Hamilton decompositions of 6-regular abelian Cayley graphs

In 1969, Lovasz asked whether every connected, vertex-transitive graph has a Hamilton path. This question has generated a considerable amount of interest, yet remains vastly open. To date, there exist no known connected, vertex-transitive graph that does not possess a Hamilton path. For the Cayley graphs, a subclass of vertex-transitive graphs, the following conjecture was made: Weak Lovász Conjecture: Every nontrivial, finite, connected Cayley graph is hamiltonian. The Chen-Quimpo Theorem proves that Cayley graphs on abelian groups flourish with Hamilton cycles, thus prompting Alspach to make the following conjecture: Alspach Conjecture: Every 2k-regular, connected Cayley graph on a finite abelian group has a Hamilton decomposition. Alspach’s conjecture is true for k = 1 and 2, but even the case k = 3 is still open. It is this case that this thesis addresses. Chapters 1–3 give introductory material and past work on the conjecture. Chapter 3 investigates the relationship between 6-regular Cayley graphs and associated quotient graphs. A proof of Alspach’s conjecture is given for the odd order case when k = 3. Chapter 4 provides a proof of the conjecture for even order graphs with 3-element connection sets that have an element generating a subgroup of index 2, and having a linear dependency among the other generators. Chapter 5 shows that if Γ = Cay(A, {s1, s2, s3}) is a connected, 6-regular, abelian Cayley graph of even order, and for some1 ≤ i ≤ 3, Δi = Cay(A/(si), {sj1 , sj2}) is 4-regular, and Δi ≄ Cay(ℤ3, {1, 1}), then Γ has a Hamilton decomposition. Alternatively stated, if Γ = Cay(A, S) is a connected, 6-regular, abelian Cayley graph of even order, then Γ has a Hamilton decomposition if S has no involutions, and for some s ∈ S, Cay(A/(s), S) is 4-regular, and of order at least 4. Finally, the Appendices give computational data resulting from C and MAGMA programs used to generate Hamilton decompositions of certain non-isomorphic Cayley graphs on low order abelian groups.

The effect of demand, tank parameters, and pumping stations on energy use in municipal drinking water distribution systems

Two of the indicators of the UN Millennium Development Goals ensuring environmental sustainability are energy use and per capita carbon dioxide emissions. The increasing urbanization and increasing world population may require increased energy use in order to transport enough safe drinking water to communities. In addition, the increase in water use would result in increased energy consumption, thereby resulting in increased green-house gas emissions that promote global climate change. The study of multiple Municipal Drinking Water Distribution Systems (MDWDSs) that relates various MDWDS aspects--system components and properties--to energy use is strongly desirable. The understanding of the relationship between system aspects and energy use aids in energy-efficient design. In this study, components of a MDWDS, and/or the characteristics associated with the component are termed as MDWDS aspects (hereafter--system aspects). There are many aspects of MDWDSs that affect the energy usage. Three system aspects (1) system-wide water demand, (2) storage tank parameters, and (3) pumping stations were analyzed in this study. The study involved seven MDWDSs to understand the relationship between the above-mentioned system aspects in relation with energy use. A MDWDSs model, EPANET 2.0, was utilized to analyze the seven systems. Six of the systems were real and one was a hypothetical system. The study presented here is unique in its statistical approach using seven municipal water distribution systems. The first system aspect studied was system-wide water demand. The analysis involved analyzing seven systems for the variation of water demand and its impact on energy use. To quantify the effects of water use reduction on energy use in a municipal water distribution system, the seven systems were modeled and the energy usage quantified for various amounts of water conservation. It was found that the effect of water conservation on energy use was linear for all seven systems and that all the average values of all the systems' energy use plotted on the same line with a high R 2 value. From this relationship, it can be ascertained that a 20% reduction in water demand results in approximately a 13% savings in energy use for all seven systems analyzed. This figure might hold true for many similar systems that are dominated by pumping and not gravity driven. The second system aspect analyzed was storage tank(s) parameters. Various tank parameters: (1) tank maximum water levels, (2) tank elevation, and (3) tank diameter were considered in this part of the study. MDWDSs use a significant amount of electrical energy for the pumping of water from low elevations (usually a source) to higher ones (usually storage tanks). The use of electrical energy has an effect on pollution emissions and, therefore, potential global climate change as well. Various values of these tank parameters were modeled on seven MDWDSs of various sizes using a network solver and the energy usage recorded. It was found that when averaged over all seven analyzed systems (1) the reduction of maximum tank water level by 50% results in a 2% energy reduction, (2) energy use for a change in tank elevation is system specific, and (2) a reduction of tank diameter of 50% results in approximately a 7% energy savings. The third system aspect analyzed in this study was pumping station parameters. A pumping station consists of one or more pumps. The seven systems were analyzed to understand the effect of the variation of pump horsepower and the number of booster stations on energy use. It was found that adding booster stations could save energy depending upon the system characteristics. For systems with flat topography, a single main pumping station was found to use less energy. In systems with a higher-elevation neighborhood, however, one or more booster pumps with a reduced main pumping station capacity used less energy. The energy savings for the seven systems was dependent on the number of boosters and ranged from 5% to 66% for the analyzed five systems with higher elevation neighborhoods (S3, S4, S5, S6, and S7). No energy savings was realized for the remaining two flat topography systems, S1, and S2. The present study analyzed and established the relationship between various system aspects and energy use in seven MDWDSs. This aids in estimating the amount of energy savings in MDWDSs. This energy savings would ultimately help reduce Greenhouse gases (GHGs) emissions including per capita CO 2 emissions thereby potentially lowering the global climate change effect. This will in turn contribute to meeting the MDG of ensuring environmental sustainability.

Performance-Based Engineering Framework for Earthquake and Fire Following Earthquake

The objective for this thesis is to outline a Performance-Based Engineering (PBE) framework to address the multiple hazards of Earthquake (EQ) and subsequent Fire Following Earthquake (FFE). Currently, fire codes for the United States are largely empirical and prescriptive in nature. The reliance on prescriptive requirements makes quantifying sustained damage due to fire difficult. Additionally, the empirical standards have resulted from individual member or individual assembly furnace testing, which have been shown to differ greatly from full structural system behavior. The very nature of fire behavior (ignition, growth, suppression, and spread) is fundamentally difficult to quantify due to the inherent randomness present in each stage of fire development. The study of interactions between earthquake damage and fire behavior is also in its infancy with essentially no available empirical testing results. This thesis will present a literature review, a discussion, and critique of the state-of-the-art, and a summary of software currently being used to estimate loss due to EQ and FFE. A generalized PBE framework for EQ and subsequent FFE is presented along with a combined hazard probability to performance objective matrix and a table of variables necessary to fully implement the proposed framework. Future research requirements and summary are also provided with discussions of the difficulties inherent in adequately describing the multiple hazards of EQ and FFE.