Blocks and Cut Vertices of the Buneman Graph

Given a set \Sg of bipartitions of some finite set $X$ of cardinality at least 2, one can associate to \Sg a canonical $X$-labeled graph \B(\Sg), called the Buneman graph. This graph has several interesting mathematical properties - for example, it is a median network and therefore an isometric subgraph of a hypercube. It is commonly used as a tool in studies of DNA sequences gathered from populations. In this paper, we present some results concerning the {\em cut vertices} of \B(\Sg), i.e., vertices whose removal disconnect the graph, as well as its {\em blocks} or 2-{\em connected components} - results that yield, in particular, an intriguing generalization of the well-known fact that \B(\Sg) is a tree if and only if any two splits in \Sg are compatible

Lassoing and corraling rooted phylogenetic trees

The construction of a dendogram on a set of individuals is a key component of a genomewide association study. However even with modern sequencing technologies the distances on the individuals required for the construction of such a structure may not always be reliable making it tempting to exclude them from an analysis. This, in turn, results in an input set for dendogram construction that consists of only partial distance information which raises the following fundamental question. For what subset of its leaf set can we reconstruct uniquely the dendogram from the distances that it induces on that subset. By formalizing a dendogram in terms of an edge-weighted, rooted phylogenetic tree on a pre-given finite set X with |X|>2 whose edge-weighting is equidistant and a set of partial distances on X in terms of a set L of 2-subsets of X, we investigate this problem in terms of when such a tree is lassoed, that is, uniquely determined by the elements in L. For this we consider four different formalizations of the idea of "uniquely determining" giving rise to four distinct types of lassos. We present characterizations for all of them in terms of the child-edge graphs of the interior vertices of such a tree. Our characterizations imply in particular that in case the tree in question is binary then all four types of lasso must coincide

1/fα1/f^\alpha spectra in elementary cellular automata and fractal signals

We systematically compute the power spectra of the one-dimensional elementary cellular automata introduced by Wolfram. On the one hand our analysis reveals that one automaton displays $1/f$ spectra though considered as trivial, and on the other hand that various automata classified as chaotic/complex display no $1/f$ spectra. We model the results generalizing the recently investigated Sierpinski signal to a class of fractal signals that are tailored to produce $1/f^{\alpha}$ spectra. From the widespread occurrence of (elementary) cellular automata patterns in chemistry, physics and computer sciences, there are various candidates to show spectra similar to our results.Comment: 4 pages (3 figs included

Cluster counting: The Hoshen-Kopelman algorithm vs. spanning tree approaches

Two basic approaches to the cluster counting task in the percolation and related models are discussed. The Hoshen-Kopelman multiple labeling technique for cluster statistics is redescribed. Modifications for random and aperiodic lattices are sketched as well as some parallelised versions of the algorithm are mentioned. The graph-theoretical basis for the spanning tree approaches is given by describing the "breadth-first search" and "depth-first search" procedures. Examples are given for extracting the elastic and geometric "backbone" of a percolation cluster. An implementation of the "pebble game" algorithm using a depth-first search method is also described.Comment: LaTeX, uses ijmpc1.sty(included), 18 pages, 3 figures, submitted to Intern. J. of Modern Physics

On the Number of Facets of Three-Dimensional Dirichlet Stereohedra III: Full Cubic Groups

We are interested in the maximum possible number of facets that Dirichlet stereohedra for three-dimensional crystallographic groups can have. The problem for non-cubic groups was studied in previous papers by D. Bochis and the second author (Discrete Comput. Geom. 25:3 (2001), 419-444, and Beitr. Algebra Geom., 47:1 (2006), 89-120). This paper deals with ''full'' cubic groups, while ''quarter'' cubic groups are left for a subsequent paper. Here, ''full'' and ''quarter'' refers to the recent classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston (math.MG/9911185, Beitr. Algebra Geom. 42.2 (2001), 475-507). Our main result in this paper is that Dirichlet stereohedra for any of the 27 full groups cannot have more than 25 facets. We also find stereohedra with 17 facets for one of these groups.Comment: 28 pages, 12 figures. Changes from v1: apart of some editing (mostly at the end of the introduction) and addition of references, an appendix has been added, which analyzes the case where the base point does not have trivial stabilize

Absolutely Koszul algebras and the Backelin-Roos property

We study absolutely Koszul algebras, Koszul algebras with the Backelin-Roos property and their behavior under standard algebraic operations. In particular, we identify some Veronese subrings of polynomial rings that have the Backelin-Roos property and conjecture that the list is indeed complete. Among other things, we prove that every universally Koszul ring defined by monomials has the Backelin-Roos property

Can Cavitation Be Anticipated?

The major problem with cavitation in pumps and hydraulic systems is that there is no effective (conventional) method for detecting or predicting its inception. The traditional method of recognizing cavitation in a pump is to declare the event occurring when the total head drops by some arbitrary value (typically 3%) in response to a pressure reduction at the pump inlet. However, the device is already seriously cavitating when this happens. What is actually needed is a practical method to detect impending rather than incipient cavitation. Whereas the detection of incipient cavitation requires the detection of features just after cavitation starts, the anticipation of cavitation requires the detection and identification of precursor features just before it begins. Two recent advances that make this detection possible. The first is acoustic sensors with a bandwidth of 1 MHz and a dynamic range of 80 dB that preserve the fine details of the features when subjected to coarse vibrations. The second is the application of Bayesian parameter estimation which makes it possible to separate weak signals, such as those present in cavitation precursors, from strong signals, such as pump vibration. Bayesian parameter estimation derives a model based on cavitation hydrodynamics and produces a figure of merit of how well it fits the acquired data. Applying this model to an anticipatory engine should lead to a reliable method of anticipating cavitation before it occurs. This paper reports the findings of precursor features using high-performance sensors and Bayesian analysis of weak acoustic emissions in the 100-1000kHz band from an experimental flow loop

Multi-triangulations as complexes of star polygons

Maximal $(k+1)$-crossing-free graphs on a planar point set in convex position, that is, $k$-triangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at $k$-triangulations, namely as complexes of star polygons. With this tool we give new, direct, proofs of the fundamental properties of $k$-triangulations, as well as some new results. This interpretation also opens-up new avenues of research, that we briefly explore in the last section.Comment: 40 pages, 24 figures; added references, update Section

Graphical representations and cluster algorithms for critical points with fields

A two-replica graphical representation and associated cluster algorithm is described that is applicable to ferromagnetic Ising systems with arbitrary fields. Critical points are associated with the percolation threshold of the graphical representation. Results from numerical simulations of the Ising model in a staggered field are presented. The dynamic exponent for the algorithm is measured to be less than 0.5.Comment: Revtex, 12 pages with 2 figure