http://ir.mu.ac.ke:8080/jspui/handle/123456789/23 2026-08-27T02:59:55Z http://ir.mu.ac.ke:8080/jspui/handle/123456789/10460 Numerical ranges and Spetra of inner product type integral transfomers on Hilbert Spaces Ohuru, Priscah Moraa The theory of linear operators in Hilbert spaces has seen substantial progress, particularly for integral operators of inner product-type, which are known to satisfy fundamental in equalities like Cauchy-Schwarz and Landau within unitarily invariant norm ideals. While these inequalities provide explicit formulas for operator norms, they do not inherently characterize or exclude the study of numerical range and spectrum- rather, they offer complementary information that bounds these spectral properties without fully describing them. This reveals a significant gap: explicit norm formulas exist but leave the numerical range, spectrum, and their relationship uncharacterized. This research addresses these gaps by systematically investigating the numerical range and spectrum of inner product type integral operators. Using operator-theoretic techniques including rank-one operators and tensor products, we construct the algebraic numerical range. Through spectral anal ysis employing abelian properties, commutators, and von Neumann algebra methods, we classify the spectrum into point, continuous, and residual components. Convexity arguments via the Toeplitz-Hausdorff theorem and norm inequalities establish bounds for the numerical range and ensure spectral inclusion. Our results demonstrate that the numerical range maintains essential classical properties, with the spectrum contained within its closure. The applications are substantial in quantum mechanics, providing a rigorous framework for characterizing quantum states and observables, with direct impli cations for evaluating system stability and dynamical behavior. We recommend future work develop more refined techniques- specifically, sharper spectral decomposition and numerical range characterization- to enable precise quantum dynamical predictions and extend these structures to open quantum systems. 2025-01-01T00:00:00Z http://ir.mu.ac.ke:8080/jspui/handle/123456789/10257 Analytical screening of organic chemicals of emerging concern in western Kenya and their contribution to the prevalence of schistosomiasis Kandie, Faith Jebiwot In the past decades, the use and production of chemicals has been on the rise globally due to increasing industrialization and intensive agriculture; resulting in the occurrence and ecotoxicological risks of chemicals of emerging concern (CECs) in the aquatic compartments. Risks include changes in community structure resulting in the dominance of one species and ecosystem imbalance. When dominant disease-causing organisms are in the environment, the disease transmission is increased. For example, host snails for the schistosomiasis, a human trematode disease, are known to be tolerant to pesticide exposure compared to the predators. This would therefore result in an increased abundance of snails which consequently increase the disease transmission in the human population. Kenya, being a low income country faces a lot of challenges with provision of clean water, diseases and sanitation facilities, and increasing population which results in intensive agriculture coupled with pesticide use. Although a lot of research has been carried out on the environmental occurrence and risk of CECs (Chapter 1), most of these studies have been done in developed countries with limited information from Africa. Additionally, research in Africa focused on urban areas with limited number of compounds analyzed and mostly in the water phase, and inadequate information on the effects of CECs on the aquatic organisms. In order to reduce this knowledge gap, this dissertation focused on identification and quantification of CECs present in water, sediment and snails from western Kenya, and the contribution of pesticides to the transmission of schistosomiasis. Chapter 2 gives a summary of the results and discussion of the dissertation. In Chapter 3, a comprehensive chemical analysis was carried out on 48 water samples to identify compounds, spatial patterns and associated risks for fish, crustacean and algae using toxic unit (TU) approach. A total of 78 compounds were detected with pesticides and biocides being the compounds most frequently detected. Spatial pattern analysis revealed limited compound grouping based on land use. Acute risk for crustaceans and algae were driven by one to three individual compounds. These compounds responsible Abstract viii for toxicity were prioritized as candidate compounds for monitoring and regulation in Kenya. In Chapter 4, an extension of Chapter 3 was done to cover the CECs present in snails and sediment from the 48 sites. A total of 30 compounds were found in snails and 78 in sediments with 68 additional compounds being found which were not previously detected in water. Higher contaminant concentrations were found in agricultural sites than in areas without anthropogenic activities. The highest acute toxicity (TU 0.99) was determined for crustaceans based on compounds in sediment samples. The risk was driven by diazinon and pirimiphos-methyl. Acute and chronic risks to algae were driven by diuron whereas fish were found to be at low to no acute risk. In Chapter 5, the effect of pesticide contamination on schistosomiasis transmission was evaluated by applying complimentary laboratory and field studies. In the field studies, the ecological mechanisms through which pesticides and physical chemical parameters affect host snails, predators and competitors were investigated. Pesticide data was obtained from the results in chapter 3. The overall distribution of grazers and predators was not affected by pesticide pollution. However, within the grazers, pesticide pollution increased dominance of host snails. On the contrary, the host-snail competitors were highly sensitive to pesticide exposure. For the laboratory studies, macroinvertebrates including Schistosoma-host snails, competitors and predators were exposed to 6 concentrations levels of imidacloprid and diazinon. Snails showed higher insecticide tolerance compared to competitors and predators. Finally, Chapter 6 summarizes the conclusions of this dissertation, placing it in a broader context. In this dissertation, a comprehensive chemical characterization and risk assessment of CECs has been carried out in freshwater systems; together with the effects of pesticides on schistosomiasis transmission in rural western Kenya. Results of this dissertation showed that rural areas are contaminated posing a risk to aquatic organisms which contribute to schistosomiasis transmission. This shows the need for regular monitoring and policy formulation to reduce pollutant emissions which contributes negatively to both ecological and human health effects 2020-01-01T00:00:00Z http://ir.mu.ac.ke:8080/jspui/handle/123456789/10001 Mathematical modeling and parameter estimation for an optimal Solar food dryer Korkoren, Kenneth Cheruiyot Food shortage in most countries is not only associated with unfavorable weather conditions, but also significantly blamed on ineffective post-harvest handling of food. This calls for an urgent need to address food insecurity in Kenya, in line with Vision 2030 and Government Big 4 Agenda. Eminent threat caused by post-harvest losses due to inadequate drying and poor storage is responsible for up to 40-60% loses of agricultural produce each season. In order to address this issue, this project seeks to model and simulate the characteristics of a solar dryer for the purpose of designing an effective and sustainable, low-cost thermal solar dryer suitable for dehydrating a variety of agricultural produce to ensure prolonged shelf life hence reduce losses. The proposed model is to be formulated using mathematical equations describing integration of four divisions, namely; solar heat collector, circulation of fluid in insulated closed loop pipe network, heat exchangers to generate heated air supplied to the drying chamber and dryer, equipped with humidity control systems, temperature, mass flow rate and energy balance. The mathematical model was formulated and simulation done in order to realize the objectives of delivering a solar drier suitable for drying a wide variety of food products. The simulation results showed that, a solar panel of it was found that a solar collector with aperture area of 𝐴𝑐 = 14.4𝑚2 and volume of 𝑉𝑐 = 500𝑙, when exposed to solar irradiation of 𝐼𝑐 = 1.367𝐾𝑊/𝑚2 at 𝜂𝑐 = 80% efficiency is able to heat water from 𝑇𝑖𝑛 = 220𝐶 to 𝑇𝑐𝑜 = 700𝐶 in 12 hours at a flow rate of 𝑣𝑐 = 1.128𝑙/𝑠, and cumulatively to1300𝐶 in 6 days. This energy if transmitted by insulated pipes to a set of 5 heat exchangers each of 𝐴 = 1𝑚2, and radiative heat transfer coefficient ℎ𝑟 = 100𝑊/𝑚2𝐾 cumulatively dissipates hot air of 2300𝐶 at 𝑣= 250𝑐𝑚3/𝑠, 1300𝐶 at � �=1000𝑐𝑚3/𝑠 and 900𝐶 at 𝑣= 2000𝑐𝑚3/𝑠 air mass flow rate. This output temperatures of dry air are regulated as desired according to the specifications of the food products to be dried. During the night or on cloudy day with minimum or no solar insolation, alternative supplementary source of heat is obtained from petroleum cooking gas, which is regulated automatically depending on the level of solar insolation. It is found that the optimal cost of the gas is 𝐾𝑠ℎ 180/𝑑𝑎𝑦 as opposed to 𝐾𝑠ℎ 560/𝑑𝑎𝑦 when used alone. This is over 67.86% reduction in cost, which makes the use of solar an ideal green energy. 2025-01-01T00:00:00Z http://ir.mu.ac.ke:8080/jspui/handle/123456789/10000 The minimal nilpotent cover of the finite symmetric, alternating and dihedral groups Kimeu, Arphaxad Ngwava Given a group G, a minimal nilpotent cover is a family of subgroups of G of minimal size subject to the property that all of the subgroups are nilpotent and their union is equal to the group G. A maximal non-nilpotent subset is a subset of G of largest size such that for any two distinct elements, the subgroup they generate is not nilpotent. There has been e ort to investigate covers of groups with particular structure, speci cally normal and abelian covers; much has not been done on nilpotent covers. The main objective of this study was to conduct an analysis of minimal nilpotent cover for each of the three families of groups; nite symmetric group Sn, alternating group An and dihedral group D2k, which are ubiquitous throughout the discipline of mathematics. The speci c objectives were to; investigate nilpotent covers, analyse their relationship with non-nilpotent sub sets, investigate the size of minimal nilpotent cover and maximal non-nilpotent subset and apply group theoretic properties in determining a general formula for the size of the minimal nilpotent cover for each family. The analysis was carried out by means of mathematical proofs based on logical approach together with results from computer algebra package GAP and properties of permutations as well as group action on a k-gon . It has been established that except for A9, the size of minimal nilpotent cover for each of Sn, An for n = 34 10, and D2k, coincide with size of maximal non-nilpotent subset. For D2k, a general formula is produced for k = 34 . The results of this study bene t scientists in identi fying redundancy in the analysis of possible organic molecular structures, solve molecular conformation problems as well as mathematicians in expanding knowl edge in group theoretic concepts. For Sn, only for n = 34 10 were analyzed since as n grows large, it becomes complex and even computer algebra package GAPcannot generate any results. We therefore recommend an investigation on a general formula for minimal nilpotent cover of the nite symmetric group Sn as well as the nite alternating group, An 2025-01-01T00:00:00Z