Toric ideals of homogeneous phylogenetic models

We consider the phylogenetic tree model in which every node of the tree is observed and binary and the transitions are given by the same matrix on each edge of the tree. We are able to compute the Grobner basis and Markov basis of the toric ideal of invariants for trees with up to 11 nodes. These are perhaps the first non-trivial Grobner bases calculations in 2^11 indeterminates. We conjecture that there is a quadratic Grobner basis for binary trees. Finally, we give a explicit description of the polytope associated to this toric ideal for an infinite family of binary trees and conjecture that there is a universal bound on the number of vertices of this polytope for binary trees.Comment: 6 pages, 17 figure

Conjunctive Bayesian networks

Conjunctive Bayesian networks (CBNs) are graphical models that describe the accumulation of events which are constrained in the order of their occurrence. A CBN is given by a partial order on a (finite) set of events. CBNs generalize the oncogenetic tree models of Desper et al. by allowing the occurrence of an event to depend on more than one predecessor event. The present paper studies the statistical and algebraic properties of CBNs. We determine the maximum likelihood parameters and present a combinatorial solution to the model selection problem. Our method performs well on two datasets where the events are HIV mutations associated with drug resistance. Concluding with a study of the algebraic properties of CBNs, we show that CBNs are toric varieties after a coordinate transformation and that their ideals possess a quadratic Gr\"{o}bner basis.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6133 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Phase Transformations and Microstructural Evolution in the U-10 wt.% Mo Alloy with Various Zr Additions at 900C and 650C

The Reduced Enrichment for Research and Test Reactor (RERTR) now known as the Material Minimization and Management Reactor Control program (MMMRC) seeks to replace the use of highly enriched uranium (HEU) fuels used in research and test nuclear reactors around the world. The low enriched uranium (LEU) fuels must have fissionable uranium densities comparable to the HEU fuels. After extensive investigation by various researchers around the world, the U-Mo alloys were selected as a promising candidate. The Mo alloyed with U allows for the stabilization of the face-centered cubic ?-U phase, which demonstrated favorable irradiation behavior. However, deleterious diffusional interaction between the fuel and the cladding, typically Al-base alloy, remain a challenge to overcome for application of U-Mo alloys as the LEU fuel. Zr has been identified as a potential diffusion barrier between monolithic U-10 wt.% Mo (U10Mo) metallic fuel and AA6061 cladding alloys for the development of a LEU fuel system. However, interdiffusion and reaction between the Zr barrier and U10Mo fuel can produce phases such as Mo2Zr, and promote the destabilization of ?-U phase into ?\u27-U (U2Mo) and ?-U. In order to better understand this phenomenon, this study examined the phases that are present in the U10Mo alloys with varying Zr concentration, 0, 0.5, 1.0, 2.0, 5.0, 10.0, 20.0 wt.% at room temperature after heat treatment at 900°C for 168 hours and 650°C for 3 hours. These two temperatures are relevant to fuel plate fabrication process of homogenization and hot-rolling, respectively. Scanning electron microscopy and X-ray diffraction were employed to identify and quantitatively document the constituent phases and microstructure to elucidate the nature of phase transformations. For U10Mo alloys containing less than 1.0 wt.% Zr, there was no significant formation of Mo2Zr after 900?C homogenization and subsequent heat treatment at 650?C for 3 hours. The ?-U phase also remained stable correspondingly for these alloys containing less than 1.0 wt.% Zr. For U10Mo alloys containing 2 wt.% or more Zr, a significant amount of Mo2Zr formation was observed after 900?C homogenization and subsequent heat treatment at 650?C for 3 hours. For these alloys, destabilization of ?-U into ?\u27-U (U2Mo), UZr2 and ?-U was observed. The alloy containing 20 wt.% Zr, however, did not demonstrate ?-U decomposition even though Mo2Zr was observed after heat treatments. The formation of Mo2Zr effectively reduced the stability of the metastable ?-U phase by depleting the ?-stabilizing Mo. The destabilization of ?-U phase into the ?-U phase is not favorable due to anisotropic and poor irradiation behavior of ?-U phase. Therefore the formation of Mo2Zr at the interface between U10Mo fuel and Zr diffusion barrier must be carefully controlled during the fabrication of monolithic LEU fuel system for successful implementation

Phylogenetic Algebraic Geometry

Phylogenetic algebraic geometry is concerned with certain complex projective algebraic varieties derived from finite trees. Real positive points on these varieties represent probabilistic models of evolution. For small trees, we recover classical geometric objects, such as toric and determinantal varieties and their secant varieties, but larger trees lead to new and largely unexplored territory. This paper gives a self-contained introduction to this subject and offers numerous open problems for algebraic geometers.Comment: 15 pages, 7 figure

Genome-wide analysis points to roles for extracellular matrix remodeling, the visual cycle, and neuronal development in myopia

Myopia, or nearsightedness, is the most common eye disorder, resulting primarily from excess elongation of the eye. The etiology of myopia, although known to be complex, is poorly understood. Here we report the largest ever genome-wide association study (43,360 participants) on myopia in Europeans. We performed a survival analysis on age of myopia onset and identified 19 significant associations (p < 5e-8), two of which are replications of earlier associations with refractive error. These 19 associations in total explain 2.7% of the variance in myopia age of onset, and point towards a number of different mechanisms behind the development of myopia. One association is in the gene PRSS56, which has previously been linked to abnormally small eyes; one is in a gene that forms part of the extracellular matrix (LAMA2); two are in or near genes involved in the regeneration of 11-cis-retinal (RGR and RDH5); two are near genes known to be involved in the growth and guidance of retinal ganglion cells (ZIC2, SFRP1); and five are in or near genes involved in neuronal signaling or development. These novel findings point towards multiple genetic factors involved in the development of myopia and suggest that complex interactions between extracellular matrix remodeling, neuronal development, and visual signals from the retina may underlie the development of myopia in humans