On the gravitational energy problem and the energy of photons

The lack of a well-established solution for the gravitational energy problem might be one of the reasons why a clear road to quantum gravity does not exist. In this paper, the gravitational energy is studied in detail with the help of the teleparallel approach that is equivalent to general relativity. This approach is applied to the solutions of the Einstein-Maxwell equations known as $pp$-wave spacetimes. The quantization of the electromagnetic energy is assumed and it is shown that the proper area measured by an observer must satisfy an equation for consistency. The meaning of this equation is discussed and it is argued that the spacetime geometry should become discrete once all matter fields are quantized, including the constituents of the frame; it is shown that for a harmonic oscillation with wavelength $\lambda_0$, the area and the volume take the form $A=4(N+1/2)l_p^2/n$ and $V=2(N+1/2)l_p^2\lambda_0$, where $N$ is the number of photons, $l_p$ the Planck length, and $n$ is a natural number associated with the length along the $z$-axis of a box with cross-sectional area $A$. The localization of the gravitational energy problem is also discussed. The stress-energy tensors for the gravitational and electromagnetic fields are decomposed into energy density, pressures and heat flow. The resultant expressions are consistent with the properties of the fields, thus indicating that one can have a well-defined energy density for the gravitational field regardless of the principle of equivalence.Comment: It has been published in Physical Review D under the title Gravitational energy problem and the energy of photon

A calcareous nannofossil and organic geochemical study of marine palaeoenvironmental changes across the Sinemurian/Pliensbachian (early Jurassic, ~191Ma) in Portugal

The Sinemurian/Pliensbachian boundary (~ 191 Ma) is acknowledged as one of the most important steps in the radiation of planktonic organisms, especially primary producers such as dinoflagellates and coccolithophores. To date, there is no detailed study documenting changes in planktonic assemblages related to palaeoceanographic changes across this boundary. The aim of this study is to characterize the palaeoenvironmental changes occurring across the Sinemurian/Pliensbachian boundary at the São Pedro de Moel section (Lusitanian Basin, Portugal) using micropalaeontology and organic geochemistry approaches. Combined calcareous nannofossil assemblage and lipid biomarker data document for a decrease in primary productivity in relation to a major sea-level rise occurring above the boundary. The Lusitanian Basin was particularly restricted during the late Sinemurian with a relatively low sea level, a configuration that led to the recurrent development of black shales. After a sharp sea-level fall, the basin became progressively deeper and more open during the earliest Pliensbachian, subsequently to a major transgression. This sea-level increase seems to have been a global feature and could have been related to the opening of the Hispanic Corridor that connected the Tethys and palaeo-Pacific oceans. The palaeoceanographic and palaeoclimatic changes induced by this opening may have played a role in the diversification of coccolithophores with the first occurrence or colonization of Tethyan waters by placolith-type coccoliths

Analysis of financial data series using fractional Fourier transform and multidimensional scaling

The goal of this study is the analysis of the dynamical properties of financial data series from worldwide stock market indexes during the period 2000–2009. We analyze, under a regional criterium, ten main indexes at a daily time horizon. The methods and algorithms that have been explored for the description of dynamical phenomena become an effective background in the analysis of economical data. We start by applying the classical concepts of signal analysis, fractional Fourier transform, and methods of fractional calculus. In a second phase we adopt the multidimensional scaling approach. Stock market indexes are examples of complex interacting systems for which a huge amount of data exists. Therefore, these indexes, viewed from a different perspectives, lead to new classification patterns

Analysis of stock market indices with multidimensional scaling and wavelets

Stock market indices SMIs are important measures of financial and economical performance. Considerable research efforts during the last years demonstrated that these signals have a chaotic nature and require sophisticated mathematical tools for analyzing their characteristics. Classical methods, such as the Fourier transform, reveal considerable limitations in discriminating different periods of time. This paper studies the dynamics of SMI by combining the wavelet transform and the multidimensional scaling MDS . Six continuous wavelets are tested for analyzing the information content of the stock signals. In a first phase, the real Shannon wavelet is adopted for performing the evaluation of the SMI dynamics, while their comparison is visualized by means of the MDS. In a second phase, the other wavelets are also tested, and the corresponding MDS plots are analyzed

Analysis of stock market indices through multidimensional scaling

We propose a graphical method to visualize possible time-varying correlations between fifteen stock market values. The method is useful for observing stable or emerging clusters of stock markets with similar behaviour. The graphs, originated from applying multidimensional scaling techniques (MDS), may also guide the construction of multivariate econometric models

A Method to Tackle First Order Differential Equations with Liouvillian Functions in the Solution - II

We present a semi-decision procedure to tackle first order differential equations, with Liouvillian functions in the solution (LFOODEs). As in the case of the Prelle-Singer procedure, this method is based on the knowledge of the integrating factor structure.Comment: 11 pages, late

Power law analysis of financial index dynamics

Power law PL and fractional calculus are two faces of phenomena with long memory behavior. This paper applies PL description to analyze different periods of the business cycle. With such purpose the evolution of ten important stock market indices DAX, Dow Jones, NASDAQ, Nikkei, NYSE, S&P500, SSEC, HSI, TWII, and BSE over time is studied. An evolutionary algorithm is used for the fitting of the PL parameters. It is observed that the PL curve fitting constitutes a good tool for revealing the signal main characteristics leading to the emergence of the global financial dynamic evolution

Analysis of financial indices by means of the windowed Fourier transform

The goal of this study is to analyze the dynamical properties of financial data series from nineteen worldwide stock market indices (SMI) during the period 1995–2009. SMI reveal a complex behavior that can be explored since it is available a considerable volume of data. In this paper is applied the window Fourier transform and methods of fractional calculus. The results reveal classification patterns typical of fractional order systems

Analytic quasi-steady evolution of a marginally unstable wave in the presence of drag and scattering

The 1D bump-on-tail problem is studied in order to determine the influence of drag on quasi-steady solutions near marginal stability ($1-\gamma_d/\gamma_L\ll 1$) when effective collisions are much larger than the instability growth rate ($\nu \gg \gamma$). In this common tokamak regime, it is rigorously shown that the paradigmatic Berk-Breizman cubic equation for the nonlinear mode evolution reduces to a much simpler differential equation, dubbed the time-local cubic equation, which can be solved directly. It is found that in addition to increasing the saturation amplitude, drag introduces a shift in the apparent oscillation frequency by modulating the saturated wave envelope. Excellent agreement is found between the analytic solution for the mode evolution and both the numerically integrated Berk-Breizman cubic equation and fully nonlinear 1D Vlasov simulations. Experimentally isolating the contribution of drag to the saturated mode amplitude for verification purposes is explored but complicated by the reality that the amount of drag can not be varied independently of other key parameters in realistic scenarios. While the effect of drag is modest when the ratio of drag to scattering $\alpha/\nu$ is very small, it can become substantial when $\alpha/\nu \gtrsim 0.5$, suggesting that drag should be accounted for in quantitative models of fast-ion-driven instabilities in fusion plasmas.Comment: 22 pages, 11 figure