Proper velocity by Hermann Minkowski's flat-space metric equation . Here a reference map frame of yardsticks and synchronized clocks define map position x and map time "t" respectively, and the "d" preceding a coordinate means infinitesimal change. A bit of manipulation allows one to show that proper velocity where as usual coordinate velocity . Thus finite "w" ensures that "v" is less than lightspeed "c". By grouping "γ" with v in the expression for relativistic momentum p, proper velocity also extends the Newtonian form of momentum as mass times velocity to high speeds without a need for relativistic mass. The proper velocity
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