Physical sciences

Permanent URI for this collectionhttps://repository.chuka.ac.ke/handle/chuka/264

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Formulation of Schrödinger Equation Using The Hilbert Space Operators Approach
(Chuka University, 2019-09) Mbatha, Elizabeth
Operators in Hilbert space have properties which are useful in the study of mathematical abstract areas such as approximation theory, Banach Fixed point theory, the spectral theory as well as Quantum Mechanics. Schrödinger equation is a fundamental entity with many applications in Quantum Mechanics. This equation was initially derived by applying the knowledge of electromagnetic wave function and Einstein theory of relativity. Later, it was derived by applying the knowledge of Newtonian mechanics. It was also derived by extending the wave equation for classical fields to photons and simplified using approximations consistent with generalized non-zero rest mass. However, from the existing literature no study has been done on deriving Schrödinger equation using properties of Hilbert space operators. In this study, Hilbert space operators that include unitary operators, self adjoint operators and compact operators, norms of linear operators, Hilbert Schmidt operator, normal operators together with Lebesque Integral, Neumann Integral and spectrum are used in place of the existing concepts of electromagnetic wave function, Einstein theory of relativity and approximation consistent with generalized non zero mass to derive the Schrödinger equation. Furthermore, this study has established the correlation between the electromagnetic wave function and Einstein theory of relativity in relation with Hilbert space operators. Application of Hilbert space operators on Quantum observables such as position, momentum and energy of a particle has been done in these study. The derivation of Schrödinger enhances equation and its application using Hilbert space operators have enhanced a better understanding of the concept of Schrödinger equation. The results of this work will be useful in quantum mechanics as well as in mathematical operator theory.
Modelling and Application of Response Surface Methodology for Optimization of Weight Gain of Eight Weeks Old Kenbro Served With Pumpkin (Cucurbita Pepo L) Seeds Extract
(Chuka University, 2019-09) Kinyua, Charles Mugendi
Extensive use of synthetic growth promoting antibiotics have become a threat to public health. Therefore, there is a need to search for bioactive phytochemicals that have positive effects on immune, growth and appetite status of the chicken. Such bioactive compounds can be obtained from plants. Pumpkin seeds have been found to be a valuable source of protein and bioactive compounds. Mathematical models have been used to study the effect of pumpkin extracts on the weight gain of chicken. However, most of these studies have focused on the levels of the application of pumpkin extract, rather than on optimising the levels of application on the weight gain of the chicken. Optimization is important since it provides information on the amount of pumpkin seed extracts that should be fed to the chicken without wastage. This study was conducted to evaluate the weight gained by the Kenbro chicken served with pumpkin (Cucurbita pepo L) seeds extract and examined the possible combination of the number of weeks and quantity of pumpkin seed extract that can result in maximum weight gain of the Kenbro chicken. The experiment was conducted using Random Complete Block Design at the poultry demonstration unit of Chuka University. Sixty- day old Kenbro chicks were randomly distributed to five treatments which contained 0, 50, 100, 150 and 200 ml of pumpkin seed extracts per litre of water. Each treatment was replicated three times with four chicken per replicate. The birds were fed with the same diets of finisher and starter feeds. The chickens were provided with enough feeds and water ad libitum in the morning at 8 am. The birds were weighed at the beginning of the experiment and thereafter at an interval of three days for eight weeks. Data was analysed using R statistical software. A Response surface model was fitted to the data and subjected to contour plots to characterize the nature of its turning point and to capture the combination of the number of weeks and quantity of pumpkin seed extract that brings maximum weight gain of the Kenbro chicken. The results showed that the average body weight gained was significant (p < 0.05). The birds served with 200 ml pumpkin seed extract for 4 weeks had the highest weight gain. The fitted Response Surface Model indicated that the number of weeks and quantity of pumpkin seed extract together with their interaction significantly (p < 0.05) determined the weight gain of Kenbro chickens. The study found that the quadratic model fitted using the data had an adjusted R-Squared value of 0.78. The optimal weight gain of 0.23 kg was achieved when the number of weeks was 3.18 with 192.40 ml of pumpkin seed extract. The number of weeks, pumpkin seed extract and their interaction play a key role in obtaining maximum weight gain of the Kenbro chicken. These factors should be put into consideration when developing a feeding system for Kenbro chickens. The study also guides the farmers on the optimization of Kenbro chicken production without incurring an extra cost in the input.
Modelling the Effects of Mindfulness Based Stress on Breast Cancer Survival Rate among Women in Meru and Nyeri Counties, Kenya, using Cox Proportional Hazard Model
(Chuka University, 2019-09) Mutwiri, Mwendwa N.
Breast cancer remains the most commonly diagnosed cancer among women, affecting 34 women per every 100,000. This has led to high number of fatalities annually, which need to be mitigated. The main interest among many cancer survivors and their families is establishing other conventional therapies they can engage in to improve their prognosis and survival. Among some of the key therapies is the interest in working on mindfulness-based stress (MBS) that patients undergo after diagnosis as complementary and alternate measures. Regardless of this, there is little that is known about the effects of MBS factors on breast cancer survival. Management of breast cancer can be enhanced through modelling the effects of MBS on breast cancer survival rate. However, there is limited information on accuracy of existing models. This study focused on developing a model to predict the effect MBS factors have on breast cancer survival rate among women in Meru and Nyeri Counties. Both Primary data and Secondary data were used. Primary data was obtained using a structured questionnaire from the breast cancer survivors and the medical practioners while secondary data was obtained from records at Meru teaching and referral hospital and Nyeri level five hospital on the MBS variables (cost burden of treatment, stress on diagnosis, prolonged time taken to access treatment, poor diet, alcohol use, physical activity and lack of awareness) among breast cancer patients for the period 2012 to 2017. Mixed method research design was used in the study. Both quantitative and qualitative data used in the study was analysed using R software. Cox proportional hazard model was used in establishing the survival rates, with the breast cancer survival rate being dependent variable while MBS factors were the independent variables. Kaplain-Meier estimators were used in determining the varying effects which the MBS factors have on survival rate. Log-rank test was used to perform comparisons of survival curves using hypothesis tests on the patients‘ survival rate considering age. The likelihood ratio test showed that MBS factors are significant in predicting hazard rates ( = 66.7, p = 0.0000119). Treatment period was highly statistically significant (p = 0.00014) as compared to other covariates. Lack of awareness (p = 0.0010124), ease of coping with stress (p = 0.000514) and observing the right diet (p = 0.04092) were also found to significantly affect survival rate. Access of treatment immediately after diagnosis, availing the right information to the patients, helping patients to cope easily with stress and observing the right diet were found to be the best estimators in increasing breast cancer survival rate. The study therefore recommends use of the model in predicting breast cancer survival rates which can greatly improve breast cancer prognosis.
STRUCTURAL, OPTICAL, AND SURFACE POTENTIAL CHARACTERIZATION OF NICKEL, ZINC DOPED MAGNESIUM FERRITES SYNTHESIZED USING SOL-GEL AUTO-COMBUSTION METHOD
(Chuka University, 2022-09) ALEX, NJOROGE
Nano-sized spinel ferrites have attracted heightened scientific focus due to their interesting structural, morphological, magnetic, optical, and electrical properties. Metal substitution in the structural equation of the spinel ferrite (MFe2O4) plays important roles in the modification of its physical properties in order to suits specific applications, like gas sensing. Recently, air pollution has greatly increased, e.g., leakage of liquid petroleum gas (LPG). This lead to deaths, fire outbreaks and other health complication. As a result, there is demand for effective gas sensors with lower limits of detection. Sensing material especially the ferrites are limited in their sensitivity and operate at high temperatures. Among the ferrites, Mg-Zn ferrites are most important magnetic oxide due to their properties. Substitution of any magnetic ion by replacing Zn2+ in Mg-Zn ferrite plays an important role in modifying its properties and hence enhancing its sensing properties. This work, aimed at contributing to the effort focused to improve ferrite sensing properties and overcome these limitations by ternary ionic substitution. Consequently, NixZn0.5-xMg0.5Fe2O4 (x = 0, 0.1, 0.3 and 0.5) ferrite nanoparticles were synthesized by Citra gel auto-combustion method at a pH 7 and characterized for their structural, elemental composition, electronic and optical properties. Pre-elemental analysis by x-ray fluorescence revealed the expected stoichiometry of the synthesized nanoferrites with low amount of impurities. X-ray Diffraction analysis confirmed cubic spinel nature of the samples with crystalline size between 26.90-43.34 nm using Debye Scherrer equation, 8.94 – 26.38 nm using Modified Debye Scherrer plots and 11.60 – 23.86 nm using Halder Wagner plots. The most intense peak was at a miller indices (311), which is a characteristic of a spinel ferrites. Nickel substitution brought variation in the lattice constants which were in the range of 8.3436 – 8.4149 Å. Both Williamson – Hall and Halder- Wagner plots revealed all sample to possess compressive micro strain. Further structural properties by Fourier transform spectroscopy showed prominent peak at range of 350 – 450 cm-1 which are vibration of metal oxygen bond in octahedral sites while vibrations at 500 – 600 cm-1 which are vibration in tetrahedral sites, a characteristics of spinel ferrites with metal oxygen bond stretching. Optical properties by UV-Vis showed the samples optical band gap in the range of 4.19 – 4.21 eV, refractive index in the range of 2.048 – 2.562 and optical dielectric constant in the range 5.062 – 5.070. Kelvin probe scans revealed a low potential material and more interesting was the negative surface potential. The work function for the area scan was in the range 4.536 – 4.588 eV while for the line scan was in the range 4.341 – 4.5673 eV. Electrical conductivities of the samples were found to increase with increase in nickel content, this was determined from the calculation of length jump which was found to decrease from 2.975 Å to 2.942 Å. Similar results of variation of electrical conductivity were found by calculation of unshared edges which were found to decrease with nickel content from 6.9773 – 6.9181 Å. In summary, the synthesized nanoferrites have properties which could find their applications in fabrications of effective gas sensors, based on their size, band gap, length jump, and surface potential obtained.

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