Computationally efficient algorithms for the two-dimensional Kolmogorov-Smirnov test

Goodness-of-fit statistics measure the compatibility of random samples against some theoretical or reference probability distribution function. The classical one-dimensional Kolmogorov-Smirnov test is a non-parametric statistic for comparing two empirical distributions which defines the largest absolute difference between the two cumulative distribution functions as a measure of disagreement. Adapting this test to more than one dimension is a challenge because there are 2^d-1 independent ways of ordering a cumulative distribution function in d dimensions. We discuss Peacock's version of the Kolmogorov-Smirnov test for two-dimensional data sets which computes the differences between cumulative distribution functions in 4n^2 quadrants. We also examine Fasano and Franceschini's variation of Peacock's test, Cooke's algorithm for Peacock's test, and ROOT's version of the two-dimensional Kolmogorov-Smirnov test. We establish a lower-bound limit on the work for computing Peacock's test of Omega(n^2.lg(n)), introducing optimal algorithms for both this and Fasano and Franceschini's test, and show that Cooke's algorithm is not a faithful implementation of Peacock's test. We also discuss and evaluate parallel algorithms for Peacock's test

The development of air shower in the iron absorber

The iron open-sandwich experiments to observe one dimensional development of individual air showers were carried out at Akeno Observatory. One dimensional energy flow, incident energy and production height of shower is estimated using the data of size and age obtained from the above experiment and simple calculation

Asymmetric line shapes for medium energy H and He ions undergoing a large angle collision

Asymmetric line shapes for medium energy H and He ions backscattered from topmost adatoms such as Si 111 3x 3 Sb and Ni 111 2x2 O are measured by a toroidal electrostatic analyzer with an excellent energy resolution. The spectra exhibit a pronounced asymmetric nature and are well fitted by an exponentially modified Gaussian profile. It is found that the nonperturbative coupled channel calculations reproduce well the observed asymmetric line shapes for He impact on different materials, although slightly overestimate the asymmetry for H impact on Au. On the other hand, the CASP 3.2 program involving additional approximations gives large underestimates for He ions and overestimates for H ions. This problem has been partially solved by modifying the order of the implementation of the shell corrections and higher order effects in the CASP mode