Branch-depth: Generalizing tree-depth of graphs

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A$ of $E$ we let $\lambda_G (A)$ be the number of vertices incident with an edge in $A$ and an edge in $E \setminus A$. For a subset $X$ of $V$, let $\rho_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $\lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $\rho_G$ has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.Comment: 36 pages, 2 figures. Final versio

Avoiding negative depth in inverse depth bearing-only SLAM

In this paper we consider ways to alleviate negative estimated depth for the inverse depth parameterisation of bearing-only SLAM. This problem, which can arise even if the beacons are far from the platform, can cause catastrophic failure of the filter.We consider three strategies to overcome this difficulty: applying inequality constraints, the use of truncated second order filters, and a reparameterisation using the negative logarithm of depth. We show that both a simple inequality method and the use of truncated second order filters are succesful. However, the most robust peformance is achieved using the negative log parameterisation. ©2008 IEEE

Depth, Stanley depth and regularity of ideals associated to graphs

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Let $G$ be a graph with $n$ vertices. Assume that $I=I(G)$ is the edge ideal of $G$ and $J=J(G)$ is its cover ideal. We prove that ${\rm sdepth}(J)\geq n-\nu_{o}(G)$ and ${\rm sdepth}(S/J)\geq n-\nu_{o}(G)-1$, where $\nu_{o}(G)$ is the ordered matching number of $G$. We also prove the inequalities ${\rm sdepth}(J^k)\geq {\rm depth}(J^k)$ and ${\rm sdepth}(S/J^k)\geq {\rm depth}(S/J^k)$, for every integer $k\gg 0$, when $G$ is a bipartite graph. Moreover, we provide an elementary proof for the known inequality ${\rm reg}(S/I)\leq \nu_{o}(G)$

Colourful Simplicial Depth

Inspired by Barany's colourful Caratheodory theorem, we introduce a colourful generalization of Liu's simplicial depth. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d^2+1 and that the maximum is d^(d+1)+1. We exhibit configurations attaining each of these depths and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.Comment: 18 pages, 5 figues. Minor polishin

Water depth influences the head depth of competitive racing starts

Recent research suggests that swimmers perform deeper starts in deeper water (Blitvich, McElroy, Blanksby, Clothier, & Pearson, 2000; Cornett, White, Wright, Willmott, & Stager, 2011). To provide additional information relevant to the depth adjustments swimmers make as a function of water depth and the validity of values reported in prior literature, 11 collegiate swimmers were asked to execute racing starts in three water depths (1.53 m, 2.14 m, and 3.66 m). One-way repeated measures ANOVA revealed that the maximum depth of the center of the head was significantly deeper in 3.66 m as compared to the shallower water depths. No differences due to water depth were detected in head speed at maximum head depth or in the distance from the wall at which maximum head depth occurred. We concluded that swimmers can and do make head depth adjustments as a function of water depth. Earlier research performed in deep water may provide overestimates of maximum head depth following the execution of a racing start in water depth typical of competitive venues