\Large Oscillations in the Tsallis income distribution
Everton M.\ C.\ Abreu
Newton J. Moura Jr.
Abner D. Soares
Marcelo B. Ribeiro
\noindent Oscillations in the cumulative individual income distribution have been found in the data of various countries studied by different authors at different time periods, but the dynamical origins of this behavior are currently unknown. These datasets can be fitted by different functions at different income ranges, but recently the Tsallis distribution has been found capable of fitting the whole distribution by means of only two parameters, procedure which showed even more clearly such oscillatory features in the entire income range. This behavior can be described by assuming log-periodic functions, however a different approach to naturally disclose such oscillatory characteristics is to allow the Tsallis $q$-parameter to become complex. In this paper we have used these ideas in order to describe the behavior of the complementary cumulative distribution function of the personal income of Brazil recently studied empirically by Soares et al.\ (2016). Typical elements of periodic motion, such as amplitude and angular frequency coupled to this income analysis, were obtained.
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\begin{document}

\title{\Large Oscillations in the Tsallis income distribution}
\author{Everton M.\ C.\ Abreu}\email{Corresponding author:
        evertonabreu@ufrrj.br}
\affiliation{Departamento de F\'{i}sica, Universidade Federal Rural do Rio de
             Janeiro--UFRRJ, Serop\'edica, RJ 23890-971, Brasil}
\affiliation{Departamento de F\'{i}sica, Universidade Federal de Juiz de
             Fora--UFJF, Juiz de Fora, Brasil}
\author{Newton J. Moura Jr.}
\affiliation{Instituto Brasileiro de Geografia e Estat\' istica--IBGE,
	    Rio de Janeiro, Brasil}
\author{Abner D. Soares}
\affiliation{Comiss\~ao Nacional de Energia Nuclear--CNEN, Rio de Janeiro,
            Brasil}
\author{Marcelo B. Ribeiro}
\affiliation{Instituto de F\' isica, Universidade Federal do Rio de
            Janeiro--UFRJ, Rio de Janeiro, Brasil}
\affiliation{Observat\'orio do Valongo, Universidade Federal do Rio de
            Janeiro--UFRJ, Rio de Janeiro, Brasil}
\date{\today}

\begin{abstract}
\noindent Oscillations in the cumulative individual income distribution 
have been found in the data of various countries studied by different 
authors at different time periods, but the dynamical origins of this 
behavior are currently unknown. These datasets can be fitted by different
functions at different income ranges, but recently the Tsallis distribution
has been found capable of fitting the whole distribution by means of only
two parameters, procedure which showed even more clearly such oscillatory
features in the entire income range. This behavior can be described by
assuming log-periodic functions, however a different approach to naturally
disclose such oscillatory characteristics is to allow the Tsallis
$q$-parameter to become complex. In this paper we have used these ideas
in order to describe the behavior of the complementary cumulative
distribution function of the personal income of Brazil recently studied
empirically by Soares et al.\ (2016). Typical elements of periodic motion,
such as amplitude and angular frequency coupled to this income analysis,
were obtained.
\end{abstract}


\pacs{}
\pacs{89.65.-s; 89.65.Gh; 05.90.+m}
\keywords{income distribution, Tsallis statistics, complex $q$-parameter}
% 89.65.-s Social and economic systems
% 89.65.Gh Economics; econophysics, financial markets, business and management

\maketitle

\section{Introduction}

The study of the individual income distribution of populations has a long
history. Vilfredo Pareto (1848-1927), the pioneer of this type of analysis,
studied the distribution of personal income at the end of 19th century
for some regions and countries in specific years and sets of years. Pareto
looked at the problem systematically and concluded that, individually
speaking, the richest people in a society have the \textit{complementary
cumulative distribution function} (CCDF) of income obeying a power law
function \cite{pareto}. Consequently, the \textit{probability density
function} (PDF) $p(x)$ of the personal income $x$ of the richest persons
may be given by,
\bee
p(x) = \beta\,x^{-(1+\alpha)}\,\,,
\label{paret}
\eee
where $\beta$ is a normalization constant. Through the years, this power
law behavior has become known as the \textit{Pareto power-law} and,
consequently, the exponent $\alpha$ is now known as the \textit{Pareto
index.} This law has been interpreted later as being a classic example
of a fractal distributions, where the Pareto index plays the role of the
single fractal dimension of the distribution \cite{mandelbrot}. Higher
values of the Pareto index imply more uneven distribution of the personal
income. The interesting detail is that this result has not been disputed
by different investigations carried out since then, which considered
several different samples obtained at different times for different
populations in distinct countries or groups of countries \cite[and
references therein]{k80,nm09,fnm10,dy01,ferrero2,s05,yr09,crbh08,by10,
nm13,cyc05}.

Despite its empirical success, the Pareto power-law does not work for
the overwhelmingly majority, and less rich, part of the population.
Namely, it only describes well the income data of those belonging to the
narrow ``slice'' of the richest population. To consider the income data
of the group composed by the less rich people, the method that has been
used since shortly after Pareto's time is to fit the data by various
other function, like the exponential, the log-normal, the gamma function,
the Gompertz curve, as well as other ones \cite{k80,nm09,fnm10,ferrero2,
nm13,cyc05}. There are also succesful approaches for analyzing the whole
data range using less than simple functions with many parameters, but
usually such approaches require four or more parameters in order to fit
the entire distribution. In addition, these methodologies mean the
assumption that societies are divided in two classes only: on one side
the very rich, formed by about 1\% of the whole population, and on the
other side the remaining 99\%. One basic problem of this metodology is
the absence of the middle class, which certainly exists in between
these two groups. In addition, the question remains of whether or
not this class division is a real feature of societies or basically a
mathematical artifact convenient for data fitting. Considering these
objections, a simple function able to fit the entire income distribution
which, at the same time, allows for various features to emerge at
different income range is certainly preferable.

A recent approach for representing the whole income data is to fit the
data using the Tsallis functions instead of the combination of functions
depicted above. The individual income distribution was classified in
terms of the well known Tsallis parameter, \textit{i.e.}, the
$q$-parameter, and another normalization constant. Borges \cite{borges}
used two $q$-parameter, where one controls the slope of the intermediate
income range and the other describes the tail of the distribution. He was
then able to analyze the income distribution concerning some
\textit{counties} of the USA from 1970 to 2000, Brazil from 1970 to 1996,
Germany from 1992 to 1998 and the United Kingdom from 1993 to 1998. The
conclusion was that an increase in $q$ with time points to growing
inequality. Greater values of $q$ indicate greater probability to find
counties much richer than others. Ferrero \cite{c5,ferrero1} used the
Tsallis function to fit the entire income data of several
\textit{countries}, but only at single years, not being able though to
indicate an evolution in the Tsallis parameters.

Recently three of us \cite{tsallis} have used the Tsallis formalism to
empirically study the income distribution of Brazil during a relatively
large yearly time window, from 1978 to 2014. The results showed that the
two fitted parameters of the Tsallis function follow a cycling behavior.
Moreover, a linear fit of the distribution in each year showed that the
data oscillate periodically around the fitted straight line with an
amplitude that grows with the income values. A closer look at the fitted
data made by other authors using different methods applied to different
samples collected at different time periods showed a similar oscillatory
pattern, which means that there must be indeed a second order dynamical
effect not previously identified in the income data \cite[p.\ 164]{smr}.
This kind of oscillatory behavior has not been noted before in the
income distribution data, although it has been observed in financial
markets and other systems \cite{smr}.

In this work we have analyzed this periodic oscillation through the
alternative approach of allowing the Tsallis parameter to become
complex, as suggested by Ref.\ \cite{ww}. Under this methodology the
$q$-parameter discloses such periodic behavior in the income
distribution curves, allowing us to defined periodic elements that are
ordinary in physics, like the amplitude and angular frequency. This paper
is organized as follows. Sect.\ 2 presents the Tsallis functions and
some of their properties required in our analysis. Sect.\ 3 presents
and discusses the complexification process and its influence in the
analysis of the individual income distribution. Sect.\ 4 presents our
conclusions.

\section{Tsallis functions}

It is well known that the Tsallis thermostatistics \cite{tsallis1,tsallis2}
is based on both the $q$-logarithm and $q$-exponential functions, given by,
\be
\ln_q x \equiv \frac{x^{(1-q)}-1}{1-q},
\lb{qlog}
\ee
\be
{{\mathrm{e}}_q}^x \equiv {\bigglb[1+(1-q)x \biggrb]}^{1/(1-q)}.
\lb{qexp}
\ee

\ni These functions are defined such that for $q=1$ both expressions
become the standard logarithm and exponential functions, namely,
${\mathrm{e}_1}^x={\mathrm{e}}^x$ and $\ln_1 x=\ln x$. Hence, the
Tsallis $q$-functions are in fact the usual exponential and logarithmic
expressions twisted in such a way as to be used in Tsallis' theory of
nonextensive statistical mechanics \cite{tsallis2}.

At this point it should be noted that there are other ways to deform
these two common functions viewing other application, such as the
personal income distribution. This is the case of  the
$\kappa$-generalized exponential, introduced by Ref.\ \cite{cgk07},
which can be used to fit the entire income data range in similar manner
as the Tsallis $q$-functions. This is especially significant because
both of them have the power-law and exponential as their limiting cases.
The interested reader can find more applications of the
$\kappa$-generalized function in Refs.\ \cite{cmgk08,cgk09,cgk12}.

From the definitions above it is clear that,
\be
{{\mathrm{e}}_q}^{(\, \ln_q x)}= \ln_q \Big({\mathrm{e}_q}^x \,\Big)=x.
\lb{recip}
\ee
Moreover, $\ln_q 1=0$ for any $q$. Hence, if we have a value $x_0$ such
that $x/x_0=1$, as a result $\ln_q\,(x/x_0)=0$. Two other properties of
the $q$-exponential useful for our purposes here are as follows
\cite{yamano},
\be
{\left[ {\mathrm{e}_q}^{f(x)} \right] }^{\up{a}}=
{\mathrm{e}}^{\up{a f(x)}}_{\dn{\dn{{1-(1-q)/a}}}} \, ,
\lb{qexpa}
\ee
\be
\frac{\dd}{\dd x} \left[ {\mathrm{e}_q}^{f(x)} \right]=
\dfrac{{\left[ {\mathrm{e}_q}^{f(x)} \right]
}^{q}}{f {\displaystyle \,'}(x)}.
\lb{dfeqfx}
\ee
These results will be useful when we explore the fact that the Tsallis
parameter can be represented in the complex plane, as discussed by
Wilk and W{\l}odarczyk in Ref.\ \cite{ww}. It is worth noting that from
Eqs.\ \eqref{qexp}, \eqref{qexpa} and \eqref{dfeqfx}, it is not obvious
that a complex $q$ can help us disclose some income distribution details
that are hidden in these functions, such as a periodic behavior, as we
shall show below.

\section{$\mathbf{q}$-parameter complexification}

\subsection{Complex heat capacity}

The fact that the nonextensivity parameter $q$ can be represented by a
complex formulation is not new. In Ref.\ \cite{bbv} the $q$-parameter
can be seen as a measure of the thermal bath heat capacity $C$, where 
\bee
\label{q-C}
C=\frac{1}{q-1}\,\,.
\eee
Moreover, such complex $C$ is well known in the literature \cite{sg}
and can be written as follows, 
\bee
\label{capacity}
C\,=\,C_\infty \,+\,\frac{C_0-C_\infty}{1+(\omega \tau)^2}\,
(1\,-\,i\omega\tau)\,\,,
\eee 
where $C_\infty$ is the heat capacity for infinitely fast degrees of
freedom (DOF), $\omega$ is the frequency, $C_0$ is the heat capacity
at equilibrium of DOF where the frequency is set to zero and $\tau$,
the time constant, is the kinetic relaxation time constant of a
certain DOF. The form of Eq.\ \eqref{capacity} suggests us that we
can write it as $C=C'\,+\,i\,C''$, where
\bee
C'=C_\infty \,+\,\frac{C_0-C_\infty}{1+(\omega\tau)^2}
\label{8.1}
\eee
and 
\bee
C''=\frac{(C_0-C_\infty)\omega\tau}{1+(\omega\tau)^2}\,\,.
\label{9.1}
\eee

Hence, let us write a complex form of $q$ as $q=q_r\,+\,i\,q_i$.
Substituting in Eq.\ \eqref{q-C} and associating with Eq.\
Eq.\ \eqref{capacity}, we have
that
\bee
\frac{q_r -1}{q_i}\,=\,\frac{C_0+C_\infty (\omega\tau)^2}
{(C_0-C_\infty)\omega\tau}\,\,,
\label{10.1}
\eee
and for $C_0<<C_\infty$ we have $(1-q_r)/q_i\,\simeq \omega\tau$,
which shows that the relationship between the real and imaginary
parts of $q$ is proportional to the frequency. More details of
this discussion can be found in Ref.\ \cite{bbv}.

\subsection{Complex income distribution means periodic behavior}

Let us start with the suggestion of Wilk and W{\l}odarczyk \cite{ww}
for the complexification of the $q$-parameter. A three-parameters
Tsallis distribution (TD) may be written as follows,
\bee
\label{1}
f(x) =C\,\mathrm{e}_q^{-x/T}= C\,\bigglb( 1-\frac{x}{mT}
\biggrb)^{-m}\,\,,
\eee
where
\bee
m=\frac{1}{(q-1)},
\label{mm}
\eee
is a real power index, $T$ is a scale parameter identified in
thermodynamical applications, in general the standard temperature,
and $C$ is a normalization constant. The proposal is to consider
$m$, or $q$, complex. Hence, the TD keeps its
main quasi-power like form, however, that brings about some
log-periodic oscillations.  As examples, one can mention that such
behavior has been encountered in many subjects, such as earthquakes
\cite{3}, chaos \cite{4}, tracers on random systems \cite{5},
random quenched and fractals \cite{6,8}, specific heat \cite{7},
clusters \cite{9}, growth models \cite{10}, stock markets \cite{11}
and, finally, non-extensive statistical mechanics log-periodic
oscillations \cite{12}.
When $m \rightarrow \infty$, namely, $q \rightarrow 1$, we have that
this power-like distribution is analogous to the standard exponential
distribution $f(x)=C\mathrm{e}^{-x/T}$.

%%%%%%%%%%%%%%
%-----------------------------

%CONFESSO QUE NAO ENTENDI ESSE ARGUMENTO

%In fact, we can consider the
%TD a generalization to real power $m$, or $q$, of well known
%distributions, where $x=T^m$, $m=(\nu+1)/2\,,\nu=\,$ integer. When
%$\nu \rightarrow \infty$ we have a Gaussian distribution; for $\nu=1$,
%a Cauchy one.

%-----------------------------
%%%%%%%%%%%%%%

The complexification proposal means turning $m$ (or $q$) complex 
in Eq.\ \eqref{1}, yielding,  
\bee
\label{2}
m=m'+im''
\eee
which means that we can also have a complex nonextensive $q$-parameter
written as below,
\bee
\label{3}
q=1\,+\,\frac 1m = q' + iq''.
\eee
It is simple to see that
\bee
\label{4}
q'=1+\frac{m'}{|m|^2} \qquad \mbox{and} \qquad q'' = -\frac{m''}{|m|^2}
\eee
where
\bee
|m|^2\,=\,m'^2\,+\, m''^2.
\label{mmod}
\eee

Hence, the goal here is to analyze the results obtained in Ref.\
\cite{smr}, where the personal income distribution of Brazil shows a
periodic behavior as a function of the income variable for each yearly
sample, in the light of the complexification of the $q$-parameter.
Therefore, we wish to describe mathematically such an oscillatory
behavior.

\subsection{Nonextensive analysis of the income distribution of Brazil}

Let us now turn our attention to the main issue of this article. As
discussed by Soares \textit{et al.\ } \cite{smr}, if the entire income
distribution range can be fitted by just one function using only two
parameters, a well-defined two-classes-base income structure implicitly
assumed when the income range is described by two distinct functions may
be open question. Hence, such income-class division could possibly be
only a result of fitting choices and not of an intrinsic feature of
societies.

The TD is known to become a pure power-law for large values of its
independent variable $x$, and exponential when $x$ tends to zero.
However, it has not the same behavior as assuming from the start a
two-classes approach to the income distribution problem because the TD
will only have power-law and exponential like behaviors as limiting
cases. Thus, a possible different behavior at the intermediate level
might not be described by neither of these functions. Hence, the TD
does not necessarily imply in two very distinct classes based on
well-defined income domain ranges, but possibly having an intermediate
income range of unknown size which might behave as neither of them. With
these points in mind, let us now proceed with the description of the
income distribution in terms of the TD and its subsequent complexification.

Let  ${\cal F}(x)$ be the \textit{cumulative distribution function} (CDF)
of the personal income, representing the proportionality, or probability,
that a person receives an income less than or equal to $x$. Let us now
denote its complementary version, the CCDF, by $F(x)$, which then
describes the probability that a person receives an income greater or
equal to $x$. It is then clear that,
\bee
\label{5}
{\cal F}(x) + F(x) = 100\,\,.
\eee
Here the maximum probability is normalized to $100{\%}$ instead of the
standard unity value. The boundary conditions involved in both functions
are ${\cal F}(x)=F(\infty)\cong 0$ and ${\cal F}(\infty)=F(0) \cong 100$.
In addition, the following properties apply to these income functions,
\bee
\frac{\dd {\cal F}(x)}{\dd x}=-\frac{\dd F(x)}{\dd x}=f(x),
\label{fff}
\eee
\bee
\label{6}
\int^{\;\infty}_0 f(x)\,\dd x = 100 \,\,,
\eee
where $f(x)$ is the PDF \cite{nm13,fnm10}.

The empirical suggestions that the income distribution can be modeled
by the TD comes from the fact that when $F(x)$ is obtained from income
data and plotted in a log-log scale, its functional curve decreases as
the income $x$ increases. In addition, the general shape of the
empirical CCDF function, particularly its ``belly'', is analogous
to the behavior of $\mathrm{e}^{-x}_{q}$ for $q > 1$ when plotted in a
log-log scale \cite[see Fig.\ 3.4 at p.\ 40 of Ref.][]{tsallis2}. Besides,
as mentioned above, the Tsallis functions behave like a power-law for high
income values, agreeing then with the Pareto power-law. Taking together
these observations into account, Ref.\ \cite{smr} advanced the
following description for the individual income distribution, 
\bee
\label{7}
F(x)=A\mathrm{e}^{-Bx}_q \,\,,
\eee
where $A$ and $B$ are positive parameters. Since we have a boundary
condition in the form of $F(0)=100$, from Eq.\ \eqref{7} it is
straightforward to conclude that $A=100$. Hence, Eq.\ \eqref{7} becomes
as below, 
\bee
\label{8}
F(x)=100\,\mathrm{e}^{-Bx}_q \,\,.
\eee
which will be the starting point for the complexification procedure.
Note that a cursory examination of Eq.\ \eqref{8} does not present any
obvious evidence of a periodic behavior, or that the complexification
of $q$ will disclose any oscillatory feature, since the $q$-parameter
is just an index. We shall show below that allowing $q$ to become
complex will expose such features.

\subsection{Periodic behavior of income distribution function}

Let us start with the definition \eqref{qexp} in order to rewrite
Eq.\ \eqref{8} as below,
\bee
\label{9}
F(x)\,=\,100\,\Big[1\,+\,(1-q)(-Bx) \Big]^{1/(1-q)} \,\,.
\eee
This equation can also be expressed as a function of $m$ using Eq.\
\eqref{mm}, 
\bee
\label{10}
F(x)\,=\,100\, \bigglb[ 1\,+\, \frac Bm x \biggrb]^{-m}
\eee
Following the complexification suggested in Eqs.\ \eqref{2} to
\eqref{mmod}, Eq.\ \eqref{10} may be written as below,
\bee
\label{D}
F(x)\,=\,100\,\Big[ a(x)\,+\,i\,b(x) \Big]^{-(m'+im'')},
\eee
where
\bee
a(x)\,=\,1\,+\,\frac{Bm'}{|m|^2}\,x\;,
\label{ax}
\eee
\bee
b(x)\,=\,-\,\frac{Bm''}{|m|^2}\,x\,\,.
\label{bx}
\eee
Expanding the terms in Eq.\ \eqref{D} results in the following expression,
\bee
\label{final}
F(x)\,=100\,{\cal A}(x)\,\mathrm{e}^{-i\,\omega (x)}\,=100\,{\cal A}(x)\,\Big[ \cos \omega(x)
\,-\,i\,\sin \omega (x) \Big],
\eee
where
\begin{empheq}[left=\empheqlbrace]{align}
{\cal A}(x)&=\exp \left[m''\varphi(x)-m'\theta(x)\right],
\label{Ax} \\[0.25cm]
\omega(x)&=m'\varphi(x)+m''\theta(x),
\label{omegax}
\end{empheq}
and
\begin{empheq}[left=\empheqlbrace]{align}
\varphi(x)&=\tan^{-1} \left[\frac{-B\,m''x}
{\left(|m|^2+B\,m'x\right)}\right],
\label{phix} \\[0.30cm]
\theta(x)&=\ln\sqrt{1+\frac{Bx}{|m|^2}\left( Bx+2m' \right)}\;.
\label{tetax}
\end{empheq}
Now, if we consider only the real part of Eq.\ \eqref{final}, the
original description of the income data in terms of the TD, as given by
Eq.\ \eqref{8}, turns out to be written as follows, 
\bee
F(x)=100\,\mathrm{e}^{-Bx}_q=100\,{\cal A}(x)\,\cos \omega(x).
\label{final2}
\eee

This result shows clearly that the periodic behavior empirically
observed by Ref.\ \cite{smr} in the income data seems to be described
by the expression above. Or that the observed oscillatory behavior can
at least be expected, since it is built in the TD. Note that the original
$q$-parameter is present in both ${\cal A}(x)$ and ${\omega}(x)$ through $m'$
and $m''$, which are the respective real and imaginary parts of $q$.

The fact that the individual income distribution function can be written
in the standard exponential form given by Eq.\ \eqref{final} shows us
that we can define a periodic function where ${\cal A}(x)$ can be seen as the
amplitude of this oscillatory motion and ${\omega}(x)$ its angular
frequency. Considering the $q$-parameter only by a real term hides this
periodic behavior. The complexification adds new components such as the
kind derived above so that it is able of revealing a periodic behavior
that does appear in the empirical curves obtained from the income
distribution data.

The issue of always considering the $q$-parameter as an entire complex
number, namely, with both real and imaginary parts, is an open question
although we believe that it depends on the problem we are dealing with.
We can imagine an analogy of this feature with the dual characteristic
of the electron in quantum physics, since it has a wave-particle
duality behavioral feature that depends on the experience that we are
analyzing. In this way we could talk about a $q$-duality, which would
deserves alternative interpretations and could  certainly be a target of
further research.

\section{conclusions}

Since the Pareto's work the study of the income distribution of the whole
population has been a target for the economic experts, and for some time
now, to the econophysics literature. Despite the relative great number of
parameters used in several functions to fit the income data, from three to
five in some cases, some mathematical approaches have thrived in the
description of the entire data range. In this work we have used Tsallis'
nonextensive point of view with a complexification mode where its 
$q$-parameter is represented by a complex number. The objective here was
to use this complexification in order to justify analytically the results
obtained in Ref.\ \cite{smr}, which show a periodic behavior in the income
data.

As shown above, in doing this we, however, increased the number of
parameters required to fit the data, from the original two to three.
Although increasing the number of unknown parameters in a problem is a
bad procedure, in our case the complexification has disclosed an extra
behavior of the income distribution in the form of a periodic motion in
the income distribution, motion which was already present in several
studies of income distribution, but was only explicitly acknowledged by
Soares {\it et al.\ } \cite{smr}. By interpreting this oscillatory
motion of the income distribution as a typical periodic motion allowed
us to define commonly used oscillatory parameters such as amplitude and
angular frequency.

So, from the results obtained here we can say that ``the highs and lows''
of the yearly income distribution samples are an expected behavior,
confirmed using the complex form of the Tsallis $q$-parameter. However,
it not all clear that this periodic oscillation could lead us to the
understanding of the real nature of the $q$-parameter, namely, if it is
complex or not depends on the features of the problem we are dealing
with.

Finally, this work shows us a glimpse of the task ahead as far as the
empirical studies of income distribution are concerned. The data give
us $F(x), x, B, q$ and $m$, so the empirical task is to determine both
$m'$ and $m''$ from the data in order to end up with only three parameters
required to characterize the whole yearly distribution, oscillatory
feature included, namely $B$, $m'$ and $m''$. 

\section{Acknowledgments}

\ni E.M.C.A. thanks the Brazilian Federal Agency CNPq for partial financial
support, Grants numbers 302155/2015-5 and 442369/2014-0 and the hospitality
of Theoretical Physics Department at Federal University of Rio de Janeiro
-- UFRJ where part of this work was carried out. M.B.R. acknowledges partial
financial support from the Rio de Janeiro State Scientific Funding Agency
FAPERJ.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% References of numbers
\begin{thebibliography}{99}
\bibitem{pareto} V.\ Pareto, \textit{``Cours d'\'{E}conomie Politique''},
	Lausanne, 1897.
\bibitem{mandelbrot} B.\ B.\ Mandelbrot, \textit{``The Fractal Geometry of
	Nature"}, Freeman, San Francisco, 1982.
\bibitem{k80} N.\ C.\ Kakwani, \textit{``Income Inequality and Poverty''},
	Oxford University Press, 1980.
\bibitem{nm09} N.\ J.\ Moura Jr., M.B.\ Ribeiro, \textit{``Evidence for
	the Gompertz Curve in the Income Distribution of Brazil
        1978-2005''}, Eur.\ Phys.\ J.\ B, 67 (2009) 101-120, arXiv:0812.2664.
\bibitem{fnm10} F.\ Chami Figueira, N.J.\ Moura Jr., M.B.\ Ribeiro,
	\textit{``The Gompertz-Pareto Income Distribution''}, Physica A,
	390 (2011) 689-698, arXiv:1010.1994. 
\bibitem{dy01} A.\ Dr\u{a}gulescu, V.M.\ Yakovenko, \textit{``Evidence
	for the Exponential Distribution of Income in the USA''}, Eur.\
	Phys.\ J.\ B, 20 (2001) 585, arXiv:cond-mat/0008305. 
\bibitem{ferrero2} J.\ C.\ Ferrero, \textit{``The Statistical Distribution
	of Money and the Rate of Money Transference''}, Physica A, 341
	(2004) 575-585.
\bibitem{s05} A.\ Christian Silva, \textit{``Applications of Physics to
	Finance and Economics: Returns, Trading Activity and Income''},
	PhD thesis, University of Maryland, 2005, arXiv:physics/0507022.
\bibitem{yr09} V.\ M.\ Yakovenko, J.B.\ Rosser, \textit{``Colloquium:
	Statistical Mechanics of Money, Wealth, and Income''}, Rev.\ Mod.\
	Phys., 81 (2009) 1703-1725, arXiv:0905.1518.
\bibitem{crbh08} R.\ Coelho, P.\ Richmond, J.\ Barry, S.\ Hutzler,
	\textit{``Double Power Laws in Income and Wealth Distributions''},
	Physica A, 387 (2008) 3847-3851, arXiv:0710.0917.
\bibitem{by10} A.\ Banerjee, V. M.\ Yakovenko, \textit{``Universal
	Patterns of Inequality''}, New J.\ Phys., 12 (2010) 075032,
	arXiv:0912.4898.
\bibitem{nm13} N.\ J.\ Moura Jr., M. B.\ Ribeiro, \textit{``Testing the
	Goodwin Growth-Cycle Macroeconomic Dynamics in Brazil''}, Physica
	A, 392 (2013) 2088-2103, arXiv:1301.1090. 
\bibitem{cyc05} Econophys-Kolkata I Workshop, \textit{``Econophysics of
	Wealth Distributions''}, A.\ Chatterjee, S.\ Yarlagadda, B.\ K.\
	Chakrabarti (Eds.), Springer, 2005.
\bibitem{borges} E. P.\ Borges, \textit{``Empirical nonextensive laws
	for the county distribution of total personal income and
	gross domestic product''}, Physica A, 334 (2004) 255.
\bibitem{c5} B.\ K.\ Chakrabarti, A.\ Chakraborti, S.R.\ Chakravarty,
	A.\ Chatterjee, \textit{``Econophysics of Income and Wealth
	Distributions''}, Cambridge University Press, 2013.
\bibitem{ferrero1} J.\ C.\ Ferrero, \textit{``The Monomodal, Polymodal,
	Equilibrium and Nonequilibrium Distribution of Money''}, In
	\cite{cyc05}, pp.\ 159, (2005).
\bibitem{tsallis} C.\ Tsallis, {\it ``Possible generalization of Boltzmann-Gibbs 
	statistics"}, J.\ Statist. Phys. 52 (1988) 479; {\it ``Nonadditive entropy: 
	The concept and its use"}, Eur.\
	Phys.\ J.\ A 40 (2009) 257; \\ V.\ Schw\"ammle and C. Tsallis, {\it 
	``Two-parameter generalization of the logarithm and exponential functions 
	and Boltzmann-Gibbs-Shannon entropy"}, 
	J.\ Math.\ Phys.\ 48 (2007) 113301.
\bibitem{smr} A.\ D.\ Soares, N.\ J.\ Moura Jr.\ and M.\ B.\ Ribeiro, {\it 
	``Tsallis statistics in the income distribution of Brazil"}, 
	Chaos, Solitons and Fractas 88 (2016) 158.
\bibitem{ww} G.\ Wilk and Z.\ W{\l}odraczyk, {\it ``Tsallis distribution 
	with complex nonextensivity parameter $q$"}, Physica A 413 (2014) 53.
%\bibitem{ww} G.\ Wilk, Z.\ W{\l}odarczyk, \textit{``Tsallis Distribution
%        Decorated with Log-Periodic Oscillation''}, Entropy 17 (2015)
%        384-400, arXiv:1501.02608v1.
\bibitem{tsallis1} C.\ Tsallis, \textit{``What are the Numbers that
	Experiments Provide?''}, Qu\'{\i}mica Nova, 17 (1994) 468-471.
\bibitem{tsallis2} C.\ Tsallis, \textit{``Introduction to Nonextensive
	Statistical Mechanics''}, Springer, 2009.
\bibitem{cgk07}F.\ Clementi, M.\ Gallegati, G.\ Kaniadakis,
	\textit{``$\kappa$-Generalised Statistics in Personal Income
	Distribution''} Eur.\ Phys.\ J.\ B, 57 (2007) 187-193,
	arXiv:physics/0607293.
\bibitem{cmgk08}F.\ Clementi, T.\ Di Matteo, M.\ Gallegati, G.\
	Kaniadakis, \textit{``The $\kappa$-Generalised Distribution: a
	New Descriptive Model for the Size Distribution of Incomes''},
	Physica A, 387 (2008) 3201-3208, arXiv:0710.3645.
\bibitem{cgk09}F.\ Clementi, M.\ Gallegati, G.\ Kaniadakis,
	\textit{``A $\kappa$-Generalized Statistical Mechanics Approach
	to Income Analysis''}, J.\ Stat.\ Mech., February (2009) P02037,
	arXiv:0902.0075.
\bibitem{cgk12}F.\ Clementi, M.\ Gallegati, G.\ Kaniadakis, \textit{``A
	generalized statistical model for the size distribution of
        wealth''}, J.\ Stat.\ Mech., December (2012) P12006,
	arXiv:1209.4787.
\bibitem{yamano} T.\ Yamano, \textit{``Some properties of $q$-logarithm
	and $q$-exponential functions in Tsallis statistics''}, Physica A
	305 (2002) 486-496.
\bibitem{bbv} T.\ S.\ Bir\'o, G.\ G.\ Barnaf\"oldi and P.\ V\'an, {\it 
	``Quark-gluon plasma connected to finite heat bath"}, Eur.\
	Phys.\ J.\ A 49 (2013) 110.
\bibitem{sg} J.\ E.\ K.\ Schawe, {\it ``A comparison of different evaluation 
	methods in modulated temperature DSC"}, Thermochim.\ Acta 260 (1995) 
	1;\\ J.-L.\
	Garden,{\it ``Simple derivation of the frequency dependent complex 
	heat capacity"},  Thermochim.\ Acta 460 (1995) 85.
\bibitem{3} Y.\ Huang, H.\ Saleur, C.\ Sammis and D.\ Sornette, {\it ``Precursors, 
	aftershocks, criticality and self-organized criticality"}, 
        Europhys.\ Lett.\ 41 (1998) 43;\\ H.\ Saleur, C.\ G.\ Sammis and D.\ 
	Sornette, {\it ``Discrete scale invariance, complex fractal dimensions, 
	and log-periodic fluctuations in seismicity"}, 
	J.\ Geophys.\ Res.\ 101 (1996) 17661.
\bibitem{4} A.\ Krawiecki, K.\ Kacperski, S.\ Matyjaskiewicz and J.\ A.\
	Holyst, {\it ``Log-periodic oscillations and noise-free stochastic 
	multiresonance due to self-similarity of fractals"}, Chaos, Solitons and 
	Fractals 18 (2003) 89.
\bibitem{5} J.\ Bernasconi and W.\ R.\ Schneider, {\it ``Diffusion in random 
	one-dimensional systems"}, J.\ Stat.\ Phys.\ 30
	(1983) 355; \\ D.\ Stauffer and D.\ Sornette, {\it ``Log-periodic oscillations 
	for biased diffusion on random lattice"}, Physica A 252 (1998)
	271; \\ D.\ Stauffer, {\it ``New simulations on old biased diffusion"}, 
	Physica A 266 (1999) 35.
\bibitem{6} B.\ Kutnjak-Urbanc, S.\ Zapperi, S.\ Milosevic and H.\ E.\
	Stanley, {\it ``Sandpile model on the Sierpinski gasket fractal"}, 
	Phys.\ Rev.\ E 54 (1996) 272; 
	R.\ F.\ S.\ Andrade, {\it ``Detailed characterization of log-periodic oscillations 
	for an aperiodic Ising model"}, Phys.\
	Rev.\ E 61 (2000) 7196; 
	M.\ A.\ Bab, G.\ Fabricius and E.\ V.\
	Albano, {\it ``Critical behavior of an Ising system on the Sierpinski carpet: 
	A short-time dynamics study"}, Phys.\ Rev.\ E 71 (2005) 036139; 
	H.\ Saleur and D.\ Sornette, {\it ``Complex Exponents and Log-Periodic Corrections 
	in Frustrated Systems"},  
	J.\ Physique I 6 (1996) 327.
\bibitem{8} C.\ Tsallis, L.\ R.\ da Silva, R.\ S.\ Mendes, R.\ O.\ Vallejos
	and A.\ M.\ Mariz, {\it ``Specific heat anomalies associated with Cantor-set 
	energy spectra"}, Phys.\ Rev.\ E 56 (1997) R4922.
\bibitem{7} R.\ O.\ Vallejos, R.\ S.\ Mendes, L.\ R.\ da Silva and C.\
	Tsallis, {\it ``Connection between energy spectrum, self-similarity, and 
	specific heat log-periodicity"}, Phys.\ Rev.\ E 58 (1998) 1346. 
\bibitem{9} D.\ Sornette, A.\ Johansen, A.\ Arneodo, J.-F.\ Muzy and H.\
	Saleur, {\it ``Complex Fractal Dimensions Describe the Hierarchical Structure
	of Diffusion-Limited-Aggregate Clusters"}, Phys.\ Rev.\ Lett.\ 76 (1996) 251.
\bibitem{10} Y.\ Huang, G.\ Ouillon, H.\ Saleur and D.\ Sornette, {\it ``Spontaneous 
	generation of discrete scale invariance in growth models"}, Phys.\
	Rev.\ E 55 (1997) 6433.
\bibitem{11} D.\ Sornette, A.\ Johansen and J.-P.\ Bouchaud, {\it ``Stock Market Crashes, 
	Precursors and Replicas"},
	J.\ Physique I 6 (1996) 167; \\ N.\ Vanderwalle, P.\ Boveroux, A.\ Minguet and
	M.\ Ausloos, {\it ``The crash of October 1987 seen as a phase transition: amplitude 
	and universality"}, Physica A 255 (1998) 201; \\ N.\ Vanderwalle and M.\
	Ausloos, {\it ``How the financial crash of October 1997 could have been predicted"}, 
	Eur.\ J.\ Phys.\ B 4 (1998) 139; \\ N.\ Vanderwalle, M.\
	Ausloos, P.\ Boveroux and A.\ Minguet, {\it ``Visualizing the log-periodic pattern 
	before crashes"}, Eur.\ J.\ Phys.\ B 9 (1999)
	355; \\ J.\ H.\ Wosnitza and J.\ Leker, {\it ``Can log-periodic power law structures 
	arise from random fluctuations?"}, Physica A 401 (2014) 228.
\bibitem{12} F.\ A.\ B.\ F.\ de Moura, U.\ Tirnakli and M.\ L.\ Lyra, {\it ``Convergence 
	to the critical attractor of dissipative maps: Log-periodic oscillations, fractality, 
	and nonextensivity"}, 
	Phys.\ Rev.\ E 62 (2000) 6361.
%\bibitem{mr} F. Chami Figueira, N. J. Moura Jr and M. B. Ribeiro, {\it ``The Gompertz-Pareto Income Distribution"}, 
%	Physica A, 390 (2011) 689, arXiv:1010.1994. 
	%{\it ``Testing the Goodwin growth-cycle macroeconomic dynamics in 
	%Brazil"}, Physica A 392 (2013) 208.
%\bibitem{rrr} F.\ M.\ Ramos, C.\ Rodrigues Neto, R.\ R.\ Rosa,
%	{\it ``Generalized thermostatistical description of intermittency
%	and non-extensivity in turbulence and Dnancial markets,"} e-print,
%	2000, arXiv:cond-mat/0010435.
%\bibitem{tab} C.\ Tsallis, C.\ Anteneodo, L.\ Borland, R.\ Osorio,
%	Physica A 324 (2003) 89.
%\bibitem{ferrero3} J. C.\ Ferrero, \textit{``An Statistical Analysis of
%       Stratification and Inequality in the Income Distribution''},
%       Eur.\ Phys.\ J.\ B 80 (2011) 255.
%\bibitem{tsallis-book}  C.\ Tsallis, \textit{``Introduction to
%        Nonextensive    Statistical Mechanics,"} Springer, 2009.
%\bibitem{kls04} G.\ Kaniadakis, M.\ Lissia, A.M.\ Scarfone,
%        \textit{``Deformed logarithms and entropies''}, Physica A 340 (2004)
%        41-49
%\bibitem{huang} Y.\ Huang, H.\ Saleur, C.\ Sammis, D.\ Sornette,
%        \textit{``Precursors, aftershocks, criticality and self-organized
%        criticality''}, Europhys.\ Lett.\ 41 (1998) 43-48
%\bibitem{sornette} D.\ Sornette, A.\ Johansen, J.-P.\ Bouchaud,
%        \textit{``Stock market crashes, precursors and replicas''},
%        J.\ Phys.\ I France 6 (1996) 167-175, arXiv:cond-mat/9510036v1 
%\bibitem{vande1} N.\ Vandewalle, Ph.\ Boveroux, A.\ Minguet, M.\
%        Ausloos, \textit{``The crash of October 1987 seen as a phase
%        transition: amplitude and universality''}, Physica A 255 (1998)
%        201-210
%\bibitem{vande2} N.\ Vandewalle, M.\ Ausloos, Ph.\ Boveroux, A.\ Minguet,
%       \textit{``How the financial crash of October 1987 could have been
%       predicted''}, Eur.\ Phys. J.\ B 4 (1998) 139-141
%\bibitem{wosnitza} J.H.\ Wosnitza, J.\ Leker, \textit{``Can log-periodic
%       power-law structures arise from random fluctuations?''}, 
%       Physica A 401 (2014) 228-250
\end{thebibliography}
\end{document}

1706.10141.extracted_bib
{pareto V.\ Pareto, ``Cours d'\'{Economie Politique'', Lausanne, 1897.
1706.10141.extracted_bib
{mandelbrot B.\ B.\ Mandelbrot, ``The Fractal Geometry of Nature", Freeman, San Francisco, 1982.
1706.10141.extracted_bib
{k80 N.\ C.\ Kakwani, ``Income Inequality and Poverty'', Oxford University Press, 1980.
1706.10141.extracted_bib
{nm09 N.\ J.\ Moura Jr., M.B.\ Ribeiro, ``Evidence for the Gompertz Curve in the Income Distribution of Brazil 1978-2005'', Eur.\ Phys.\ J.\ B, 67 (2009) 101-120, arXiv:0812.2664.
1706.10141.extracted_bib
{fnm10 F.\ Chami Figueira, N.J.\ Moura Jr., M.B.\ Ribeiro, ``The Gompertz-Pareto Income Distribution'', Physica A, 390 (2011) 689-698, arXiv:1010.1994.
1706.10141.extracted_bib
{dy01 A.\ Dr\u{agulescu, V.M.\ Yakovenko, ``Evidence for the Exponential Distribution of Income in the USA'', Eur.\ Phys.\ J.\ B, 20 (2001) 585, arXiv:cond-mat/0008305.
1706.10141.extracted_bib
{ferrero2 J.\ C.\ Ferrero, ``The Statistical Distribution of Money and the Rate of Money Transference'', Physica A, 341 (2004) 575-585.
1706.10141.extracted_bib
{s05 A.\ Christian Silva, ``Applications of Physics to Finance and Economics: Returns, Trading Activity and Income'', PhD thesis, University of Maryland, 2005, arXiv:physics/0507022.
1706.10141.extracted_bib
{yr09 V.\ M.\ Yakovenko, J.B.\ Rosser, ``Colloquium: Statistical Mechanics of Money, Wealth, and Income'', Rev.\ Mod.\ Phys., 81 (2009) 1703-1725, arXiv:0905.1518.
1706.10141.extracted_bib
{crbh08 R.\ Coelho, P.\ Richmond, J.\ Barry, S.\ Hutzler, ``Double Power Laws in Income and Wealth Distributions'', Physica A, 387 (2008) 3847-3851, arXiv:0710.0917.
1706.10141.extracted_bib
{by10 A.\ Banerjee, V. M.\ Yakovenko, ``Universal Patterns of Inequality'', New J.\ Phys., 12 (2010) 075032, arXiv:0912.4898.
1706.10141.extracted_bib
{nm13 N.\ J.\ Moura Jr., M. B.\ Ribeiro, ``Testing the Goodwin Growth-Cycle Macroeconomic Dynamics in Brazil'', Physica A, 392 (2013) 2088-2103, arXiv:1301.1090.
1706.10141.extracted_bib
{cyc05 Econophys-Kolkata I Workshop, ``Econophysics of Wealth Distributions'', A.\ Chatterjee, S.\ Yarlagadda, B.\ K.\ Chakrabarti (Eds.), Springer, 2005.
1706.10141.extracted_bib
{borges E. P.\ Borges, ``Empirical nonextensive laws for the county distribution of total personal income and gross domestic product'', Physica A, 334 (2004) 255.
1706.10141.extracted_bib
{c5 B.\ K.\ Chakrabarti, A.\ Chakraborti, S.R.\ Chakravarty, A.\ Chatterjee, ``Econophysics of Income and Wealth Distributions'', Cambridge University Press, 2013.
1706.10141.extracted_bib
{ferrero1 J.\ C.\ Ferrero, ``The Monomodal, Polymodal, Equilibrium and Nonequilibrium Distribution of Money'', In \cite{cyc05, pp.\ 159, (2005).
1706.10141.extracted_bib
{tsallis C.\ Tsallis, {``Possible generalization of Boltzmann-Gibbs statistics", J.\ Statist. Phys. 52 (1988) 479; {``Nonadditive entropy: The concept and its use", Eur.\ Phys.\ J.\ A 40 (2009) 257; \\ V.\ Schw\"ammle and C. Tsallis, {``Two-parameter generalization of the logarithm and exponential functions and Boltzmann-Gibbs-Shannon entropy", J.\ Math.\ Phys.\ 48 (2007) 113301.
1706.10141.extracted_bib
{smr A.\ D.\ Soares, N.\ J.\ Moura Jr.\ and M.\ B.\ Ribeiro, {``Tsallis statistics in the income distribution of Brazil", Chaos, Solitons and Fractas 88 (2016) 158.
1706.10141.extracted_bib
{ww G.\ Wilk and Z.\ W{\lodraczyk, {``Tsallis distribution with complex nonextensivity parameter $q$", Physica A 413 (2014) 53. %
1706.10141.extracted_bib
{ww G.\ Wilk, Z.\ W{\lodarczyk, ``Tsallis Distribution % Decorated with Log-Periodic Oscillation'', Entropy 17 (2015) % 384-400, arXiv:1501.02608v1.
1706.10141.extracted_bib
{tsallis1 C.\ Tsallis, ``What are the Numbers that Experiments Provide?'', Qu\'{\imica Nova, 17 (1994) 468-471.
1706.10141.extracted_bib
{tsallis2 C.\ Tsallis, ``Introduction to Nonextensive Statistical Mechanics'', Springer, 2009.
1706.10141.extracted_bib
{cgk07F.\ Clementi, M.\ Gallegati, G.\ Kaniadakis, ``$\kappa$-Generalised Statistics in Personal Income Distribution'' Eur.\ Phys.\ J.\ B, 57 (2007) 187-193, arXiv:physics/0607293.
1706.10141.extracted_bib
{cmgk08F.\ Clementi, T.\ Di Matteo, M.\ Gallegati, G.\ Kaniadakis, ``The $\kappa$-Generalised Distribution: a New Descriptive Model for the Size Distribution of Incomes'', Physica A, 387 (2008) 3201-3208, arXiv:0710.3645.
1706.10141.extracted_bib
{cgk09F.\ Clementi, M.\ Gallegati, G.\ Kaniadakis, ``A $\kappa$-Generalized Statistical Mechanics Approach to Income Analysis'', J.\ Stat.\ Mech., February (2009) P02037, arXiv:0902.0075.
1706.10141.extracted_bib
{cgk12F.\ Clementi, M.\ Gallegati, G.\ Kaniadakis, ``A generalized statistical model for the size distribution of wealth'', J.\ Stat.\ Mech., December (2012) P12006, arXiv:1209.4787.
1706.10141.extracted_bib
{yamano T.\ Yamano, ``Some properties of $q$-logarithm and $q$-exponential functions in Tsallis statistics'', Physica A 305 (2002) 486-496.
1706.10141.extracted_bib
{bbv T.\ S.\ Bir\'o, G.\ G.\ Barnaf\"oldi and P.\ V\'an, {``Quark-gluon plasma connected to finite heat bath", Eur.\ Phys.\ J.\ A 49 (2013) 110.
1706.10141.extracted_bib
{sg J.\ E.\ K.\ Schawe, {``A comparison of different evaluation methods in modulated temperature DSC", Thermochim.\ Acta 260 (1995) 1;\\ J.-L.\ Garden,{``Simple derivation of the frequency dependent complex heat capacity", Thermochim.\ Acta 460 (1995) 85.
1706.10141.extracted_bib
{3 Y.\ Huang, H.\ Saleur, C.\ Sammis and D.\ Sornette, {``Precursors, aftershocks, criticality and self-organized criticality", Europhys.\ Lett.\ 41 (1998) 43;\\ H.\ Saleur, C.\ G.\ Sammis and D.\ Sornette, {``Discrete scale invariance, complex fractal dimensions, and log-periodic fluctuations in seismicity", J.\ Geophys.\ Res.\ 101 (1996) 17661.
1706.10141.extracted_bib
{4 A.\ Krawiecki, K.\ Kacperski, S.\ Matyjaskiewicz and J.\ A.\ Holyst, {``Log-periodic oscillations and noise-free stochastic multiresonance due to self-similarity of fractals", Chaos, Solitons and Fractals 18 (2003) 89.
1706.10141.extracted_bib
{5 J.\ Bernasconi and W.\ R.\ Schneider, {``Diffusion in random one-dimensional systems", J.\ Stat.\ Phys.\ 30 (1983) 355; \\ D.\ Stauffer and D.\ Sornette, {``Log-periodic oscillations for biased diffusion on random lattice", Physica A 252 (1998) 271; \\ D.\ Stauffer, {``New simulations on old biased diffusion", Physica A 266 (1999) 35.
1706.10141.extracted_bib
{6 B.\ Kutnjak-Urbanc, S.\ Zapperi, S.\ Milosevic and H.\ E.\ Stanley, {``Sandpile model on the Sierpinski gasket fractal", Phys.\ Rev.\ E 54 (1996) 272; R.\ F.\ S.\ Andrade, {``Detailed characterization of log-periodic oscillations for an aperiodic Ising model", Phys.\ Rev.\ E 61 (2000) 7196; M.\ A.\ Bab, G.\ Fabricius and E.\ V.\ Albano, {``Critical behavior of an Ising system on the Sierpinski carpet: A short-time dynamics study", Phys.\ Rev.\ E 71 (2005) 036139; H.\ Saleur and D.\ Sornette, {``Complex Exponents and Log-Periodic Corrections in Frustrated Systems", J.\ Physique I 6 (1996) 327.
1706.10141.extracted_bib
{8 C.\ Tsallis, L.\ R.\ da Silva, R.\ S.\ Mendes, R.\ O.\ Vallejos and A.\ M.\ Mariz, {``Specific heat anomalies associated with Cantor-set energy spectra", Phys.\ Rev.\ E 56 (1997) R4922.
1706.10141.extracted_bib
{7 R.\ O.\ Vallejos, R.\ S.\ Mendes, L.\ R.\ da Silva and C.\ Tsallis, {``Connection between energy spectrum, self-similarity, and specific heat log-periodicity", Phys.\ Rev.\ E 58 (1998) 1346.
1706.10141.extracted_bib
{9 D.\ Sornette, A.\ Johansen, A.\ Arneodo, J.-F.\ Muzy and H.\ Saleur, {``Complex Fractal Dimensions Describe the Hierarchical Structure of Diffusion-Limited-Aggregate Clusters", Phys.\ Rev.\ Lett.\ 76 (1996) 251.
1706.10141.extracted_bib
{10 Y.\ Huang, G.\ Ouillon, H.\ Saleur and D.\ Sornette, {``Spontaneous generation of discrete scale invariance in growth models", Phys.\ Rev.\ E 55 (1997) 6433.
1706.10141.extracted_bib
{11 D.\ Sornette, A.\ Johansen and J.-P.\ Bouchaud, {``Stock Market Crashes, Precursors and Replicas", J.\ Physique I 6 (1996) 167; \\ N.\ Vanderwalle, P.\ Boveroux, A.\ Minguet and M.\ Ausloos, {``The crash of October 1987 seen as a phase transition: amplitude and universality", Physica A 255 (1998) 201; \\ N.\ Vanderwalle and M.\ Ausloos, {``How the financial crash of October 1997 could have been predicted", Eur.\ J.\ Phys.\ B 4 (1998) 139; \\ N.\ Vanderwalle, M.\ Ausloos, P.\ Boveroux and A.\ Minguet, {``Visualizing the log-periodic pattern before crashes", Eur.\ J.\ Phys.\ B 9 (1999) 355; \\ J.\ H.\ Wosnitza and J.\ Leker, {``Can log-periodic power law structures arise from random fluctuations?", Physica A 401 (2014) 228.
1706.10141.extracted_bib
{12 F.\ A.\ B.\ F.\ de Moura, U.\ Tirnakli and M.\ L.\ Lyra, {``Convergence to the critical attractor of dissipative maps: Log-periodic oscillations, fractality, and nonextensivity", Phys.\ Rev.\ E 62 (2000) 6361. %
1706.10141.extracted_bib
{mr F. Chami Figueira, N. J. Moura Jr and M. B. Ribeiro, {``The Gompertz-Pareto Income Distribution", % Physica A, 390 (2011) 689, arXiv:1010.1994. %{``Testing the Goodwin growth-cycle macroeconomic dynamics in %Brazil", Physica A 392 (2013) 208. %
1706.10141.extracted_bib
{rrr F.\ M.\ Ramos, C.\ Rodrigues Neto, R.\ R.\ Rosa, % {``Generalized thermostatistical description of intermittency % and non-extensivity in turbulence and Dnancial markets," e-print, % 2000, arXiv:cond-mat/0010435. %
1706.10141.extracted_bib
{tab C.\ Tsallis, C.\ Anteneodo, L.\ Borland, R.\ Osorio, % Physica A 324 (2003) 89. %
1706.10141.extracted_bib
{ferrero3 J. C.\ Ferrero, ``An Statistical Analysis of % Stratification and Inequality in the Income Distribution'', % Eur.\ Phys.\ J.\ B 80 (2011) 255. %
1706.10141.extracted_bib
{tsallis-book C.\ Tsallis, ``Introduction to % Nonextensive Statistical Mechanics," Springer, 2009. %
1706.10141.extracted_bib
{kls04 G.\ Kaniadakis, M.\ Lissia, A.M.\ Scarfone, % ``Deformed logarithms and entropies'', Physica A 340 (2004) % 41-49 %
1706.10141.extracted_bib
{huang Y.\ Huang, H.\ Saleur, C.\ Sammis, D.\ Sornette, % ``Precursors, aftershocks, criticality and self-organized % criticality'', Europhys.\ Lett.\ 41 (1998) 43-48 %
1706.10141.extracted_bib
{sornette D.\ Sornette, A.\ Johansen, J.-P.\ Bouchaud, % ``Stock market crashes, precursors and replicas'', % J.\ Phys.\ I France 6 (1996) 167-175, arXiv:cond-mat/9510036v1 %
1706.10141.extracted_bib
{vande1 N.\ Vandewalle, Ph.\ Boveroux, A.\ Minguet, M.\ % Ausloos, ``The crash of October 1987 seen as a phase % transition: amplitude and universality'', Physica A 255 (1998) % 201-210 %
1706.10141.extracted_bib
{vande2 N.\ Vandewalle, M.\ Ausloos, Ph.\ Boveroux, A.\ Minguet, % ``How the financial crash of October 1987 could have been % predicted'', Eur.\ Phys. J.\ B 4 (1998) 139-141 %
1706.10141.extracted_bib
{wosnitza J.H.\ Wosnitza, J.\ Leker, ``Can log-periodic % power-law structures arise from random fluctuations?'', % Physica A 401 (2014) 228-250