The only thing that is certain and absolute about Nature is its patchiness. Patchiness is ubiquitous, occurring across systems, organizational levels, and spatio-temporal scales. Traditional modeling approaches in ecology often fail to recognize spatial patchiness because they usually assume spatial homogeneity. A landscape may be viewed as a hierarchical mosaic system of patches that are diffe...

Patchy particles are a popular paradigm for the design and synthesis of nanoparticles and colloids for self-assembly. In "traditional" patchy particles, anisotropic interactions arising from patterned coatings, functionalized molecules, DNA, and other enthalpic means create the possibility for directional binding of particles into higher-ordered structures. Although the anisotropic geometry of ...

We introduce and develop new techniques to quantify DNA patchiness, and to quantify characteristics of its mosaic structure. These techniques, which involve calculating two functions, alpha(l) and beta(l), measure correlations at length scale l and detect distinct characteristic patch sizes embedded in scale-invariant patch size distributions. Using these new methods, we address a number of iss...

We investigate the class of splitting distributions as composition a singular multivariate distribution and univariate distribution. It will be shown that most common parametric count (multinomial, negative multinomial, hypergeometric, …) can written with separate parameters for both components, thus facilitating their interpretation, inference, study probabilistic characteristics extensions to...