Increasing United States Investment in Foreign Securities: An Evaluation of SEC Rule 144A

The dimension of a poset $P$ is the minimum number of total orders whose intersection is $P$. We prove that the dimension of every poset whose comparability graph has maximum degree $\Delta$ is at most $\Delta \log ^{1+o(1)} \Delta$. This result improves on a 30-year old bound of Füredi and Kahn and is within a $\log ^{o(1)}\Delta$ factor of optimal. We prove this result via the notion of boxicity. The boxicity of a graph $G$ is the minimum integer $d$ such that $G$ is the intersection graph of $d$-dimensional axis-aligned boxes. We prove that every graph with maximum degree $\Delta$ has boxicity at most $\Delta \log ^{1+o(1)} \Delta$, which is also within a $\log ^{o(1)}\Delta$ factor of optimal. We also show that the maximum boxicity of graphs with Euler genus $g$ is $\Theta (\sqrt {g \log g})$, which solves an open problem of Esperet and Joret and is tight up to a constant factor

Bad News for Chordal Partitions

Reed and Seymour [1998] asked whether every graph has a partition into induced connected non-empty bipartite subgraphs such that the quotient graph is chordal. If true, this would have significant ramifications for Hadwiger's Conjecture. We prove that the answer is `no'. In fact, we show that the answer is still `no' for several relaxations of the question

COMPUTATIONAL APPROACHES TO UNDERSTAND PHENOTYPIC STRUCTURE AND CONSTITUTIVE MECHANICS RELATIONSHIPS OF SINGLE CELLS

The goal of this work is to better understand the relationship between the structure and function of biological cells by simulating their nonlinear mechanical behavior under static and dynamic loading using image structure-based finite element modeling (FEM). Vascular smooth muscle cells (VSMCs) are chosen for this study due to the strong correlation of the geometric arrangement of their structural components on their mechanical behavior and the implications of that behavior on diseases such as atherosclerosis. VSMCs are modeled here using a linear elastic material model together with truss elements, which simulate the cytoskeletal fiber network that provides the cells with much of their internal structural support. Geometric characterization of single VSMCs of two physiologically relevant phenotypes in 2D cell culture is achieved using confocal microscopy in conjunction with novel image processing techniques. These computer vision techniques use image segmentation, 2D frequency analysis, and linear programming approaches to create representative 3D model structures consisting of the cell nucleus, cytoplasm, and actin stress fiber network of each cell. These structures are then imported into MSC Patran for structural analysis with Marc. Mechanical characterization is achieved using atomic force microscopy (AFM) indentation. Material properties for each VSMC model are input based on values individually obtained through experimentation, and the results of each model are compared against those experimental values. This study is believed to be a significant step towards the viability of finite element models in the field of cellular mechanics because the geometries of the cells in the model are based on confocal microscopy images of actual cells and thus, the results of the model can be compared against experimental data for those same cells

Clustered Colouring in Minor-Closed Classes

The "clustered chromatic number" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We prove that for every graph $H$, the clustered chromatic number of the class of $H$-minor-free graphs is tied to the tree-depth of $H$. In particular, if $H$ is connected with tree-depth $t$ then every $H$-minor-free graph is $(2^{t+1}-4)$-colourable with monochromatic components of size at most $c(H)$. This provides the first evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016) about defective colouring of $H$-minor-free graphs. If $t=3$ then we prove that 4 colours suffice, which is best possible. We also determine those minor-closed graph classes with clustered chromatic number 2. Finally, we develop a conjecture for the clustered chromatic number of an arbitrary minor-closed class

Development of a System to Optimise Water Recharge and Timber Production from \u3cem\u3ePinus pinaster\u3c/em\u3e Aiton Plantations on the Gnangara Water Mound

Degree Name: Doctor of Natural Resource Management In 1996, the Western Australian Government decided to progressively liquidate 21,000 hectares of Pinus pinaster Aiton plantations located 20 kilometres north of Perth on the Gnangara Water Mound. The decision was made for a number of reasons but one of the principal concerns was the conflict between water use and timber production. The decision to liquidate the plantation over a prescribed period while meeting a legal requirement to produce timber for a laminated veneer lumber plant provided the opportunity to integrate the large database on plantation growth and water recharge and use operational research models to determine the optimum liquidation scenario. Pinus pinaster Aiton growth on the Gnangara Water Mound on sites with shallow water tables was found to be less than on sites with deep water tables. This indicates that the trees on these shallow sites, with depths to water table of five to ten metres, are not accessing the water table. Consequently, it is concluded that it is the volume of the water in the unsaturated zone, beneath the trees that is the factor limiting growth not the depth of the water table. Leaf area index (LAI) can be derived from satellite photography except in stands with very open canopies. The relationship between LAI and the extent to which groundwater is recharged is critical. The relationship between Leaf Area Index and Basal Area was not linear as has been assumed in previous recharge models but there was a strong relationship between tree growth and Leaf Area Index. LAI is subject to a number of influences which include amongst other species, age and stand density. Site maximum LAI for Pinus pinaster Aiton was 3.3 on the Gnangara Mound. Leaf Area Index regrowth for Pinus pinaster Aiton on the Gnangara Mound after thinning for a whole stand could be explained by: LAI regrowth = z0 + a(SPH) + b(AGE) + c(SPH*SPH) + d(AGE*AGE) (where SPH is stems per hectare and AGE is time since planting). The LAI growth curve of unthinned young (old) stands was sigmoidal and explained by LAI = 3/(1-81^((3.5+12/2-year)/12)). This pattern is probably caused by increasing reductions of water availability. Because of this LAI pattern for young stands (old) it was hypothesized that the water recharge /age relationship for these stands is a negative sigmoid. That is it inversely mirrors the growth rates of LAI. Consequently the proportion of total rainfall used by plantations would increase with increasing LAI until it reaches the site maximum LAI and then plateaus. Combining the recharge relationship with LAI growth for the same ages of young stands (below 20 years of age) results in a negative linear relationship where percentage recharge = -15*LAI +45. This differs for the LAI to recharge relationship used in PRAMS( Perth Regional Aquifer Model) which proposes that above LAI 1.5 there is a complete extinguishment of recharge. The hypothesised negative linear relationship of LAI to water recharge is more readily able to explain the LAI to age pattern for young stands(old) whereas a full extinguishment at LAI 1.5 is not able to explain the ongoing LAI growth to 3.3. It is not conceivable that a plant would double its transpiration and rainfall interception area if it has already reached the limitation of full site water usage as proposed in PRAMS. There will always be some uncertainty in the LAI to water recharge relationship because it is impossible to measure directly. Sensitivity analysis was used to accommodate any departure from the actual relationship. LAI does have a strong inverse relation to water recharge. Consequently, LAI minimization within a model would have the same effect as maximizing the water recharge. A Gnangara Mound Model was developed to provide a high level decision support tool that could effectively evaluate the relationship between water recharge and timber production. The model required a number of scenarios to test sensitivity to growth rates, recovery percentage of Laminated Veneer Lumber (LVL) from gross volumes, increasing volume requirements and shorter liquidation periods. Changing the objective to one of minimizing LAI (equivalent to maximizing ground water recharge) was tested to determine if it would give a significantly different result to that obtained with a maximize volume objective. The effect of a number of harvesting regimes and other management options on the change in average per annum water recharge derived from the Gnangara Mound Model are listed below:- • Increasing the frequency of prescribed burning native vegetation, increased recharge by 33.5 GL/yr. • A reduction of private and public abstraction from 370 to 304GL/yr, increased recharge by 66 GL/yr. • Harvesting regime for Sustainable timber volume supply to 2026, increased recharge by 45.5 to 55 GL/yr. • Harvesting regime for Minimize LAI with 100000 m3 minimum annual timber volume until 2026, increased recharge by 47.6 to 58.2 GL/yr. • Harvesting regime for Sustainable timber volume supply to 2016, increased recharge by 50.5 to 62 GL/yr. An equal area of Banksia, at LAI 1, to the pine plantations would recharge annually 39.4 to 36.2 GL. A continuing lower rainfall of 100mm less than long term average would mean a reduction of recharge annually of 107 GL to that achieved prior to 1970. Within the constraints of the State Agreement Act a small increase in water recharge could be achieved by using a harvesting schedule with an objective that minimizes LAI provided the volumes from 2008 to 2026 are no lower than 117000 m3 per annum. It is also possible to combine this scenario with maximizing the liquidation of Gnangara plantation rather than the Pinjar and Yanchep plantations within the next 5 years. The combination of these two scenarios would result in the best water outcome (3.2 GL per annum) within the timber production constraints of achieving a minimum of 117000m3 per annum. Early liquidation of the plantation will increase the water recharge but the water outcomes are small (5-7 GL per annum) in comparison to the timber volume loss and potentially could result in significant costs. Far greater savings could be achieved at a lower cost by using other management strategies such as reduction in Banksia water use through prescribed burning or less water extraction for private use. Small gains in water production would be lost by only a small reduction in annual rainfall due to continuing climate variability

Long-term Effects of Tillage on the Retention and Transport of Soil Water

Quantitative measurements were made of the physical and chemical properties of two virgin prairie soils, Crowley and Jay, that remain in their native Arkansas environments and of similar soils that had been tilled extensively. Comparisons were made of soil properties at several depths. When compared with the tilled soils the virgin soils had higher organic matter contents, saturated hydraulic conductivities and water retained at several applied pressures. Bulk densities and hydraulic resistances were lower in the virgin soils. For the Crowley silt loam, values of pH and elemental contents of the virgin soil were higher than those of the tilled soil. Determinations also were made of the effects of a 14-year addition of winter cover crops on a Dubbs-Dundee soil in continuous cotton production. In general, the winter cover crops tended to increase hydraulic conductivity~. ·porosity and organic matter content. These results indicated that the detrimental effects of long-term tillage on soil hydraulic properties could partially be overcome with the planting of these crops during the winter. However, the rate of improvement in the hydraulic properties was not dramatic