Birch and Tverberg partitions are closely related concepts from discrete geometry. We show two properties for the number of Birch partitions: Evenness, and a lower bound. This implies the first non-trivial lower bound for the number of Tverberg partitions that holds for arbitrary q, where q is the number of partition blocks. The proofs are based on direct arguments, and do not use the equivaria...

If a1, a2, . . . , ak and n are positive integers such that n = a1+a2+· · ·+ak, then the sum a1 + a2 + · · ·+ ak is said to be a partition of n of length k, and a1, a2, . . . , ak are said to be the parts of the partition. Two partitions that differ only in the order of their parts are considered to be the same partition. Let Pn be the set of partitions of n, and let Pn,k be the set of partitio...

In 2009, Blagojević, Matschke & Ziegler established the first tight colored Tverberg theorem. We develop a colored version of our previous results (2008): Evenness and non-trivial lower bounds for the number of colored Tverberg partitions. Both properties follow from similar results on the number of colored Birch partitions.

Contaminant releases from upland areas can have adverse water quality and stream ecology impacts. TREX (Two-dimensional, Runoff, Erosion, and Export) is a spatially distributed, physically-based model to simulate chemical transport and fate at the watershed scale. TREX combines surface hydrology and sediment transport features from the CASC2D watershed model with chemical transport features fro...

Recently the authors have proposed tests for the one-sample and the ksample problem, and a test for independence. All three tests are based on sample space partitions, but they were originally developed in different papers. Here we give an overview of the construction of these tests, stressing the common underlying concept of “sample space partitions.”

In this paper a morphological contrast measure is introduced. The quantification of the contrast is based on the analysis of the edges, which are associated with substantial changes in luminance. Due to this, the contrast measure is used to detect the image that presents a high visual contrast when a set of output images is analyzed. The set of output images is obtained by application of morpho...

An M-partition of a positive integer m is a partition of m with as few parts as possible such that every positive integer less than m can be written as a sum of parts taken from the partition. This type of partition is a variation of MacMahon’s perfect partition, and was recently introduced and studied by O’Shea, who showed that for half the numbers m, the number of M-partitions of m is equal t...