# Browsing by Subject "Densities"

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Item Densities in graphs and matroids(2009-05-15) Kannan, LavanyaShow more Certain graphs can be described by the distribution of the edges in its subgraphs. For example, a cycle C is a graph that satisfies |E(H)| |V (H)| < |E(C)| |V (C)| = 1 for all non-trivial subgraphs of C. Similarly, a tree T is a graph that satisfies |E(H)| |V (H)|?1 ? |E(T)| |V (T)|?1 = 1 for all non-trivial subgraphs of T. In general, a balanced graph G is a graph such that |E(H)| |V (H)| ? |E(G)| |V (G)| and a 1-balanced graph is a graph such that |E(H)| |V (H)|?1 ? |E(G)| |V (G)|?1 for all non-trivial subgraphs of G. Apart from these, for integers k and l, graphs G that satisfy the property |E(H)| ? k|V (H)| ? l for all non-trivial subgraphs H of G play important roles in defining rigid structures. This dissertation is a formal study of a class of density functions that extends the above mentioned ideas. For a rational number r ? 1, a graph G is said to be r-balanced if and only if for each non-trivial subgraph H of G, we have |E(H)| |V (H)|?r ? |E(G)| |V (G)|?r . For r > 1, similar definitions are given. Weaker forms of r-balanced graphs are defined and the existence of these graphs is discussed. We also define a class of vulnerability measures on graphs similar to the edge-connectivity of graphs and show how it is related to r-balanced graphs. All these definitions are matroidal and the definitions of r-balanced matroids naturally extend the definitions of r-balanced graphs. The vulnerability measures in graphs that we define are ranked and are lesser than the edge-connectivity. Due to the relationship of the r-balanced graphs with the vulnerability measures defined in the dissertation, identifying r-balanced graphs and calculating the vulnerability measures in graphs prove to be useful in the area of network survivability. Relationships between the various classes of r-balanced matroids and their weak forms are discussed. For r ? {0, 1}, we give a method to construct big r-balanced graphs from small r-balanced graphs. This construction is a generalization of the construction of Cartesian product of two graphs. We present an algorithmic solution of the problem of transforming any given graph into a 1-balanced graph on the same number of vertices and edges as the given graph. This result is extended to a density function defined on the power set of any set E via a pair of matroid rank functions defined on the power set of E. Many interesting results may be derived in the future by choosing suitable pairs of matroid rank functions and applying the above result.Show more Item Planting density effects on lint yield and quality of three stacked gene cotton cultivars(Texas A&M University, 2006-08-16) Halfmann, Shane WilliamShow more The increased cost of planting transgenic or stacked gene cotton cultivars has stimulated interest in determining the optimal planting density for commercial production. If seeding rates can be reduced without adversely affecting lint yield and fiber quality, producers could regulate initial inputs by fluctuating seeding rates. However, manipulating plant density per unit area can affect the growth and development of the crop. This altered growth throughout the season could potentially affect fiber quality. Fiber properties, which dictate price discounts, are determined by maturity, diameter and length, as well as by physiological activity at the cellular level. These fiber properties are also affected by genetics and environmental conditions, which ultimately can impact lint production as well as the location of bolls set throughout the plant and the maturation period. The objective of this study was to examine the impact of plant density (including high, ideal and low densities) on growth and development of transgenic cotton cultivars. Field experiments were conducted in 2003 and 2004 at the Texas Agricultural Experiment Station in Burleson County, Texas to assess the effects of plant density on lint yield and fiber quality. Experimental design was a spit-plot design with four replications of three cultivars (SG 215 BG/RR, DP 555 BG/RR, ST 4892 BG/RR) in densities ranging from 74 to 222 thousand plants hectare-1. Plant density had no significant effect on lint yield in 2003 or 2004. However, low plant density treatments contained significantly more bolls plant-1 as a result of the plant??s compensatory ability to produce the same number of bolls in a given area. These low density treatments also produced more vegetative biomass plant-1. Due to lower boll numbers and lower ginout percentage, ST 4892 produced the lowest lint yield each year. Lint quality was not significantly affected by density or cultivar treatments either year. However, in 2003 micronaire values were within the discount ranges for ST 4892, and the two lowest density treatments.Show more Item Variation of density with composition for natural gas mixtures in the supercritical region(Texas A&M University, 2004-11-15) WidiaShow more The densities of three different natural gas mixtures (Case A, Case B, and Case C) were evaluated at pressures from 14 to 38 MPa (2000 to 5500 psia) and temperatures from 230 K to 350 K by using SonicWare? and NIST-14 software packages. The chosen pressures and temperatures were based on the phase diagrams for each composition and the probability of encountering such conditions in reservoir or pipeline environment. For each isotherm, the heaviest hydrocarbon was varied from 0 to 1 mole percent in increments of 0.001 (Dx=0.001) and the density calculated for each composition. After the densities were obtained, the partial derivatives of the densities with respect to composition, were calculated numerically at fixed pressure and temperature. The results and calculations suggest that it is very difficult to obtain the desired accuracy (+ 0.1 %) in densities when using a combination of composition measurements and equation of state calculations.Show more