Teleparallel Stringy Black Holes and Topological Nieh-Yan Charges

Recently Banerjee [Class Quantum Gravity 2010] has investigated the Nieh-Yan (NY) topological charges in static black holes (BH) of Schwarzschild type. In this paper we extend Banerjee computations to a static BH with a cosmic string inside it and check how this modifies the NY anomaly. In orthonormal coordinates it is shown that NY topological invariant does not produce any contribution to anomaly. Nevertheless since other topological invariants as the Pontryagin density may appear in teleparallel gravity, since there the full Riemann-Cartan (RC) tensor vanishes and the Riemann tensor can be expressed in terms of torsion. Hence, Pontryagin topological charge may be computed in terms of torsion. The horizons of black holes and singularities are examined. The vanishing of torsion flux along Strings inside BH indicates that the string is confined inside the BH. This similarity is between the NY topological invariant $N=d(T^{i}{\wedge}e_{i})\sim{T^{i}{\wedge}T_{i}}$ and the torsion scalar defined here as $T^{2}= T_{ijk}T^{ijk}$ where T represents torsion differential forms and tensors. It is also shown that the Kerr BH can pursue a NY form invariant.the same problem in some metric forms.

Interpretation of Compositional Regression with Application to Time Budget Analysis

Regression with compositional response or covariates, or even regression between parts of a composition, is frequently employed in social sciences. Among other possible applications, it may help to reveal interesting features in time allocation analysis. As individual activities represent relative contributions to the total amount of time, statistical processing of raw data (frequently represented directly as proportions or percentages) using standard methods may lead to biased results. Specific geometrical features of time budget variables are captured by the logratio methodology of compositional data, whose aim is to build (preferably orthonormal) coordinates to be applied with popular statistical methods. The aim of this paper is to present recent tools of regression analysis within the logratio methodology and apply them to reveal potential relationships among psychometric indicators in a real-world data set. In particular, orthogonal logratio coordinates have been introduced to enhance the interpretability of coefficients in regression models.

Practical Aspects of Log-ratio Coordinate Representations in Regression with Compositional Response

Regression analysis with compositional response, observations carrying relative information, is an appropriate tool for statistical modelling in many scientific areas (e.g. medicine, geochemistry, geology, economics). Even though this technique has been recently intensively studied, there are still some practical aspects that deserve to be further analysed. Here we discuss the issue related to the coordinate representation of compositional data. It is shown that linear relation between particular orthonormal coordinates and centred log-ratio coordinates can be utilized to simplify the computation concerning regression parameters estimation and hypothesis testing. To enhance interpretation of regression parameters, the orthogonal coordinates and their relation with orthonormal and centred log-ratio coordinates are presented. Further we discuss the quality of prediction in different coordinate system. It is shown that the mean squared error (MSE) for orthonormal coordinates is less or equal to the MSE for log-transformed data. Finally, an illustrative real-world example from geology is presented.

Covariance Structure of Compositional Tables

Recent experiences with interpretation of orthonormal coordinates in compositionaldata show clearly a necessity of their better understanding in terms of logratios that formthe primary source of information within the logratio methodology. This is even morecrucial in the special case of compositional tables, where both balances and coordinateswith odds ratio interpretation are involved. The aim of the paper is to provide a decompo-sition of covariance structure of orthonormal coordinates in compositional tables in termsof logratio variances that could serve for this purpose. For their better interpretability,the formulas are also accompanied with appropriate comments and graphical illustrations,and implications for the prominent case of 2 2 compositional tables are discussed.