Mathematics
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Item Norm of elementary operator in tensor product of c∗-algebras(Chuka University, 2024) Muiruri Peter GuchuMany properties of elementary operators, including spectrum, numerical ranges, compactness, rank, and norm have been studied in depth with the norm property attracting many researchers due to its wide range of applications. Generally, the calculation of norms involves finding a formula that describes the norms in terms of their coefficients. The discussion on the norm of the elementary operator can be traced to Stampfli’s theorem of 1970 which created a proper base for the study of norms. The determination of the norm of the elementary operator in C∗-algebras, JB*-algebras, standard operator algebras, Cartan factor, prime C∗-algebras, two-dimensional complex Hilbert space and tensor product have been studied and based on earlier research, the norm of basic elementary operator in tensor product of C∗-algebras have been evaluated using the concept of finite rank operator and Stampfli’s maximal numerical range. The norms of other types of elementary operators in tensor product of C∗-algebras was not determined. This study extended the study of the norm of elementary operator in tensor product of C∗-algebras to the general finite length elementary operator. The study determined the norm of finite length elementary operator in tensor product of C∗-algebras and found for a finite length elementary operator, Tn, in a tensor product of C∗algebras . Consequently, the conditions under which the norm of an arbitrary finite length elementary operator in tensor products of C∗-algebras as is expressible in terms of the norms of its coefficient operators were also established and found that if ||Ai ⊗ Bi|| ∈ W◦(Ai ⊗ Bi) and ||Ci ⊗ Di|| ∈ W◦(Ci ⊗ Di) ∀ i = 1,2,...,n. Finally, the research determined the norm of Jordan elementary operator in the tensor products of C∗algebras where for every tensor product X ⊗ Y ∈ B(H ⊗ K) with ∥X ⊗ Y ∥ = 1, then ∥UA⊗B,C⊗D∥ = 2∥A∥∥B∥∥C∥∥D∥, A,C ∈ B(H), B,D ∈ B(K) and UA⊗B,C⊗D is the Jordan elementary operator in a tensor product of C∗-algebras. The techniques of rank one operator,finite rank operator, the definition of the elementary operator in tensor product, properties of tensor product, inner product, properties of functionals, and the norm were used to achieve the objectives of this study. The finding of this research can be used in areas of functional analysis, linear algebra, operator theory and mathematical physics by utilizing various properties of an elementary operator under the study of C∗-algebra.Item Modelling fluid flow in an open horseshoe channel with lateral inflow channels(Chuka University, 2024) Jomba JasonFlooding has been an issue for many years, particularly during periods of heavy rainfall. Channels have been constructed to reduce flooding by directing water to rivers, lakes and oceans. Engineers have faced a challenge of designing a hydraulically efficient chan- nel for conveying maximum amount of water for generating electric power, drainage ditches, floodways, navigation channels and irrigation canals. Horseshoe-shaped chan- nels with lateral inputs have received less attention in open channel flow studies than rectangular, parabolic, trapezoidal, and circular channels. This study aims at modeling a uniform flow with horseshoe cross-section with lateral inflows. The choice of a lat- eral horseshoe shape is due to its distinctive combination of geometric and functional advantages. In fluid dynamics, this shape is particularly valuable for promoting stream- lined flow. Its curved form allows fluids to move around it with reduced turbulence and drag, making it ideal for minimizing flow resistance. This study investigates the effects of increasing lateral inflows, varying lateral inflow channel angles, increasing lateral inflows’ cross-sectional areas, and increasing lateral inflows’ lengths on the velocity of the primary channel’s flow. To obtain particular governing equations, the physical conditions of the flow problem were applied to conservation equations. The finite dif- ference technique is utilized to resolve the differential equations governing the flow due to its accuracy, consistency and convergence. The equations are first represented in dimensionless form. Numerical values are simulated using the Python program, and the results obtained from this study are represented graphically. The findings showed that when lateral inflows increased, the main channel’s velocity reduced. Furthermore, it was noted that the velocity decreased as the lateral inflows’ angles increased. Simi- larly, the velocity in the primary channel decreased as the cross-sectional area of the lat- eral inflows increases. Conversely, the results showed that when the length of the lat- eral inflows increased, it also increased the main channel’s velocity. This research will help engineers construct open horseshoe channels with lateral inflows that are hydrauli- cally efficient and can transfer the most water possible for a variety of uses. One of the main applications is irrigation, because efficient water flow is necessary for productive farming. The findings of the study will also be helpful in the production of hydroelectric power since they offer information on how to optimize water flow to improve energy output. The research also intends to improve channel designs that can manage high water volumes in an effort to aid in flood reduction efforts.Item Improving accuracy of compressed mixed linear model: an application to genome-wide association studies(Chuka University, 2024) Obare Dominic Mong’areMixed linear models are very popular in various disciplines due to their robustness in handling complex datasets and taking into account the data structures. Genome wide association studies (GWAs) are key to success in genomic prediction and statistical modelling of genotype-phenotype relationships. Genomic wide association and genomic prediction combines molecular markers and statistical models to detect variants of interest. Though several statistical models have been used in GWAS, advancement in phenotyping and sequencing technologies necessitates improvement of the existing ones in order to increase their statistical power. The general objective of this study was to develop an improved enriched compressed linear mixed model that addresses aspects of accuracy and statistical power. This study took into account cumulative genetic variants causing phenotypic differences at different developmental stages of the plant. Secondary data obtained from the database at IPK-Gatersleben, Germany, was used in this study. The data set consists of phenotypic data from 252 maize inbred lines and 50,000 Single Nucleotide Polymorphism (SNPs) markers. Data analysis was done on R-statistical software Version 4.4.1. Analysis was done on three developmental stages, at 11, 26 and 42 days after sowing (DAS). Plant phenotypic features such as volume, side area and height were used to predict plant biomass. Single trait analysis was done first (plant side area, height and volume) followed by a combination of two traits (plant volume+Plant height, Plant height + Plant side area, Plant volume+Plant Side area) then lastly a combination of all the three traits (plant Plant volume+Plant height+ Plant side area). On plant side area total number of SNPs detected were 6, on volume 8 SNPs were detected, plant height 8 SNPs were detected. On plant volume+ Plant height 20 SNPs were detected, on plant volume+ Plant side area 11 SNPs and on plant volume+ Plant height + Plant area 22 SNPs were detected across the entire analysis on different developmental stages. The results of this study underscored the significance of considering multiple composite traits simultaneously in GWAs to unravel complex genetic correlations and synergistic effects that influence plant architecture and performance. The study revealed dynamic shifts in significant SNP associations as plants progressed through different growth stages, highlighting the evolving genetic landscape during plant development. The study demonstrated the efficiency of the Compressed Mixed Linear Model (CMLM) proved to be highly efficient in clustering individuals and identifying putative quantitative trait nucleotides (QTNs). Incorporating composite phenotypic variables (plant volume, surface area and height) in the model produced the lowest AIC and BIC 1967.630 and 1999.870, respectively, indicating a well-fitting and parsimonious model. Based on the results, the study recommends using machine learning techniques like Random Forest and Lasso to select the most significant phenotypic features for predicting plant biomass. By combining predicted biomass values from multiple variables through standardization aggregation and summation statistical technique, a more informative composite feature can be generated. The composite variable provides a robust input for trait-SNPs association in GWAS, as demonstrated by the enhanced results in this study.Item Characterization and commutation relations on square normal and class Q∗ operators in Hilbert spaces(Chuka University, 2024) Nyaga Edith WarueMany researchers have widely studied operators in Hilbert spaces due to their wide application in areas like computer programming, financial mathematics and quantum physics. The study of operators in Hilbert space has been categorized according to their properties, the relation between different classes and their spectral properties. Researchers have studied various operators in Hilbert spaces, examining their algebraic properties, commutation relations, independence and inclusions. The classical normal operators has played an important role in the development, study and generalization of these classes of operators in Hilbert spaces. This study focused on the extension of properties of normal operators to two classes of operators, the square normal operators and class Q∗ operators. The aim was to determine their characterization, algebraic properties and relationship with other operators in Hilbert space. By use of the relationships with normal operators, this study has established that for any square normal operator T ∈ B(H), then T∗, T−1 and any other operator unitarily equivalent to T are square normal operators. Furthermore, it has been shown that for square normal operators T,S and scalar λ ∈ C, then (λT), (λ + T), (T + S) and (TS) are square normal operators provided some certain conditions are met. The study shows that class Q∗ operators are not convex and establishes that if two class Q∗ operators T and S commute, then their sum T + S is in class Q∗ and their product TS is in class Q∗ if TS∗ = S∗T and T∗S = ST∗. The research also found that 2-normal operators are both square normal and class Q∗ operators while the class of 3-normal and square normal operators are independent. Furthermore, the study observed that while class Q∗ operators are square normal, the converse is not necessarily true. These findings contribute to the existing knowledge in operators theory and functional analysis and offer potential applications in practical domains such as computer science, finance, and quantum mechanics.Item Modeling open channel fluid flow past a trapezoidal cross-section with a segment base having lateral inflow channel(Chuka University, 2024) Nyaga Charles MwanikiA fluid flow in an enclosed conduit is termed as an open channel flow if it has a free surface. A free surface is the surface of a fluid that is subject to zero parallel shear stress such as interface between two homogeneous fluids. Floods particularly in flood- stricken areas have been a major threat to the survival of lives and livelihoods in various aspects. For instance, increased pot-holes, road disconnection and tearing off as well as bridges being carried away by floods have led to increased cases of accidents lead- ing to loss of lives. This has led to Government over-stretching budgetary allocations to cater for maintenance and repair of roads and bridges. This study has developed a model for fluid flow past an open trapezoidal cross-section channel with a segment base having lateral inflow channel. The parameters considered for the flow regime include the surface roughness, cross- section area, the length, angle of inclination and velocity of the lateral inflow channel while flow parameters in the main channel include velocity and depth of the fluid. The turbulent formation between the lateral inflow channel and the main channel will be assumed to be negligible and hence the flow is considered to be laminar and unsteady. The model equations, developed from the general continu- ity and momentum equations governing the fluid flow will be discretized and solved using finite-difference method and the numerical values simulated using Matlab soft- ware. The study showed that an increase in the length of the lateral channel leads to a decrease in the flow velocity of the main channel. As the coefficient of roughness of the lateral channel increases, the flow velocity in the main channel decreases. When the cross-section area of the lateral channel is increased, there is an increase in discharge to the main channel leading to an increase in flow velocity. An angle of inclination of the lateral channel at a range of 300 to 450 exhibit higher values of flow velocity in the main channel compared to other angles. However, maximum velocity at the main channel is attained at an inclination angle of 300. At this angle, there is minimum shear stress hence less resistance to the flow profile. The results of this study will be employed by engineers to optimize channel hydraulic geometries in designing efficient channels for maximum discharge. The well designed channels with optimal dimensions will be highly applied in the construction of drainage systems in roads, sewer buildings, street drainage, airport construction and dams for electric power plants.