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This paper discusses if large scale galaxy distribution samples containing almost one million objects can be characterized as fractal systems. The analysis performed by Teles et al. (2021; arXiv:2012.07164) on the UltraVISTA DR1 survey is extended here to the SPLASH and COSMOS2015 catalogs, hence adding 750k new galaxies with measured redshifts to the studied samples. The standard $\Lambda$CDM cosmology having $H_0=(70\pm5)$ km/s/Mpc and number density tools required for describing these galaxy distributions as single fractal systems with dimension $D$ are adopted. We use the luminosity distance $d_L$, redshift distance $d_z$ and galaxy area distance (transverse comoving distance) $d_G$ as relativistic distance definitions to derive galaxy number densities in the redshift interval $0.1\le z\le4$ at volume limited subsamples defined by absolute magnitudes in the K-band. Similar to the findings of Teles et al. (2021; arXiv:2012.07164), the results show two consecutive redshift scales where galaxy distribution data behave as single fractal structures. For $z<1$ we found $D=1.00\pm0.12$ for the SPLASH galaxies, and $D=1,39\pm0.19$ for the COSMOS2015. For $1\le z\le4$ we respectively found $D=0.83^{+0.36}_{-0.37}$ and $D=0.54^{+0.27}_{-0.26}$. These results were verified to be robust under the assumed Hubble constant uncertainty. Calculations considering blue and red galaxies subsamples in both surveys showed that the fractal dimensions of blue galaxies as basically unchanged, but the ones for the red galaxies changed mostly to smaller values, meaning that $D$ may be seen as a more intrinsic property of the distribution of objects in the Universe, therefore allowing for the fractal dimension to be used as a tool to study different populations of galaxies. All results confirm the decades old theoretical prediction of a decrease in the fractal dimension for $z>1$., Comment: 14 pages, 15 figures, 43 graphs, 6 tables. LaTeX. Accepted for publication in the "European Physical Journal C"

Alcubierre proposed in 1994 that the well known special relativistic limitation that particles cannot travel with velocities bigger than the light speed can be bypassed when such trips are considered globally within specific general relativistic frameworks. Although initial results indicated this scenario as being unphysical, since it would seem to require negative mass-energy density, recent theoretical analyses suggest that such an unphysical situation may not always be necessarily true. In this paper we review some solutions of the Einstein equations using the original Alcubierre warp drive metric endowed with various matter-energy sources, namely dust, perfect fluid, anisotropic fluid, and perfect fluid with a cosmological constant. A connection of some of these solutions featuring shock waves described by the Burgers equation is also shown., Comment: 16 pages, 4 tables, LaTeX. To appear in the Proceedings of the 16th Marcel Grossmann Meeting (July 5-10, 2021)

The Alcubierre metric describes a spacetime geometry that allows a massive particle inside a spacetime distortion, called warp bubble, to travel with superluminal global velocities. In this work we advance solutions of the Einstein equations with the cosmological constant for the Alcubierre warp drive metric having the perfect fluid as source. We also consider the particular dust case with the cosmological constant, which generalizes our previous dust solution (arXiv:2008.06560) and led to vacuum solutions connecting the warp drive with shock waves via the Burgers equation, as well as our perfect fluid solution without the cosmological constant (arXiv:2101.11467). All energy conditions are also analyzed. The results show that the shift vector in the direction of the warp bubble motion creates a coupling in the Einstein equations that requires off-diagonal terms in the energy-momentum source. Therefore, it seems that to achieve superluminal speeds by means of the Alcubierre warp drive spacetime geometry one may require a complex configuration and distribution of energy, matter and momentum as source in order to produce a warp drive bubble. In addition, warp speeds seem to require more complex forms of matter than dust for stable solutions and that negative matter may not be a strict requirement to achieve global superluminal speeds., Comment: 15 pages, LaTeX. Accepted for publication in The European Physical Journal Plus. arXiv admin note: text overlap with arXiv:2101.11467

We study the cosmological power spectra (PS) of the differential and integral galaxy volume number densities $\gamma_i$ and $\gamma_i^{*}$, constructed with the cosmological distances $d_i$ $(i=A,G,L,Z)$, where $d_A$ is the angular diameter distance, $d_G$ is the galaxy area distance, $d_L$ is the luminosity distance and $d_z$ is the redshift distance. Theoretical and observational quantities were obtained in the FLRW spacetime with a non-vanishing $\Lambda$. The radial correlation $\Xi_i$, as defined in the context of these densities, is discussed in the wave number domain. All observational quantities were computed using luminosity function (LF) data obtained from the FORS Deep Field galaxy survey. The theoretical and observational PS of $\gamma_i$, $\gamma_i^{\ast}$, $\Xi_i$ and $\gamma_i / \gamma_i^\ast$ were calculated by performing Fourier transforms on these densities previously derived by Iribarrem et al. (2012) from the observed values $\gamma_{obs}$ and ${\gamma^\ast}_{obs}$ obtained using the galactic absolute magnitudes and galaxy LF Schechter's parameters presented in Gabasch et al. (2004, 2006) in the range $0.5 \le z \le5.0$. The results show similar behavior of the PS obtained from $\gamma$ and $\gamma^{\ast}$ using $d_L$, $d_z$ and $d_G$ as distance measures. The PS of the densities defined with $d_A$ have a different and inconclusive behavior, as this cosmological distance reaches a maximum at $z\approx 1.6$ in the adopted cosmology. For the other distances, our results suggest that the PS of ${\gamma_i}_{obs}$, ${\gamma^\ast_i}_{obs}$ and ${\gamma_i / \gamma^{\ast}_i}_{obs}$ have a general behavior approximately similar to the PS obtained with the galaxy two-point correlation function and, by being sample size independent, they may be considered as alternative analytical tools to study the galaxy distribution., Comment: In memory of William R. Stoeger (1943-2014). LaTeX. 15 pages, 15 figures, 35 graphs. Additional comments at the end of Secs. 2.3, 3.2, 4, 5 and caption of Fig. 2. Accepted for publication in "The European Physical Journal Plus"

A spatially homogeneous Bianchi type VI0 model containing a viscous fluid in the presence of an axial magnetic field has been studied. A barotropic equation of state together with a pair of linear relations among the square root of matter density, shear scalar, and expansion scalar have been assumed. Solutions are obtained in the presence of a magnetic field, only in two special cases, which are comparatively simpler. The complete solutions for this model in the absence of a magnetic field are also obtained. The presence of a magnetic field in the former case however, does not in effect cause any major modification in the fundamental nature of the initial singularity of the expanding model., Comment: 7 pages, 0 figures

The Alcubierre warp drive metric is a spacetime construction where a massive particle located inside a spacetime distortion, called warp bubble, travels at velocities arbitrarily higher than the velocity of light. This theoretically constructed spacetime geometry is a consequence of general relativity where global superluminal velocities, also known as warp speeds, are possible, whereas local speeds are limited to subluminal ones as required by special relativity. In this work we analyze the solutions of the Einstein equations having charged dust energy-momentum tensor as source for warp velocities. The Einstein equations with the cosmological constant are written and all solutions having energy-momentum tensor components for electromagnetic fields generated by charged dust are presented, as well as the respective energy conditions. The results show an interplay between the energy conditions and the electromagnetic field such that in some cases the former can be satisfied by both positive and negative matter density. In other cases the dominant and null energy conditions are violated. A result connecting the electric energy density with the cosmological constant is also presented, as well as the effects of the electromagnetic field on the bubble dynamics., Comment: 21 pages, 1 table. LaTeX. Accepted for publication in General Relativity and Gravitation

The Alcubierre warp drive metric is a spacetime geometry featuring a spacetime distortion, called warp bubble, where a massive particle inside it acquires global superluminal velocities, or warp speeds. This work presents solutions of the Einstein equations for the Alcubierre metric having fluid matter as gravity source. The energy-momentum tensor considered two fluid contents, the perfect fluid and the parametrized perfect fluid (PPF), a tentative more flexible model whose aim is to explore the possibilities of warp drive solutions with positive matter density content. Santos-Pereira et al. (2020; arXiv:2008.06560) have already showed that the Alcubierre metric having dust as source connects this geometry to the Burgers equation, which describes shock waves moving through an inviscid fluid, but led the solutions back to vacuum. The same happened for two out of four solutions subcases for the perfect fluid. Other solutions for the perfect fluid indicate the possibility of warp drive with positive matter density, but at the cost of a complex solution for the warp drive regulating function. Regarding the PPF, solutions were also obtained indicating that warp speeds could be created with positive matter density. Weak, dominant, strong and null energy conditions were calculated for all studied subcases, being satisfied for the perfect fluid and creating constraints in the PPF quantities such that positive matter density is also possible for creating a warp bubble. Summing up all results,energy-momentum tensors describing more complex forms of matter, or field, distributions generate solutions for the Einstein equations with the warp drive metric where negative matter density might not be a strict precondition for attaining warp speeds., Comment: 39 pages, 3 tables. LaTeX. Minor changes to match proofs. Accepted for publication in the European Physical Journal C

This paper seeks to test if the large-scale galaxy distribution can be characterized as a fractal system. Tools appropriate for describing galaxy fractal structures with a single fractal dimension $D$ in relativistic settings are developed and applied to the UltraVISTA galaxy survey. A graph of volume-limited samples corresponding to the redshift limits in each redshift bins for absolute magnitude is presented. Fractal analysis using the standard $\Lambda$CDM cosmological model is applied to a reduced subsample in the range $0.1\le z \le 4$, and the entire sample within $0.1\le z\le 6$. Three relativistic distances are used, the luminosity distance $d_L$, redshift distance $d_z$ and galaxy area distance $d_G$, because for data at $z\gtrsim 0.3$ relativistic effects are such that for the same $z$ these distance definitions yield different values. The results show two consecutive and distinct redshift ranges in both the reduced and complete samples where the data behave as a single fractal galaxy structure. For the reduced subsample we found that the fractal dimension is $D=\left(1.58\pm0.20\right)$ for $z<1$, and $D=\left(0.59\pm0.28\right)$ for $1\le z\le 4$. The complete sample yielded $D=\left(1.63\pm0.20\right)$ for $z<1$ and $D=\left(0.52\pm0.29\right)$ for $1\le z\le6$. These results are consistent with those found by Conde-Saavedra et al. (2015; arXiv:1409.5409v1), where a similar analysis was applied to a much more limited survey at equivalent redshift depths, and suggest that either there are yet unclear observational biases causing such decrease in the fractal dimension, or the galaxy clustering was possibly more sparse and the universe void dominated in a not too distant past., Comment: 9 pages, 11 figures, LaTeX. Accepted for publication in Physics Letters B

The Alcubierre metric is a spacetime geometry where a massive particle inside a spacetime distortion, called warp bubble, is able to travel at velocities arbitrarily higher than the velocity of light, a feature known as the warp drive. This is a consequence of general relativity, which allows global superluminal velocities but restricts local speeds to subluminal ones as required by special relativity. In this work we solved the Einstein equations for the Alcubierre warp drive spacetime geometry considering the dust matter distribution as source, since the Alcubierre metric was not originally advanced as a solution of the Einstein equations, but as a spacetime geometry proposed without a source gravity field. We found out that all Einstein equations solutions of this geometry containing pressureless dust lead to vacuum solutions. We also concluded that these solutions connect the Alcubierre metric to the Burgers equation, which describes shock waves moving through an inviscid fluid. Our results also indicated that these shock waves behave as plane waves., Comment: 15 pages, 1 figure. LaTeX. Accepted for publication in the European Physical Journal C

The Maxwell electromagnetic theory embedded in an inhomogeneous Lema\^{\i}tre-Tolman-Bondi (LTB) spacetime background was described a few years back in the literature. However, terms concerning the mass or high-derivatives were no explored. In this work we studied the inhomogeneous spacetime effects on high-derivatives and massive electromagnetic models. We used the LTB metric and calculated the physical quantities of interest, namely the scale factor, density of the electromagnetic field and Hubble constant, for the Proca and higher-derivative Podolsky models. We found a new singularity in both models, and that the magnetic field must be zero in the Proca model., Comment: Accepted for publication in the European Physical Journal C (Particles and Fields)

Oscillations in the complementary cumulative distribution function (CCDF) of individual income data 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. Although these datasets can be fitted by different functions at different income ranges, the Tsallis distribution has recently been found capable of fitting the whole distribution by means of only two parameters. This procedure showed clearly such oscillatory feature in the entire income range feature, but made it particularly visible at the tail of the distribution. Although log-periodic functions fitted to the data are capable of describing this behavior, a different approach to naturally disclose such oscillatory characteristics is to allow the Tsallis $q$-parameter to become complex. In this paper we use this idea in order to describe the behavior of the CCDF of the Brazilian personal income 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 by means of this approach. A highly non-linear function for the CCDF was obtained through this methodology and a numerical test showed it capable of recovering the main oscillatory feature of the original CCDF of the personal income data of Brazil., Comment: 14 pages. Final version to appear in Physica A

This paper discusses the empirical evidence of Tsallis statistical functions in the personal income distribution of Brazil. Yearly samples from 1978 to 2014 were linearized by the q-logarithm and straight lines were fitted to the entire range of the income data in all samples, producing a two-parameters-only single function representation of the whole distribution in every year. The results showed that the time evolution of the parameters is periodic and plotting one in terms of the other reveals a cycle mostly clockwise. It was also found that the empirical data oscillate periodically around the fitted straight lines with the amplitude growing as the income values increase. Since the entire income data range can be fitted by a single function, this raises questions on previous results claiming that the income distribution is constituted by a well defined two-classes-base income structure, since such a division in two very distinct income classes might not be an intrinsic property of societies, but a consequence of an a priori fitting-choice procedure that may leave aside possibly important income dynamics at the intermediate levels., Comment: 10 figures, 39 graphs, 17 pages, LaTeX. Minor changes to match version sent to publisher. To appear in "Chaos, Solitons & Fractals"

Evidence is presented that the galaxy distribution can be described as a fractal system in the redshift range of the FDF galaxy survey. The fractal dimension $D$ was derived using the FDF galaxy volume number densities in the spatially homogeneous standard cosmological model with $\Omega_{m_0}=0.3$, $\Omega_{\Lambda_0}=0.7$ and $H_0=70 \; \mbox{km} \; {\mbox{s}}^{-1} \; {\mbox{Mpc}}^{-1}$. The ratio between the differential and integral number densities $\gamma$ and $\gamma^\ast$ obtained from the red and blue FDF galaxies provides a direct method to estimate $D$, implying that $\gamma$ and $\gamma^\ast$ vary as power-laws with the cosmological distances. The luminosity distance $d_{\scriptscriptstyle L}$, galaxy area distance $d_{\scriptscriptstyle G}$ and redshift distance $d_z$ were plotted against their respective number densities to calculate $D$ by linear fitting. It was found that the FDF galaxy distribution is characterized by two single fractal dimensions at successive distance ranges. Two straight lines were fitted to the data, whose slopes change at $z \approx 1.3$ or $z \approx 1.9$ depending on the chosen cosmological distance. The average fractal dimension calculated using $\gamma^\ast$ changes from $\langle D \rangle=1.4^{\scriptscriptstyle +0.7}_{\scriptscriptstyle -0.6}$ to $\langle D \rangle=0.5^{\scriptscriptstyle +1.2}_{\scriptscriptstyle -0.4}$ for all galaxies, and $D$ decreases as $z$ increases. Small values of $D$ at high $z$ mean that in the past galaxies were distributed much more sparsely and the large-scale galaxy structure was then possibly dominated by voids. Results of Iribarrem et al. (2014, arXiv:1401.6572) indicating similar fractal features with $\langle D \rangle =0.6 \pm 0.1$ in the far-infrared sources of the Herschel/PACS evolutionary probe (PEP) at $1.5 \lesssim z \lesssim 3.2$ are also mentioned., Comment: LaTex, 15 pages, 28 figures, 4 tables. To appear in "Physica A"

We study the galaxy cosmological mass function (GCMF) in a semi-empirical relativistic approach using observational data provided by galaxy redshift surveys. Starting from the theory of Ribeiro & Stoeger (2003, arXiv:astro-ph/0304094) between the mass-to-light ratio, the selection function obtained from the luminosity function (LF) data and the luminosity density, the average luminosity $L$ and the average galactic mass $\mathcal{M}_g$ are computed in terms of the redshift. $\mathcal{M}_g$ is also alternatively estimated by a method that uses the galaxy stellar mass function (GSMF). Comparison of these two forms of deriving the average galactic mass allows us to infer a possible bias introduced by the selection criteria of the survey. We used the FORS Deep Field galaxy survey sample of 5558 galaxies in the redshift range $0.5 < z < 5.0$ and its LF Schechter parameters in the B-band, as well as this sample's stellar mass-to-light ratio and its GSMF data. Assuming ${\mathcal{M}_{g_0}} \approx 10^{11} \mathcal{M}_\odot$ as the local value of the average galactic mass, the LF approach results in $L_{B} \propto (1+z)^{(2.40 \pm 0.03)}$ and $\mathcal{M}_g \propto (1+z)^{(1.1\pm0.2)}$. However, using the GSMF results produces $\mathcal{M}_g \propto (1+z)^{(-0.58 \pm 0.22)}$. We chose the latter result as it is less biased. We then obtained the theoretical quantities of interest, such as the differential number counts, to calculate the GCMF, which can be fitted by a Schechter function. The derived GCMF follows theoretical predictions in which the less massive objects form first, being followed later by more massive ones. In the range $0.5 < z < 2.0$ the GCMF has a strong variation that can be interpreted as a higher rate of galaxy mergers or as a strong evolution in the star formation history of these galaxies., Comment: In memory of William R. Stoeger (1943-2014). LaTeX, 8 pages, 7 figures. Minor changes to match version sent to publisher. To appear in "Astronomy and Astrophysics"

This work uses observed galaxy surface brightness profiles at high redshifts to determine the cosmological model best suited to interpret these observations. Theoretical predictions of galactic surface brightness profiles are compared to observational data in two cosmological models, $\Lambda$CDM and Einstein-de Sitter, to calculate the evolutionary effects of different spacetime geometries in these profiles in order to try to find out if the available data is capable of indicating the cosmology that best represents actual galactic brightness profiles observations. Starting from the connection between the angular diameter distance and the galactic surface brightness as advanced by Ellis and Perry (1979), we derived scaling relations using data from the Virgo galactic cluster in order to obtain theoretical predictions of the galactic surface brightness modeled by the Sersic profile at redshift values equal to a sample of galaxies in the range $1.5 \lesssim z \lesssim 2.3$ composed by a subset of Szomoru's et al. (2012) observations. We then calculated the difference between theory and observation in order to determine the changes required in the effective radius and effective surface brightness so that the observed galaxies may evolve to have features similar to the Virgo cluster ones. Our results show that within the data uncertainties of this particular subset of galaxies it is not possible to distinguish which of the two cosmological models used here predicts theoretical curves in better agreement with the observed ones, that is, one cannot identify a clear and detectable difference in galactic evolution incurred by the galaxies of our sample when applying each cosmology. We also concluded that the Sersic index $n$ does not seem to play a significant effect in the evolution of these galaxies., Comment: 13 pages, 11 figures, 22 graphs, 4 tables. LaTeX. Minor changes to match published version. Accepted for publication in Astrophysics and Space Science

This work presents a brief and non-technical description of the main results and concepts of the modern scientific cosmology, viewing it from an epistemological perspective which allows a dialog with other modes of thinking like e.g. history, philosophy, sociology and religion. This epistemological viewpoint is based on the philosophical theses advanced by Ludwig Boltzmann (1844-1906) which states that scientific theories are nothing more than representations, or images, of nature (arXiv:physics/0701308v1). By being representations one cannot know how nature really is because the intrinsic and indispensable properties that characterize nature are unreachable by science. In other words, the true essences that constitute nature are unknowable. Therefore, all answers proposed by science are partial, simplified and replaceable. Another way of putting forward this viewpoint is to state that all scientific truths are provisional, a result which naturally leads to the conclusion that the same set of phenomena, or scientific questions, may have various answers, or representations. This conclusion is generally known as theoretical pluralism (arXiv:physics/9806011). It is exactly such a plurality for conceiving, or representing, nature that opens the way for a possibly fruitful dialog among the various forms of thinking, since this dialog can take place in the realm of the representations. A few examples taken from cosmology, sociology and theology are discussed in the context of this epistemological framework., Comment: 27 pages. Invited talk presented at the XII Symposium of the Brazilian Association for the History of Religions. Text written in Brazilian Portuguese

This paper discusses the empirical validity of Goodwin's (1967) macroeconomic model of growth with cycles by assuming that the individual income distribution of the Brazilian society is described by the Gompertz-Pareto distribution (GPD). This is formed by the combination of the Gompertz curve, representing the overwhelming majority of the population (~99%), with the Pareto power law, representing the tiny richest part (~1%). In line with Goodwin's original model, we identify the Gompertzian part with the workers and the Paretian component with the class of capitalists. Since the GPD parameters are obtained for each year and the Goodwin macroeconomics is a time evolving model, we use previously determined, and further extended here, Brazilian GPD parameters, as well as unemployment data, to study the time evolution of these quantities in Brazil from 1981 to 2009 by means of the Goodwin dynamics. This is done in the original Goodwin model and an extension advanced by Desai et al. (2006). As far as Brazilian data is concerned, our results show partial qualitative and quantitative agreement with both models in the studied time period, although the original one provides better data fit. Nevertheless, both models fall short of a good empirical agreement as they predict single center cycles which were not found in the data. We discuss the specific points where the Goodwin dynamics must be improved in order to provide a more realistic representation of the dynamics of economic systems., Comment: 21 pages, 9 figures, 1 table, LaTeX. Minor changes to match corrected proofs. To appear in "Physica A"

This paper discusses a connection between the relativistic number counts of cosmological sources and the observed galaxy luminosity function (LF). Observational differential number densities are defined and obtained from published LF data using such connection. We observe a distortion in the observational quantities that increases with higher redshift values as compared to the theoretical predictions. The use of different cosmological distance measures plays a role in such a distortion, Comment: 3 pages, 3 figures. Abridged version of arXiv:1201.5571

This paper studies the connection between the relativistic number density of galaxies down the past light cone in a Friedmann-Lemaitre-Robertson-Walker spacetime with non-vanishing cosmological constant and the galaxy luminosity function (LF) data. It extends the redshift range of previous results presented in Albani et al. (2007:astro-ph/0611032) where the galaxy distribution was studied out to z=1. Observational inhomogeneities were detected at this range. This research also searches for LF evolution in the context of the framework advanced by Ribeiro and Stoeger (2003:astro-ph/0304094), further developing the theory linking relativistic cosmology theory and LF data. Selection functions are obtained using the Schechter parameters and redshift parametrization of the galaxy luminosity functions obtained from an I-band selected dataset of the FORS Deep Field galaxy survey in the redshift range 0.5

This work analyzes the Gompertz-Pareto distribution (GPD) of personal income, formed by the combination of the Gompertz curve, representing the overwhelming majority of the economically less favorable part of the population of a country, and the Pareto power law, which describes its tiny richest part. Equations for the Lorenz curve, Gini coefficient and the percentage share of the Gompertzian part relative to the total income are all written in this distribution. We show that only three parameters, determined by linear data fitting, are required for its complete characterization. Consistency checks are carried out using income data of Brazil from 1981 to 2007 and they lead to the conclusion that the GPD is consistent and provides a coherent and simple analytical tool to describe personal income distribution data., Comment: 13 pages, 5 figures, LaTeX. Accepted for publication in "Physica A"

The aim of this project is to determine the consistency of an assumed cosmological model by means of a detailed analysis of the brightness profiles of distant galaxies.Starting from the theory developed by Ellis and Perry (1979) connecting the angular diameter distance obtained from a relativistic cosmological model and the detailed photometry of galaxies, we assume the presently most accepted cosmological model with non-zero cosmological constant and attempt to predict the brightness profiles of galaxies of a given redshift. Then this theoretical profile can be compared to observational data already available for distant, that is, high redshift, galaxies. By comparing these two curves we may reach conclusions about the observational feasibility of the underlying cosmological model., Comment: 4 pages, no figures. Contributed paper presented at IAU 2009, Rio de Janeiro, Brazil. Typos corrected

This article reviews an approach for constructing a simple relativistic fractal cosmology whose main aim is to model the observed inhomogeneities of the distribution of galaxies by means of the Lemaitre-Tolman solution of Einstein's field equations for spherically symmetric dust in comoving coordinates. This model is based on earlier works developed by L. Pietronero and J.R. Wertz on Newtonian cosmology, whose main points are discussed. Observational relations in this spacetime are presented, together with a strategy for finding numerical solutions which approximate an averaged and smoothed out single fractal structure in the past light cone. Such fractal solutions are shown, with one of them being in agreement with some basic observational constraints, including the decay of the average density with the distance as a power law (the de Vaucouleurs' density power law) and the fractal dimension in the range 1 <= D <= 2. The spatially homogeneous Friedmann model is discussed as a special case of the Lemaitre-Tolman solution, and it is found that once we apply the observational relations developed for the fractal model we find that all Friedmann models look inhomogeneous along the backward null cone, with a departure from the observable homogeneous region at relatively close ranges. It is also shown that with these same observational relations the Einstein-de Sitter model can have an interpretation where it has zero global density, a result consistent with the "zero global density postulate" advanced by Wertz for hierarchical cosmologies and conjectured by Pietronero for fractal cosmological models. The article ends with a brief discussion on the possible link between this model and nonlinear and chaotic dynamics., Comment: 29 pages, 8 figures. LaTeX. Conference paper published in 1994. Original text preserved, but 3 very brief notes and 8 references were added to point out relevant follow-up literature produced after 1994, all clearly indicated to avoid confusion with the original text

This work presents an empirical study of the evolution of the personal income distribution in Brazil. Yearly samples available from 1978 to 2005 were studied and evidence was found that the complementary cumulative distribution of personal income for 99% of the economically less favorable population is well represented by a Gompertz curve of the form $G(x)=\exp [\exp (A-Bx)]$, where $x$ is the normalized individual income. The complementary cumulative distribution of the remaining 1% richest part of the population is well represented by a Pareto power law distribution $P(x)= \beta x^{-\alpha}$. This result means that similarly to other countries, Brazil's income distribution is characterized by a well defined two class system. The parameters $A$, $B$, $\alpha$, $\beta$ were determined by a mixture of boundary conditions, normalization and fitting methods for every year in the time span of this study. Since the Gompertz curve is characteristic of growth models, its presence here suggests that these patterns in income distribution could be a consequence of the growth dynamics of the underlying economic system. In addition, we found out that the percentage share of both the Gompertzian and Paretian components relative to the total income shows an approximate cycling pattern with periods of about 4 years and whose maximum and minimum peaks in each component alternate at about every 2 years. This finding suggests that the growth dynamics of Brazil's economic system might possibly follow a Goodwin-type class model dynamics based on the application of the Lotka-Volterra equation to economic growth and cycle., Comment: 22 pages, 15 figures, 4 tables. LaTeX. Accepted for publication in "The European Physical Journal B"

This paper presents numerical solutions of particular Lemaitre-Tolman models approximating a fractal behaviour along the past light cone, as discussed in paper I (0807.0866) of this series. The initial conditions of the numerical problem are discussed and the algorithm used to carry out the numerical integrations is presented. It was found that the numerical solutions are stiff across the flat-curved interface necessary to obtain the initial conditions of the problem. The spatially homogeneous Friedmann models are treated as special cases of the Lemaitre-Tolman solution and solved numerically. Extending the results of paper II (0807.0869) on the Einstein-de Sitter model, to the $K = \pm 1$ models, it was found that the open and closed Friedmann models also do not appear to remain homogeneous along the backward null cone, with a vanishing volume (average) density as one approaches the Big Bang singularity hypersurface. Fractal solutions, that is, solutions representing an averaged and smoothed-out single fractal, were obtained in all three classes of the Lemaitre-Tolman metric, but only the hyperbolic ones were found to be in agreement with observations, meaning that a possible Friedmann background universe would have to be an open one. The best fractal metric obtained through numerical simulations is also analysed in terms of evolution, homothetic self-similarity, comparison with the respective spatially homogeneous case and the fitting problem in cosmology. The paper finishes with a discussion on some objections raised by some authors against a fractal cosmology., Comment: 57 pages. 22 figures. LaTeX. Paper published in 1993. The numerical code used in this paper is discussed in gr-qc/0205095

This paper studies the spatially homogeneous Einstein-de Sitter cosmological model in the context of a relativistic hierarchical (fractal) cosmology as developed in paper I (0807.0866). The Einstein-de Sitter model is treated as a special case of Lemaitre-Tolman's spacetime, obtained by the appropriate choice of the latter's three arbitrary functions. The observational relations along the past light cone of the model under consideration are calculated, and an investigation of whether or not it has fractal behaviour is performed. It was found that the Einstein-de Sitter model does not seem to remain homogeneous along the geodesic and that it also has no fractal features along the backward null cone., Comment: 13 pages. 1 figure. Paper published in 1992

This work examines a relativistic model for the observed inhomogeneities of the large scale structure where the hypothesis that this structure can be described as being a self-similar fractal system is advanced. The concept of hierarchical clustering is identified with a fractal distribution and the problems raised by the use of fractal ideas in a relativistic model are discussed, as well as their relations to the Copernican and Cosmological Principles. Voids, clusters and superclusters of galaxies are assumed to be part of a smoothed-out fractal structure described by a Lemaitre-Tolman solution. The basic concepts of the Newtonian model presented by Pietronero (1987) are reinterpreted and applied to this inhomogeneous curved spacetime. This fractal system is also assumed to have a crossover to homogeneity which leads to a "Swiss cheese" type model, composed by an interior Lemaitre-Tolman metric and an exterior dust Friedmann solution. The Darmois junction conditions between the two spacetimes are calculated, and the observational relations necessary to compare the model with observations are obtained for the interior region. The differential equations of the interior spacetime are set up and a numerical strategy is devised for finding particular Tolman solutions representing a fractal behaviour along the past light cone., Comment: 22 pages. LaTeX. Paper published in 1992

This work discusses the possible empirical verification of the geometrical concept of homogeneity of the standard relativistic cosmology considering its various definitions of distance. We study the physical consequences of the distinction between the usual concept of spatial homogeneity (SH), as defined by the Cosmological Principle, and the concept of observational homogeneity (OH), arguing that OH is in principle falsifiable by means of astronomical observations, whereas verifying SH is only possible indirectly. Simulated counts of cosmological sources are produced by means of a generalized number-distance expression that can be specialized to produce either the counts of the Einstein-de Sitter (EdS) cosmology, which has SH by construction, or other types of counts, which do, or do not, have OH by construction. Expressions for observational volumes and differential densities are derived with the various cosmological distance definitions in the EdS model. Simulated counts that have OH by construction do not always exhibit SH features. The reverse situation is also true. Besides, simulated counts with no OH features at low redshift start showing OH characteristics at high redshift. The comoving distance seems to be the only distance definition where both SH and OH appear simultaneously. The results show that observations indicating possible lack of OH do not necessarily falsify the standard Friedmannian cosmology, meaning that this cosmology will not necessarily always produce observable homogeneous densities. The general conclusion is that the use of different cosmological distances in the characterization of the galaxy distribution lead to significant ambiguities in reaching conclusions about the behavior of the large-scale galaxy distribution in the Universe., Comment: 12 pages, 12 figures, LaTeX. Matches the final version sent to the journal. Accepted for publication in "Astronomy and Astrophysics"

This paper discusses how theories can be assessed within the epistemological viewpoint advanced by the Austrian physicist Ludwig E. Boltzmann. It builds upon, and further develops, the perspective of Boltzmann's thinking as advanced by Ribeiro and Videira (1998, arXiv:physics/9806011). Boltzmann's epistemological viewpoint accepts that reality is real and proposes that reality can be described by different points of view because his main philosophical thesis states that scientific theories are images of Nature. We present the historical context that witnessed the genesis of Boltzmann's ideas and expand Ribeiro and Videira's (Ibid.) perspective by arguing that later in his life Boltzmann realized the insufficiency of his thesis as justification for theoretical pluralism and avoidance of dogmatism. Consequently, his thinking went beyond epistemology, the nature of scientific knowledge, to include realism, the nature of the represented objects., Comment: 19 pages. New abstract. Sections 1 and 5 were slightly revised and enlarged. Version v1 was intended as a chapter of a book on Boltzmann, but the publisher cancelled it. This version matches the text appearing in "Interdisciplinary Science Reviews"

This paper uses data obtained from the galaxy luminosity function (LF) to calculate two types of radial number densities statistics of the galaxy distribution as discussed in Ribeiro (2005), namely the differential density $\gamma$ and the integral differential density $\gamma^\ast$. By applying the theory advanced by Ribeiro and Stoeger (2003), which connects the relativistic cosmology number counts with the astronomically derived LF, the differential number counts $dN/dz$ are extracted from the LF and used to calculate both $\gamma$ and $\gamma^\ast$ with various cosmological distance definitions, namely the area distance, luminosity distance, galaxy area distance and redshift distance. LF data are taken from the CNOC2 galaxy redshift survey and $\gamma$ and $\gamma^\ast$ are calculated for two cosmological models: Einstein-de Sitter and an $\Omega_{m_0}=0.3$, $\Omega_{\Lambda_0}=0.7$ standard cosmology. The results confirm the strong dependency of both statistics on the distance definition, as predicted in Ribeiro (2005), as well as showing that plots of $\gamma$ and $\gamma^\ast$ against the luminosity and redshift distances indicate that the CNOC2 galaxy distribution follows a power law pattern for redshifts higher than 0.1. These findings bring support to Ribeiro's (2005) theoretical proposition that using different cosmological distance measures in statistical analyses of galaxy surveys can lead to significant ambiguity in drawing conclusions about the behavior of the observed large scale distribution of galaxies., Comment: LaTeX, 37 pages, 6 tables, 10 figures. Accepted for publication in "The Astrophysical Journal"

This work studies the Zipf Law for cities in Brazil. Data from censuses of 1970, 1980, 1991 and 2000 were used to select a sample containing only cities with 30,000 inhabitants or more. The results show that the population distribution in Brazilian cities does follow a power law similar to the ones found in other countries. Estimates of the power law exponent were found to be 2.22 +/- 0.34 for the 1970 and 1980 censuses, and 2.26 +/- 0.11 for censuses of 1991 and 2000. More accurate results were obtained with the maximum likelihood estimator, showing an exponent equal to 2.41 for 1970 and 2.36 for the other three years., Comment: 12 pages, 6 figures, 3 tables, Elsevier LaTeX, accepted for publication in "Physica A". Correction of minor mistyping (eq. 8)

This paper studies the effect of the distance choice in radial (non-average) statistical tools used for fractal characterization of galaxy distribution. After reviewing the basics of measuring distances of cosmological sources, various distance definitions are used to calculate the differential density $\gamma$ and the integral differential density $\gamma^\ast$} of the dust distribution in the Einstein-de Sitter cosmology. The main results are as follows: (1) the choice of distance plays a crucial role in determining the scale where relativistic corrections must be taken into account, as both $\gamma$ and $\gamma^\ast$ are strongly affected by such a choice; (2) inappropriate distance choices may lead to failure to find evidence of a galaxy fractal structure when one calculates those quantities, even if such a structure does occur in the galaxy distribution; (3) the comoving distance and the distance given by Mattig's formula are unsuitable to probe for a possible fractal pattern as they render $\gamma$ and $\gamma^\ast$ constant for all redshifts; (4) a possible galaxy fractal system at scales larger than 100Mpc (z \~ 0.03) may only be found if those statistics are calculated with the luminosity or redshift distances, as they are the ones where $\gamma$ and $\gamma^\ast$ decrease at higher redshifts; (5) C\'el\'erier and Thieberger's (2001) critique of Ribeiro's (1995: astro-ph/9910145) earlier study are rendered impaired as their objections were based on misconceptions regarding relativistic distance definitions., Comment: 14 pages, 4 figures, A&A LaTeX macro. Minor linguistic changes to match the version sent to the publisher. Accepted for publication in "Astronomy and Astrophysics"

This paper outlines CLEW (collaborative learning environment for the Web). The project combines MUD (Multi-User Dimension), workflow, VRML (Virtual Reality Modeling Language) and educational concepts like constructivism in a learning environment where students actively participate in the learning process. The MUD shapes the environment structure. The workflow system allows the execution and coordination of the course activities. The VRML establishes the presentation format of the courses' virtual worlds. Finally, the educational concepts influence the way the didactic material is published. Topics discussed include: the educational background that guides the environment; the environment components, including the participants, regions, and objects; the six layers of the system architecture--presentation, application, management and control, adaptation, external services, and data; the course development, including structuring a course, and the participant's perspective; and future plans. A diagram of the system architecture is included. Contains 20 references. (Author/DLS)