Department of Mathematics

Permanent URI for this collectionhttp://erepository.kibu.ac.ke/handle/123456789/112

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Further Generalization of Unitary Quasi-Equivalence of Operators
(International Journal of Mathematics And its Applications, 2020-10-06) Wanjala, Victor; Nyongesa, A. M.
In this paper, we further generalize the class of n-Unitary Quasi-Equivalence by extending this study to (n,m)-Unitary Quasi-Equivalence. We investigate the properties of this class and also the relation of this equivalence class to other relations.
On Class (Q )
(International Journal of Mathematics And its Applications, 2020-11-12) Wanjala, Victor; Nyongesa, A. M.
Class (Q) operators was introduced and studied by Jibril in (quote). In this paper , we introduce and study the “cousins” to this class, namely class (Q∗). An operator T ∈ B(H) is said to belong to class (Q∗) if T ∗2T 2 = (T T ∗) 2 . We study the algebraic properties of this class. We also strike the relationship between this class and square hyponormal operators through characterization of (α, β)-class (Q) operators
On N Quasi D-Operator Operators
(International Journal of Mathematics And its Applications, 2020-01-11) Wanjala, Victor; Nyongesa, A. M.
: In this paper, we introduce the class of N quasi D-operator acting on the usual Hilbert space H over the complex plane. An operator T is said to be an N quasi D-operator if T(T ∗2 (T D) 2 ) = N(T ∗T D) 2T, where N is a bounded operator on H. We investigate the basic behavior of this class of operator.
On Some generalization of Unitary Quasi-Equivalence of Operators
(International Journal of Mathematics And its Applications, 2020-01-01) Wanjala, Victor; Nyongesa, NyoA. M
In this paper, we generalize the class of Unitary Quasi-Equivalence by extending this study to n-Unitary Quasi-Equivalence and investigate the properties of this class. We investigate the relation of this equivalence class to other relations.
On derivation of the probability of occurrence of an epidemic with application to HIV/AIDS spread given tuberculosis co-infections in the presence of treatment
(Science Publishing Group, 2017-08-22) Simwa, Richard Onyino; Muhati, Nelson L.; Chikamai, Lucy W.
Human Immunodeficiency Virus (HIV) and Mycobacterium Tuberculosis (TB) infections are two major world’s public health problems especially in developing countries. Worldwide, 13% of TB cases are estimated to be co-infected with HIV and about a third of 33 million people living with HIV are infected with the bacterium that causes TB. Deterministic models are derived and applied to estimate the basic reproduction number of HIV and TB co-infection as a single output value by treating each of the parameter input as a constant value. In this paper the basic reproduction number is modeled as a random variable, then the probability that there will bean epidemic, is derived and computed. In particular it is shown that for the sub-Saharan region, the probability of the epidemic occurring is 7.9%, as expected since an epidemic is generally a rare event. This research, thus develops the methodology for the computation of the probability of occurrence of an epidemic, which is useful for the public health policy formulation.
Hybrid single particle simulation of atmospheric pollutants over Bungoma County in Kenya
(Ethiopian Journal of Environmental Studies and Management (EJESM), 2018-12-20) Shem, Godfrey Juma; Makokha, John Wanjala; Kelonye, Festus Beru
Microscale dispersion models with different levels of complexity may be used to assess urban air quality and support decision making for pollution control. Mathematical models calculate pollutant concentrations by solving either analytically a simplified set of parametric equations or numerically a set of differential equations that describe in detail wind flow and pollutant dispersion. Air pollution has been evident at Webuye town in Bungoma Kenya and this may be attributable to the current and residual effect of the pan paper and heavy chemical processing plants. A Lagrangian trajectory modeling system for urban air pollution was used to model the flow of atmospheric pollutants in Bungoma County with emphasis to Webuye, an industrial town within the County. Results of the study reveal that pollutants from the area of study exhibit a seasonal dispersion trend over Lake Victoria and surroundings to the Eastern part of Uganda. An investigation of factors that influence this trend is recommended.
Nonparametric estimator for the standardized sum using edgeworth expansions
(International Journal of Scientific Research, 2018-02-01) Kololi, Moses Mukhwana; Orwa, George O.
Constructing an estimator for functional estimation is one of the problems in statistical inference. In this paper, the problem of constructing an estimator for a studentized sum is considered in the nonparametric set-up. In this set-up, data are used to infer to an unknown quantity while making as few assumptions as possible. This non-smooth functional lack some degree of properties traditionally relied upon in estimation. Smooth functionals converge at the rate of 𝑛−12 while non-smooth functionals converge at the rate slower than 𝑛−12 . This highlights the reason why standard techniques fail to give sharp results. A clear and accurate approximation is obtained by using an approximation that admits cumulant generating function; saddle point approximation. An optimal estimator is obtained using the MiniMax criterion where the lower and upper bounds are constructed. While working in the context of MiniMax estimation, the lower bounds are most important. The MiniMax lower bound is obtained by applying the general lower bound technique based on testing two composite hypotheses. The quality of an estimator is evaluated with the MiniMax risk. Best polynomial approximation of an absolute value function and Hermite polynomials are used to construct an optimal estimator when the means are bounded by a given value $M>0$.
School factors as correlates of secondary students’ achievement in mathematics in Bungoma County, Kenya
(International Journal of Educational Investigations, 2015-10-01) Sirengo, John Luka
There is abundant research evidence to support the view that when Mathematics is taught in an enabling environment, a lot of enjoyable learning takes place. But in reality this is not always so, the implication is that students' achievement in this subject still continue to dwindle. This study therefore aimed at finding the extent to which school factors predict secondary school students' achievement in Mathematics. The study adopted the descriptive survey research design of the ex-post facto type and made use of a sample of 206 Mathematics teachers and 206 principals selected through a multi-stage sampling procedure. Instruments were developed and validated for the study. The two validated instruments used were School-based Inventory (r = 0.89) and School-Based Factor Questionnaire (r = 0.92). Three hypotheses were tested at 0.05 level of significance. Data collected were analyzed using means, standard deviation and multiple regressions. Out of the nine variables, the two variables that contributed significantly to student’s achievement in Mathematics are conveniences and instructional materials (ß = 0.130, t = 2.381, P < 0.05), (ß = 0.134, t = 2.470; P < 0.05) respectively. Findings from the study showed that instructional materials and conveniences (toilets) have been adjudged to have contributed significantly to students’ achievement in Mathematics. Therefore, ministry of education and other stakeholders should ensure that schools are provided with effective and adequate toilet facilities. It is also recommended that instructional resources should be provided in schools as the conditions accrued to school factors improve, the performance of students in Mathematics improves. Recommendations were made based on these findings.
Projected rainfall and temperature changes over Bungoma County in western kenya by the year 2050 based on PRECIS modeling system
(Ajol: Ethiopian Journal of Environmental Studies & Management, 2016-06-19) shem, Godfrey Juma; Kelonye, Festus Beru
This study investigated projected changes in rainfall and temperature over Bungoma County by the year 2050 based on the Intergovernmental Panel on Climate Change (IPCC) Special Report on Emission Scenarios (SRES) A1B and A2B emission Scenarios (IPCC, 2007) using the Providing Regional Climates for Impacts Studies (PRECIS); (Giorgi, 2007). The PRECIS regional Climate Model (Hadley RM3P) was configured in 0.22°×0.22° horizontal grid resolution was forced at the lateral boundaries by the UKMO-HadAM3P and UKMOHadCM3 global Models. The future projection of temperature indicates warming over Bungoma County by the year 2050 coupled with reduced precipitation. Time series analysis revealed a cyclic and seasonal trend in rainfall and temperature over the area of study. Temporal characteristics revealed a warmer and colder September-October-November (SON) season under A1B and A2B scenarios respectively. The results also revealed increasing temperatures and reducing rainfall across all seasons under both scenarios except in March-April-May (MAM) season where rainfall amounts increased and temperature reduced. A two paired t-test for the two climate variables revealed a ρ value of less than 0.05 (ρ<0.05) suggesting a statistically significant relationship between each pair of the two variables. The study recommends further evaluation of the model performance in simulating the present day climate over the area of study.
Temporal analysis of drought in Mwingi sub-county of Kitui County in Kenya using the standardized precipitation index (SPI)
(discovery journals : Climate Change, 2018-12-01) Cassim, Zuberi; shem, Godfrey Juma
This study attempts to temporally characterize drought using Standardized Precipitation Index (SPI) over Mwingi Sub-County of Kenya. Rainfall data spanning 1961-2011 over the area of study was used to determine SPI values using quantitative techniques in R programming. The SPI values were temporally characterized using series graphs and trend analysis carried out. In order to enhance understanding of vegetative characteristics over the area of study, Vegetation Cover Index data was used to generate 3 month VCI spatial characteristics. Results of this study revealed that Mwingi region has been experiencing increasing mild to moderate drought events with occasional severe cases being reported since 1961. No extreme drought event was recorded during this period. The study noted that the drought events were increasingly varying in intensity during the period of study. However, no extreme drought event was recorded during this period. The study recommends correlation analysis between the SPI values and all climate variables over the area of study.
Derivation and solution of the heat equation in 1-D
(International Journal of Engineering, Science and Mathematics, 2013-06-01) Kwach, Boniface Otieno; Ongati, Omolo; Alambo, David; Okaka, Colleta A.
Heat flows in the direction of decreasing temperature, that is, from hot to cool. In this paper we derive the heat equation and consider the flow of heat along a metal rod. The rod allows us to consider the temperature, u(x,t), as one dimensional in x but changing in time, t.
Forensic estimation of time of death: a mathematical model
(International Journal of Management,IT and Engineering (IJMIE), 2013-07-01) Kwach, Boniface Otieno; Ongati, Omolo; Nyakinda, J.O.; Nyang’inja, Rachael
In this paper we establish the exact time of death of a murdered person. This leads to an ordinary differential equations whose solution has been analyzed to provide the approximate time of death. Forensic expert will try to estimate this time from body’s current temperature and calculating how long it would have taken to lose heat to reach this point. This provides an accurate approach to establish the approximate time when crime is committed.
Solutions of second-order partial differential equations in two independent variables using method of characteristics
(International Journal of Engineering and Mathematical Sciences, 2013-06-01) Kwach, Boniface Otieno; Manyonge, Alfred W.; Olambo, David; Aminer, Titus J.O.
Building Materials from Stabilized Soils: The use of under-utilized resources for Low Cost Building
(International Journal of Sustainable Construction Engineering and Technology, 2013-01-01) Kwach, Boniface Otieno
The relationship between students socio-economic status and accessto public university education in Bungoma West District, Kenya
(International Journal of Advanced Research, 2013-05-29) Mabonga, Reuben Manyonge; Aywa, Shem; Odebero, Stephen; Poipoi, Moses
To access university education in Kenya, one needs to attain grade C+ in Kenya Certificate of Secondary Education and above as a minimum qualification according to Joint Admission Board (JAB) requirements. The sample of students for this study was identified through simple random sampling while Public Universities were purposefully identified. The instruments comprised questionnaire and interviews. The validity of instruments was done through consultation with lecturers who are the experts in the Department of Education Planning and Management of Masinde Muliro University of Science and Technology and supervisors. Data analysis was done on computer based Statistical Package for Social Sciences (SPSS) programme at the ANOVA and Chi-Square value 0.05 level of significance. Both descriptive statistics involving means, mode, frequencies and inferential statistics were used. The study found out that students’ socio economic status influenced the number of those who accessed university education. There is need for the government to establish a comprehensive scholarship and loan system for needy students.
A probabilistic data encryption scheme
(international journal sharing platiform ; Journal of Natural Sciences Research, 2013-01-01) Wanambisi, Adrin W.; Maende, cleophas; Muketha, Geoffrey Muchiri; Aywa, Shem
In this paper the author presents a probabilistic encryption scheme that is polynomially secure and has the efficiency of deterministic schemes. From the theoretical construction of Brands and Gill (1996), it is clear that the proof of Pseudo randomness of the quadratic residue generator is complete if it can be shown that there exists a one-way function under the possible assumption that it is infeasible to solve the quadratic residuacity problem provided the factorization of the composite integer is unknown
Mixed Galerkin finite element solution of the homogeneous Burgers equation
(International Journal of Engineering and Scientific Research (IJESR), 2015-02-01) Adenyah, Rodgers K.; Okoya, Michael O.; shem, aywa; Oganga, Duncan
The Burgers’ equation is a very useful mathematical model and can be used in solving a variety of interesting problems in applied mathematics. It models effectively certain problems of a fluid flow nature, in which either shocks or viscous dissipation is a significant factor. The mathematical theory behind the Burgers’ equation is rich and interesting, and, in the broad sense, is a topic of active mathematical research. In this study we solve the homogeneous Burgers’ equation using Galerkin mixed finite element method with Robin’s boundary conditions. We use our results to find out the effect of removing the diffusive term and the convective term on the solution of the Burgers’ equation. Our numerical results suggest that the omission of the convective term gives linear results. It is also observed that the approximations obtained without the diffusive term are not steady and they are non-convergent
On generation of measurable covers for measurable sets using multiple integral of functions
(Science Signpost Publishing:Journal of Mathematics and Statistical Science,, 2015-06-01) Otanga, Levis Olwamba; Aywa, Shem; Owino, Maurice Oduor
In this article, we formulate an n-dimensional structure of measurable covers for measurable sets. Properties such as monotonocity, countable additivity and σ-finiteness of the projective tensor product of vector measure duality are largely applied.
Elementary partial differential equations
(Lambert, 2013-06-01) Wanjala, Charles C.; Manyonge, Alfred W.; Kwach, Boniface Otieno
The importance of partial differential equations cannot be gainsaid. They are used in science and engineering. Many natural phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow etc occurring in science and engineering are described by partial differential equations. Partial differential equations often model mathematical systems where many variables exist. They are also used in statistics especially in the field of stochastic processes.
Mathematical model for drug therapy in patients with diabetes mellitus
(International Journal of Engineering, Science and Mathematics, 2013-03-01) Kwach, Boniface Otieno; Ongati, Omolo; Okoya, Oduori; Otedo, Amos
This study presents a new mathematical model for Drug Therapy in Patients with Diabetes Mellitus which includes external rate at which blood glucose, insulin and epinephrine is being increased in the form Y= fi (g,h,e)+ri (t) . The system has been analyzed and solved to provide the systems natural frequency, ω0, which is the basic descriptor of saturation level of the drug. We establish that the resonance period for the final model, that is, T0=3.76912 hrs, agrees well with the data for the existing insulin therapy, showing that the peak, which is the time period for insulin to be most effective in lowering blood sugar, is in the acceptable therapeutic range

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