ner_tags: O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: comparison once the angular positions of the satellites are randomized around this point. From this tensor we compute its eigenvalues, λ 1 > λ 2 > λ 3 , and corresponding eigenvectors, , , . We define the size of the three ellipsoidal axis as a = λ 1 , b = λ 2 , and c = λ 3 . We also define as the vector perpendicular to the planar satellite distribution. We also define the RMS plane width w as the standard deviation of the satellite distances to the plane defined by the vector . To summarize we characterize the satellite distributionwith the following quantities obtained from the inertia tensor: RMS plane width, w . c / a axis ratio. b / a axis ratio. 3.3 Characterizing asphericity with normalized scalars We compare each satellite distribution against its own spherically randomized distribution. We keep fixed the radial position of every satellite with respect to the central galaxy and then randomize its angular position. We repeat this procedure 10 4 times for each satellite distribution and proceed to measure the quantities mentioned in the previous section: w , c / a , and b / a . This allows us to build a normalized version of all quantities of interest by subtracting the mean and dividing by the standard deviation of the values from randomized samples. We use the normalized quantities to build the analytic probability distributions for the scalars describing asphericity. 3.4 Building an analytic asphericity probability distribution After building the normalized variables for the simulated data we perform a Kolmogorov–Smirnov test with the null hypothesis of belonging to a normal distribution with mean and standard deviation computed from the mean and standard deviation estimated from the data. Although the physical quantities of interest are bound, we find that the distributions for the normalized w , c / a , and b / a are indeed consistent with Gaussian distributions. Based on this result we build a multivariate normal distribution for the joint distributions of the normalized w , c / a , and b / a where X = [ w , c / a , b / a ] T is a vector variable with the normalized quantities, μ is the vector mean, and the Σ is the covariance matrix. We compute the preferred covariance matrix and the mean distribution values with a jackknife technique. That is, out of the n pairs in each simulation, we perform n different covariance and mean value measurements using only n − 1 pairs. The reported covariance and mean values correspond to the average of all measurements, the corresponding standard deviation also helps us to estimate the uncertainty on every reported coefficient. This compact description allows us to generate samples of size N that are consistent by construction with their parent simulation. Finally, we use the generated samples to estimate how common are the deviations from sphericity that we measure in the observational data. We use a double-tailed test around the mean in this comparison, meaning that we count the fraction of