An analysis is made of the perception of musical intervals. Two kinds of stimuli were used; intervals consisting of two simultaneous simple tones (sinusoids) and intervals consisting of two simultaneous complex tones (fundamental plus harmonics). Subjects judged the stimuli by the method of triadic comparisons in an incomplete balanced design. Multidimensional analysis was performed according t...

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...

Let G and H be graphs, with |G| ≤ |H|. Consider a one-to-one map f:G→H, and define |f| = min{distH(f(x),f(y)): xy E(G)}. Now let the separation of G in H be defined by sep(G,H) = max{|f|: all one-to-one maps f:G→H}. We survey some results of the past several years on this parameter, in particular using Alon and Milman's results on the connection between the second largest eigenvalue of a graph ...

We give a short new proof of a version of the Kruskal-Katona theorem due to Lovász. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Fried...

The energy content of electrostatic perturbations about homogeneous equilibria is discussed. The calculation leading to the well-known dielectric (or as it is sometimes called, the wave) energy is revisited and interpreted in light of Vlasov theory. It is argued that this quantity is deficient because resonant particles are not correctly handled. A linear integral transform is presented that so...

We present examples of flag homology spheres whose γvectors satisfy the Kruskal-Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In another direction, we show that if a flag (d− 1)-sphere has at most 2d + 2 vertices its γ-vector satisfies the Kruska...