al., 2015) a close relationship between the Feii and the NIR Ca ii
$\lambda$8498,$\lambda$ 8542 and $\lambda$8662 Å triplet. This relation is
very well appreciable in this object: as well as the optical Feii is strong,
the NIR Ca ii triplet also is. It is the first time where we observe the Ca ii
triplet lines isolated at high redshift. Strong intensities of both ions imply
an extremely low-ionization degree ($U<$10-2; $U$: ionization parameter) and a
high density (nH$\sim$1011-13 cm-3) (Baldwin et al., 2004; Matsuoka et al.,
2007; Martínez-Aldama et al., 2015).

Figure 2:  Multicomponent analysis on the UV, optical and NIR spectra of
HE0359-3959 after continuum subtraction. Top panels: In the left side is shown
the UV spectrum, while in the right one is shown the near–infrared spectrum.
The different components (broad (BC), BLUE, and narrow (NC)) in the line
fitting are specified in each panel. Vertical lines indicate the rest-frame
obtained through H$\beta_{{\mathrm{NC}}}$. The gray line marks the Feii
contribution. The vertical scale represents the relative flux in units of
10-15 erg s-1 cm-2 Å-1. Bottom panel: Residuals of the fittings. The
horizontal scale is the radial velocity shift in km s-1. In all the panels the
horizontal scale represents the rest–frame wavelength in Å.

In the UV region, the 1900Å blend is formed by two intermediate–ionization
lines (IIL; IP$\sim$20–40 eV), Aliii$\lambda$1860 and Siiii]$\lambda$1892,
which are accompanied by Ciii]$\lambda$1909 and some Feiii transitions. In
this blend we appreciate a blueshifted component. This component should be
most likely associated with Aliii$\lambda$1860\. Respect to
Aliii$\lambda$1860, the blueshifted component has a contribution of the total
profile of 60$\%$. The centroid a half intensity is
c($\nicefrac{{1}}{{2}}$)$\approx$–3200$\pm$250 km s-1, which indicates the
presence of an outflow generated by radiation forces presented in the
intermediate–ionization lines (Marziani et al., 2017). On the other hand,
considering the high intensity of Aliii$\lambda$1860, Siiii]$\lambda$1892, Ca
ii and Feii, it could suggest a possible chemical enrichment of the BLR
(Juarez et al., 2009).

High ionization lines (HIL; IP$>$40 eV), Civ$\lambda$1549, Heii$\lambda$1640
and Siiv$\lambda$1397, show a prominent blueshifted component. We find that
the blue component has a contribution of 76$\%$, 62$\%$ and 57$\%$ to the
total flux of Civ$\lambda$1549, Heii$\lambda$1640 and Siiv$\lambda$1397
respectively. The Civ$\lambda$1549 reaches
c($\nicefrac{{1}}{{2}})\sim$–6000$\pm$500 km s-1, while Heii$\lambda$1640 and
Siiv$\lambda$1397 c($\nicefrac{{1}}{{2}})\sim$–4000$\pm$550 km s-1. The
velocities reached are ones of the highest found in the literature (Richards
et al., 2011; Coatman et al., 20016; Sulentic et al., 2017). Then, it
indicates that the full profile is dominated by an outflow and suggests the
disk plus wind scenario (Gaskell, 1982; Richards et al., 2002, 2011).

###  3.2 Physical properties of HE0359-3959

In order to study the physical properties of the quasar HE0359-3959, we built
a grid of photoionization simulations using the CLOUDY code (Ferland et al.,
1998, 2013). For our simulations we considerer a Mattews and Ferland continuum
(Mathews $\&$ Ferland, 1987), a plane-parallel geometry, a metallicity
5Z$\odot$ with an overabundance of Al and Si with respect to carbon (by a
factor of three), and a column density of Nc=1023 cm-2. See Negrete et al.
(2012) for more details. Our simulations span the density range
7.00$\leq$log(nH)$\leq$14.00 and –4.5$\leq$log($U$)$\leq$0.00 for the
ionization parameter, in intervals of 0.25 dex. More details about the cloudy
simulations can be found in Negrete et al. (2014). Using the UV lines, we
define three groups of diagnostic ratios::

  * •

The flux ratio Aliii$\lambda$1860/Siiii]$\lambda$1892 is a useful density
diagnostic.

  * •

The flux ratio Siiv$\lambda$1397/Siiii]$\lambda$1892 for the ionization
parameter.

  * •

The flux ratio Civ$\lambda$1549/Siiv$\lambda$1397 is mainly sensitive to the
relatives abundances of C and Si.

In Figure 3 is shown the result of the simulations. We obtained that the flux
ratios are intersected in log(nH)=12.32 cm-3 and log($U$)=–2.95. Compared to
not highly radiating AGNs (Negrete et al., 2013), this source shows a high
density and a low ionization parameter, which marks a different behavior in
the BLR, probably causing by the slim disk hosted in these kind of sources.
Taking into account the high intensity of Aliii$\lambda$1860, Feii and Ca ii
we conclude that effectively the low–ionization emitter zone has a high
density and low–ionization parameter.

Figure 3: Isocontours for HE0359-3959 with 5Z⊙ and an overabundance of Al and
Si. The blue line indicates the flux ratio
Aliii$\lambda$1860/Siiii]$\lambda$1892, the yellow one indicates the ratio
Siiv$\lambda$1397/Siiii]$\lambda$1892 and the orange corresponds to the
Aliii$\lambda$1860/Siiv$\lambda$1397\. Shadows associated with each line
indicate the error. The flux ratios are intersected in nH$\cdotp
U$=9.27$\pm$0.39

.

Negrete et al. (2012) proposed a new method to determine the size of the BLR
(rBLR) and the black hole mass (MBH) based on the product nH$\cdotp U$ and
independently of redshift. This method gives similar results to the obtained
from the classical methods such as reverberation mapping at low–$z$ (Negrete
et al., 2014). Knowing the product of nH$\cdotp U$ obtained from the cloudy
simulations, we compute the size of the BLR (rBLR) and considering the FWHM of
the broad components as the velocity dispersion, we can get the black hole
mass (MBH) and the Eddington ratio. The size of the BLR is
log(rBLR)=18.37$\pm$0.04 cm and the black hole mass is log(MBH)=9.52$\pm$0.41
M⊙. These values are in agreement with the ones found for a large xA sample at
high-redshift (Martínez-Aldama et al. in prep.).

The Eddington ratio for this source is L/LEdd=0.74$\pm$0.11. Considering that
it shows a c($\nicefrac{{1}}{{2}}$)$\sim$–6000$\pm$500 km s-1 for
Civ$\lambda$1549, we confirm the directly proportional relation between
c($\nicefrac{{1}}{{2}}$) and L/LEdd. Indicating that L/LEddcould be the driver
of the outflows (Sulentic et al., 2017).

##  4 Conclusions

The information given by the multiwavelength analysis indicates that in
HE0359–3959 there is coexistence of substructures in the broad line region.
Low and intermediate–ionization regions, where H$\beta$, Aliii$\lambda$1860
and Siiii]$\lambda$1892 are emitted, are dense (nH$\sim$1011-12 cm-3) and
optically thick ($U\sim$10-2.5). They are mainly governed by virial motions
and the presence of a blueshifted component indicates the influence of
radiation forces. On the other hand, according to Marziani et al. (2010) the
high–ionization region is less dense (nH$\sim$1010 cm-3, $U\sim$10-1),
pointing out a difference with the physical conditions shown by the low and
intermediate–ionization lines.

High ionization lines are dominated by strong radiation forces, producing
outflows in high–ionization lines like Civ$\lambda$1549, Heii$\lambda$1640 and
Siiv$\lambda$1397\. The high Eddington ratio value suggests the presence of a
slim optically thick disk which could be related to the extreme outflow
properties observed in HE0359-3959. The presence of strong outflows has been
related with the co–evolution of the active galactic nuclei and the host
galaxy.

Acknowledgements. M.L.M.A acknowledge the postdoctoral grant from CONACyT.
M.L.M.A., A.d.O. and M.A.M.C. acknowledge financial support from Spanish
Ministry for Economy and Competitiveness through grants AYA2013-42227-P and
AYA2016-76682-C3-3-1-P.

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