We develop a new approach to local nonlinear effects in cosmic microwave background anisotropies, and discuss the qualitative features of these effects. New couplings of the baryonic velocity to radiation multipoles are found, arising from nonlinear Thomson scattering effects. We also find a new nonlinear shear effect on small angular scales. The full set of evolution and constraint equations is derived, including the nonlinear generalizations of the radiation multipole hierarchy, and of the dynamics of multi-fluids. These equations govern radiation anisotropies in any inhomogeneous spacetime, but their main application is to second-order effects in a universe that is close to the Friedmann models. Qualitative analysis is given here, and quantitative calculations are taken up in further papers.

This is the first of a series of papers extending a 1+3 covariant and and gauge invariant treatment of kinetic theory in curved space-times to a treatment of Cosmic Background Radiation (CBR) temperature anisotropies arising from inhomogeneities in the early universe. This paper deals with algebraic issues, both generically and in the context of models linearised about Robertson-Walker geometries.The approach represents radiation anisotropies by Projected Symmetric and Trace-Free tensors. The Angular correlation functions for the mode coefficients are found in terms of these quantities, following the Wilson-Silk approach, but derived and dealt with in 1+3 covariant and gauge invariant (CGI) form. The covariant multipole and mode-expanded angular correlation functions are related to the usual treatments in the literature. The CGI mode expansion is related to the coordinate approach by linking the Legendre functions to the Projected Symmetric Trace-free representation, using a covariant addition theorem for the tensors to generate the Legendre Polynomial recursion relation. This paper lays the foundation for further papers in the series, which use this formalism in a CGI approach to developing solutions of the Boltzmann and Liouville equations for the CBR before and after decoupling, thus providing a unified CGI derivation of the variety of approaches to CBR anisotropies in the current literature.

We apply random matrix theory to compare correlation matrix estimators C obtained from emerging market data. The correlation matrices are constructed from 10 years of daily data for stocks listed on the Johannesburg Stock Exchange (JSE) from January 1993 to December 2002. We test the spectral properties of C against random matrix predictions and find some agreement between the distributions of eigenvalues, nearest neighbour spacings, distributions of eigenvector components and the inverse participation ratios for eigenvectors. We show that interpolating both missing data and illiquid trading days with a zero-order hold increases agreement with RMT predictions. For the more realistic estimation of correlations in an emerging market, we suggest a pairwise measured-data correlation matrix. For the data set used, this approach suggests greater temporal stability for the leading eigenvectors. An interpretation of eigenvectors in terms of trading strategies is given, as opposed to classification by economic sectors.

Assuming that the cosmological principle holds, Maartens, Ellis and Stoeger (MES) recently constructed a detailed scheme linking anisotropies in the cosmic background radiation (CMB) with anisotropies and inhomogeneities in the large scale structure of the universe and showed how to place limits on those anisotropies and inhomogeneities simply by using CMB quadrupole and octupole limits. First we indicate and discuss the connection between the covariant multipole moments of the temperature anisotropy used in the MES scheme and the quadrupole and octupole results from COBE. Then we introduce those results into the MES limit equations to obtain definite quantitative limits on the complete set of cosmological measures of anisotropy and inhomogeneity. We find that all the anisotropy measures are less than 10 −4 in the case of those not affected by the expansion rate H, and less than 10 −6 M pc −1 in the case of those which are. These results quantitatively demonstrate that the observable universe is indeed close to Friedmann-Lemaître-Robertson-Walker (FLRW) on the largest scales, and can be adequately modelled by an almost-FLRW model -that is, the anisotropies and inhomogeneities characterizing the observable universe on the largest scales are not too large to be considered perturbations to FLRW. 1

We propose the application of a high-speed maximum likelihood clustering algorithm to detect temporal financial market states, using correlation matrices estimated from intraday market microstructure features. We first determine the ex-ante intraday temporal cluster configurations to identify market states, and then study the identified temporal state features to extract state signature vectors which enable online state detection. The state signature vectors serve as low-dimensional state descriptors which can be used in learning algorithms for optimal planning in the high-frequency trading domain. We present a feasible scheme for real-time intraday state detection from streaming market data feeds. This study identifies an interesting hierarchy of system behaviour which motivates the need for timescale specific state space reduction for participating agents.

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We implement a master-slave parallel genetic algorithm (PGA) with a bespoke log-likelihood fitness function to identify emergent clusters within price evolutions. We use graphics processing units (GPUs) to implement a PGA and visualise the results using disjoint minimal spanning trees (MSTs). We demonstrate that our GPU PGA, implemented on a commercially available general purpose GPU, is able to recover stock clusters in sub-second speed, based on a subset of stocks in the South African market. This represents a pragmatic choice for low-cost, scalable parallel computing and is significantly faster than a prototype serial implementation in an optimised C-based fourth-generation programming language, although the results are not directly comparable due to compiler differences. Combined with fast online intraday correlation matrix estimation from high frequency data for cluster identification, the proposed implementation offers cost-effective, near-real-time risk assessment for financial practitioners.

This is the second of a series of papers extending the 1+3 covariant and gauge-invariant treatment of kinetic theory to an examination of cosmic microwave background temperature anisotropies arising from inhomogeneities in the early universe. The first paper (Paper I) dealt with algebraic issues, representing anisotropies in a covariant and gauge-invariant way by means of projected symmetric and trace-free tensors. Here we derive the mode form of the integrated Boltzmann equations, first, giving a covariant version of the standard derivation using the mode recursion relations, second, demonstrating the link to the the multipole divergence equations and finally various analytic ways of solving the resulting equations are discussed. A general integral form of solution is obtained for the equations with Thomson scattering. The covariant Friedmann Lema@^tre multipole form of the transport equations are found near tight-coupling using the covariant and gauge-invariant generalization of the Peebles and Yu expansion in Thompson scattering time. The dispersion relations and damping scale are then obtained from the covariant approach. The equations are integrated to give the covariant and gauge-invariant equivalent of the canonical scalar sourced anisotropies in the K=0 (flat background) case. We carry out a simple treatment of the matter dominated free-streaming projection, slow-decoupling, and tight-coupling cases in covariant and gaugeinvariant theory, with the aim of both giving a unified transparent derivation of this range of results and clarifying the formal connection between the usual approaches (for example, works by Hu and Sugiyama) and the covariant and gauge-invariant like treatments for scalar perturbations (for example, works by Challinor and Lasenby).