Variable-mass system events. The following derivation is for a body that is gaining mass (accretion). A body of time-varying mass "m" moves at a velocity v at an initial time "t". In the same instant, a particle of mass dm moves with velocity u. The initial momentum can be written as Now at a time "t" + d"t", let both the main body and the particle accrete into a body of velocity v + dv. Thus the new momentum of the system can be written as Since d"m"dv is the product of two small values, it can be ignored, meaning during d"t"
Rank product determine how likely a given RP value or better is observed in a random experiment. Permutation re-sampling requires a computationally demanding number of permutations to get reliable estimates of the "p"-values for the most differentially expressed genes, if "n" is large. Eisinga, Breitling and Heskes (2013) provide the exact probability mass distribution of the rank product statistic. Calculation of the exact "p"-values offers a substantial improvement over permutation approximation, most significantly for that part of the distribution rank product analysis is most interested in, i.e., the thin right tail. However, exact statistical significance of large rank products may take unacceptable