Statistical tests are powerful tools for data analysis. Kruskal-Wallis test is a non-parametric statistical test that evaluates whether two or more samples are drawn from the same distribution. It is commonly used in various areas. But sometimes, the use of the method is impeded by privacy issues raised in fields such as biomedical research and clinical data analysis because of the confidential...

Approximations to the Kruskal-Katona theorem are stated and proven. These approximations are weaker than the theorem, but much easier to work with numerically.

Proof. This theorem is a corollary of the (much more general) Kruskal-Katona theorem. The Kruskal-Katona theorem has a very hands-on proof, based on iteratively modifying the graph. We will see a linear-algebraic proof. Let A be the n×n adjacency matrix of G (Auv = 1 if vertex u is adjacent to vertex v, and Auv = 0 otherwise). Note that A is symmetric. It turns out that e and t are both fundame...

Calculating the statistical linear response of turbulent dynamical systems to the change in external forcing is a problem of wide contemporary interest. Here the authors apply linear regression models with memory, AR(p) models, to approximate this statistical linear response by directly fitting the autocorrelations of the underlying turbulent dynamical system without further computational exper...

Adjacent segmental degeneration (ASD) is a major issue after pedicular fixation. This study examined the degeneration of the adjacent levels due to the insertion of the Wallis interspinous stabilization system compared with discectomy, using magnetic resonance imaging (MRI).Thirty-eight patients diagnosed with lumbar degeneration disorders at L4-L5 were reviewed: 19 patients underwent discectom...