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 (α,β)-Class (Q) Operators
(International Journal of Mathematics and its Applications, 2021-01-12) Wanjala, Victor; Nyongesa, A. M.
In this paper, we introduce a new class of operator, the class of (α, β)-Class (Q) operator acting on a complex Hilbert space H. An operator T ∈ B(H) is said to be (α, β)-Class (Q) if α 2T∗2T 2 ≤ (T∗T) 2 ≤ β 2T ∗2T 2 for 0 ≤ α ≤ 1 ≤ β. We look at some properties that this class are privileged to enjoy.
School factors as correlates of secondary students’ achievement in mathematics in Bungoma county, Kenya
(2015-10) Sirengo, John L.
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.
Nonparametric estimatorfor the standardized sum using edgeworth expansions
(iosr, 2018) Kololi, Moses.; Orwa, George.
This article makes three contributions. First, we introduce a computationally efficient estimator for the component functions in additive nonparametric regression exploiting a different motivation from the marginal integration estimator of Linton and Nielsen. Our method provides a reduction in computation of order n which is highly significant in practice. Second, we define an efficient estimator of the additive components, by inserting the preliminary estimator into a backfitting˙ algorithm but taking one step only, and establish that it is equivalent, in various senses, to the oracle estimator based on knowing the other components. Our two-step estimator is minimax superior to that considered in Opsomer and Ruppert, due to its better bias. Third, we define a bootstrap algorithm for computing pointwise confidence intervals and show that it achieves the correct coverage.
Re-orientation of Science and Mathematics teachers instructional strategies for information communication technology integration
(2015) Mutende, R.A.
This article reports on a study which explored teachers re-conceptualization and re-orientation process during their in-service training for ICT-pedagogy integration in teaching and learning. The qualitative research design was used for the study. It was found that there was a limited ICT infrastructure as well as inadequate technological access and reliability, the participating teachers were engaged in authentic hands-on learning experience and that the teachers engagement in the learning activities demonstrated they had not developed expertise in ICT usage for teaching and learning. It was therefore recommended that opportunities to acquire professional ICT integration skills for both teachers and trainers be expanded.
Derivation and solution of the heat equation in 1-D
(International journal of engineering, science and mathematics, 2013) Kwach, O.; Ongati, Naftali O.; Alambo, David O.; 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, 2013) Kwach, O.; Ongati, Naftali O.; Nyakinda, JO.; 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.
The role of teachers in implementing eﬀective play in early childhood and educational centers
(Lap lambert academic publishing, 2013) Kwach.; Alphonce, O.; Kwach, O.; Ogwan’g, Isabel.
Over the last several years, there has been an increased focus on school readiness and supporting children during the preschool years to learn the skills they need to be successful in elementary school and beyond (Bowman, Donovan, Bums, et al., 2000; Shonkoff & Phillips, 2000). The capacity to develop positive social relationships, to concentrate and persist on challenging tasks, to effectively communicate emotions, and to problem solve are just a few of the competencies young children need to be successful as they transition to school. In this article, we describe the Teaching Pyramid (Fox, Dunlap, Hemmeter, Joseph, & Strain, 2003), a model for promoting young children's social-emotional development and addressing children's challenging behavior and its link to critical outcomes for children, families, and early childhood programs. The Pyramid includes four components: building positive relationships with children, families, and colleagues; designing supportive and engaging environments; teaching social and emotional skills; and developing individualized interventions for children with the most challenging behavior. Given the unique characteristics of early childhood settings, implementation issues and implications of the model are a primary focus of the discussion.
Mathematical model for nutrient exchange across the placenta
(International journal of engineering & mathematical sciences, 2015) Odongo, Joel Olielo.; Ongati, Naftali O.
This study presents a new mathematical model for nutrient exchange across the placenta which include nutrient exchange from foetus to mother to provide a system of equations in the form, Y ሶ ൌ AY ൅ rԦሺtሻ and whose solution was analyzed for equilibrium and stability. This model introduces another parameter that takes care of waste elimination from foetus to mother. It was established that the final model is stable compared to the existing models, that is, the eigenvalues of the coefficient matrix are negative real number and complex numbers with negative real parts. This shows that the new model provides one straight line of solutions tending to the origin and a plane of solutions which spiral towards the origin. This gives a more accurate mathematical model for nutrient exchange in the placenta. This model would create a lot of insight into nutrient exchange in the placenta, the elimination of waste from the foetus and open room for further research from the mathematical concept developed.
Mathematical modeling of insulin therapy in patients with diabetes mellitus
(IJESM, 2015) Kwach, O.; Ongati, O.; Omolo.; Okoya, M.; Oduor.; Otedo, Amos.
—This study presents a Mathematical Model Insulin Therapy in Patients with Diabetes Mellitus which includes external rate at which blood glucose, insulin and epinephrine are being increased in the form, Y =AY+r  (t) and whose solution was analyzed to provide the systems natural frequency, 0 , which is the basic descriptor of saturation level of the drug. It was established that the resonance period for the final model, that is, T 0 =3.76912 hrs, is in the acceptable therapeutic range and agrees well with the data for the existing insulin therapy. By employing the model, it is shown that, the peak, which is the time period for insulin to be most effective in lowering blood sugar, is shorter than T 0 =5.3199 hrs, for the existing model. This model would help the medical practitioners to predict drug therapy in patients with Diabetes Mellitus, in such a way that the concentration of the drug remains in the therapeutic range
Mathematical modelling of the role of interference on the transmission dynamics and management of Hiv and Aids
(2018-09-22) Mutwiwa, Jacinta M.; Nthiiri, Joyce K.; Kwach, O.
In this paper, a deterministic mathematical model incorporating interference is developed andanalysed to investigate the role of interference on the transmission dynamics and managementof HIV and AIDS. The model is shown to be positively invariant as well as bounded. Theendemic state is shown to exist provided that the reproduction number is greater than unity.Furthermore, by the use of Routh-Hurwitz criterion and suitable Lyapunov functions, theendemic states are shown to be locally and globally asymptotically stable. This implies thatdisease transmission levels can be kept quite low or manageable with minimal deaths at the peaktimes of the re-occurrences. Numerical simulations indicate that minimal interference againstthe disease lowers the rate of infection and enhances the disease management.
Mathematical model for detecting diabetes in the blood
(Journal of applied mathematical sciences, 2011) Kwach, B.O.; Ongati, O.; Simwa, R.
This study presents a new mathematical model for Blood Glucose Regulatory System(BGRS) which includes epinephrine as a third variable in the form, Y˙ = AY, and whose solution has been analyzed for equilibrium and stability to provide the blood glucose concentrations for diabetics and non-diabetics. We establish that the final model is asymptotically stable compared to the existing models, that is, the eigenvalues of the coefficient matrix are complex numbers with negative real parts. Furthermore, the resonance period for the final model, that is, T0 = 2.9847134 hours, is far less than that of the existing model, showing that the glucose concentration returns to normal level within a shorter time.
A Predator-Prey Model with a Time Lag in the Migration
(2014) Wasike, A.A.M.; Lawi, G.O.; Mulati, O.N.
We develop a two-patch migration model of the classical LotkaVoltera predator-prey system with a time lag in the migration between patches. We show that when the migration rate is less than the prey growth rate the species in at least one patch survives. When the migration rate is greater than the prey growth rate, the species in both
Factors that influence teachers perception towards the implementation of strengthening mathematics and science in secondary education (SMASSE)Programme in Bungoma .
(201-03) Peter, Wamalwa; Masibo, Edwin NM.; Mutsotso, Stanley N.
Dismal performance in Mathematics and Science subjects poses a challenge towards realization of the Kenya National development goal of industrialization by the year 2020. The government of Kenya in liaison with the Japanese government came up with Strengthening of Mathematics and Science in Secondary Education (SMASSE) to remedy the situation. This was to be achieved through in-service training for mathematics and science teachers to improve on teacher classroom practices and student’s learning.This paper presents a study on motivation as a strategy that influence mathematics and science teachers’ participation in the inservice training. The study was guided by Rogers’s innovation-implementation diffusion theory. Descriptive survey design was used and thetarget population was 1450 teachers teaching in 275 secondary schools and 9 sub-county Quality Assurance Officers (QASO). Simple random sampling and purposive sampling techniques were used for a sample size of 438. Data was collected using questionnaires, interview schedules and an observation guide. It was analyzed using both descriptive and inferential statistics and it was found out that provision of motivation to teachers influence their participation. It was concluded that all teachers would attend INSETs if motivated. From the conclusion, it was recommended that the national SMASSE office and the MoEST should consider teachers’ views on motivation.
Effect of Peer Teaching among Students on their Performance in Mathematics
(International journal of scientific research and innovative technology, 2016) Elizabeth, O.; Mutsotso, AN .; Masibo, Edwin NM.
Mathematics is a key subject in the school curriculum and is considered as critical filter for learners’ career choices. However, over the years mathematics has been one of the poorly performed subjects in the Kenya Certificate of Secondary Education (KCSE). In an attempt to improve performance, great effort has been put into use of appropriate teaching and learning methods that stimulate learners’ interest in the subject. This study was done in 12 randomly selected schools in Bungoma South Sub County. The objective of this study was to determine the effect of peer teaching among students on their performance in mathematics in the teaching and learning process. The study was guided by Vygotsky’s social interaction theory of learning. This theory opines that social interaction plays a fundamental role in cognitive development and that all learning occurs in a cultural context and involves social interactions where peers assist learners in developing new ideas and skills. The target population was heads of departments, teachers of mathematics and form three students. Twelve heads of department, twenty four mathematics teachers and one hundred and seventy six form three students were drawn from the sampled schools to participate in the study. A descriptive survey design was adopted for the study. Data was collected using a teachers’ questionnaire, students’ questionnaire, interview schedule for heads of department and students’ achievement test. Data was analyzed using Statistical Package for Social Sciences (SPSS) version 17.0 and Statistical t-test. The Students’ Achievement test showed that peer teaching increases students’ achievement in mathematics and 100% of the heads of department interviewed believe that peer teaching strategy improves performance. The conclusions made from the study were that peer teaching encourages students’ motivation to learn mathematics, enhances understanding of mathematics concepts and builds confidence in the students. Students should be allowed to form discussion groups where peer teaching can be done especially at the end of every topic as it offers a great opportunity in overcoming the challenge of a demanding mathematics curriculum.
Exemplary practice and outcome-based mathematics instruction in Kenyan schools/Prácticas ejemplares basadas en resultados de instrucción de las matemáticas en las escuelas de Kenya
(Journal of science education, 2007) Wekesa, Duncan Wasike
The general performance in mathematics among secondary school students in Kenya has over the years been poor. There is evidence that the learning environment is murky and often intractable. This calls for exemplary practice and outcome based instruction where teachers have to create a learning environment in which the learners' capabilities as well as their existing ideas could be effectively used to produce change in the learners' cognitive structures through collaborative social interactions
Effectiveness of teaching preparation on mathematics achievement: The case of Kenya
(2011-11-01) Amadalo, M.M.; Wekesa, W.D.; Wambua, J.M.
Despite playing a central role in peoples’ daily life, the average Kenyan secondary school students’ mathematics score in national examinations has consistently averaged below 40%. The contribution of teachers’ lesson preparations and subsequent delivery leading to this poor result has not been investigated sufficiently in Kenyan secondary schools. This is especially so for topics deemed to be difficult. The present study investigated the effect of teacher preparations when teaching the topic “Vectors” to form three secondary school students. The instructional plans impact on achievement as well as on skills performance in Mathematics formed the objectives of the study. The Solomon- four experimental design was adopted. Professionally drawn Instructional Plans provided the treatment. Students’ achievement was determined using a Mathematics Achievement Test, MAT. The study determined that the use of the instructional plans improved students’ achievement and skill performance compared to the control group. Consequently use of instructional plans when teaching mathematics was recommended for improved students’ achievement. Emphasis on students’ stepwise skill performance rather than insistence on acquisition of correct answers during problem solution in mathematics was recommended

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