Long branch attraction is a problem that afflicts phylogenetic methods and a procedure to detect a data set suffering from this problem is the long branch extraction method[1]. This method has been well cited and used by many authors for their analysis but no strong validation has been performed as to its accuracy. We performed such an analysis by an extensive search of the branch length search...

Let P1, . . . , Pd+1 ⊂ R be d-dimensional point sets such that the convex hull of each Pi contains the origin. We call the sets Pi color classes, and we think of the points in Pi as having color i. A colorful choice is a set with at most one point of each color. The colorful Carathéodory theorem guarantees the existence of a colorful choice whose convex hull contains the origin. So far, the com...

Robots still struggle to dynamically traverse complex 3-D terrain with many large obstacles, an ability required for critical applications. Body-obstacle interaction is often inevitable and induces perturbation uncertainty in motion that challenges closed-form dynamic modeling. Here, inspired by recent discovery of a terradynamic streamlined shape, we studied how two body shapes interacting obs...