A numerical method for differential equations of second order

A new algorithm of order five is presented for the solution of the initial value problem where the system of ordinary differential equations is of second order and does not contain the first derivative.     

Discrimination of sedimentary facies by association analysis

The ability of association analysis to discriminate sedimentary facies was tested on Purdy's modal analyses of modern sediments of the Great Bahama Bank. Purdy's data set has served in the past as a standard reference for evaluating various multivariate classification algorithms. In order to adapt Purdy's data to association analysis, the percent abundance of the 12 constituents was converted to binary form by dichotomizing each variable on its mean value. The results obtained by association analysis are virtually identical to those obtained by Purdy and other authors. The same four main sedimentary facies were discriminated; 86% of the samples were identically classified (97% when misclassified borderline cases are counted as matches); the total partition variance of the classification is only negligibly greater (4%); and the grouping of the variables yielded the same four groups. The rank order of the three division-attributes responsible for the sample classification is fines, oolites, and corals. Association analysis has been employed by other authors to differentiate meaningful facies groups in studies of ancient reef carbonates, modern reef sediments, and heavy minerals in stream sediments. In all these studies, the results were found to be compatible with those obtained by using the continuous quantitative measurements, indicating that qualitative binary data may often be sufficient for the purpose of facies discrimination in many branches of geology and that association analysis is an effective method for this purpose.     

Cosmic structure growth and dark energy

Dark energy has a dramatic effect on the dynamics of the Universe, causing the recently discovered acceleration of the expansion. The dynamics are also central to the behaviour of the growth of large-scale structure, offering the possibility that observations of structure formation provide a sensitive probe of the cosmology and dark energy characteristics. In particular, dark energy with a time-varying equation of state can have an influence on structure formation stretching back well into the matter-dominated epoch. We analyse this impact, first calculating the linear perturbation results, including those for weak gravitational lensing. These dynamical models possess definite observable differences from constant equation of state models. Then we present a large-scale numerical simulation of structure formation, including the largest volume to date involving a time-varying equation of state. We find the halo mass function is well described by the Jenkins et al. mass function formula. We also show how to interpret modifications of the Friedmann equation in terms of a time-variable equation of state. The results presented here provide steps toward realistic computation of the effect of dark energy in cosmological probes involving large-scale structure, such as cluster counts, the Sunyaev–Zel'dovich effect or weak gravitational lensing.