SIMC 2017

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http://hdl.handle.net/11071/10447

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On joint essential maximal numerical ranges
(Strathmore University, 2017) Omoke, Priscah Mogotu
The study of numerical range of an operator has been an area of intense research. The motivation for the development arose from the classical theory of quadratic forms. It forms a very important aspect in functional analysis, operator theory and its applications to economics, quantum chemistry and quantum computing amongst other fields. A lot of results have been obtained on numerical ranges particularly by Fong, Khan among others. The concept of maximal numerical ranges of a bounded operator T E B (H) was studied by Stampfli who used it to derive an identity for the norm of derivation. This concept was later generalized by Ghan to the joint maximal numerical ranges of m- tuples of operator. The joint essential maximal numerical range was studied by Khan and established that the joint essential maximal numerical range can be empty. In this paper we have showed that the joint essential maximal numerical range is nonempty, compact and convex. We also established that each element in the joint essential maximal numerical range is a star centre of the joint maximal numerical range. The result obtained show that star-shapedness is related to convexity in that a convex set is starshaped with all its points being star centres.
Initial reflections on teaching and learning Mathematics using tablets in a Prison Education Centre
(Strathmore University, 2017) Njurai, Evelyn
The reflections reported in this paper involve an intervention using a tablet as a tool for enhancing teaching and learning of mathematics in a Prison Education Centre in Kenya. The tablet is connected wirelessly to a server running the tutor-web mathematics software, installed at the Centre. The prison staff-teachers first duty is security and then teaching. With these priorities, the teachers sometimes do not attend to their mathematics lessons for hours or days and hence inmate-students miss continued mathematical instructions. To address the challenge, the use of tablet was implemented as a tool to facilitate continued instructions times and subsequent learning. The participants were Form 1 students in the secondary section of the Centre. Findings show that the use of technology did not only facilitate continued learning, but also changed the inmate-students’ negative attitude towards mathematics, motivated them and signs of improved mathematics performance. Challenges of using the tablet as a tool were noted. In general, the project continues to facilitate achievement of the mission of Kenya Prison Service (KPS) of containment and rehabilitation of offenders, responsive administration of justice, social reintegration and community protection. Future improvements on the project have been suggested. The findings inform the government’s implementation of digital learning of mathematics which is key for the development of the country.
From smileys to Smiley coins -using a crypto currency in education
(Strathmore University, 2017) Lentin, Jamie; Stefansson, Gunnar
This paper describes a crypto currency to reward students for their studies. The currency bears the apt name Smiley coin or SMLY and is used within the tutor-web online learning platform. In order to make the SMLY more attractive several approaches have been used, including support from companies whose services can be purchased for SMLY. The paper describes the use of the SMLY as a reward mechanism in a large undergraduate calculus course, including student adoption, student use of SMLY and abuse tracking.
Mathematical modelling of Hepatitis E Virus (HEV) and Chronic Myeloid Leukemia (CML) co-infection dynamics
(Strathmore University, 2017) Katakara, Denis Kagyera; Faraimunashe, Uganda Chirove
There are major advances which have been made to understand HEV and CML transmission dynamics but none of these have considered the effects of transmission parameters on the burden of HEV on CML prevalence in a co-infection scenario. In this paper we formulated a mathematical model for the co-infection of HEV and CML using a system of ordinary differential equations, in order to understand the effects of the co-infection on HEV and CML and vice versa in a human population. The model was analysed and steady state conditions were derived. Our results showed that the disease free equilibrium was both locally stable and globally stable if the basic reproduction number, R0 ≤ 1 and unstable if the basic reproduction number, R0 > 1. Our results also suggest that, (i) HEV reduces the CML infectives and accelerates the co-infection, (ii) CML enhances the progression of both HEV infection and the co-infection and, (iii) there is an increase in HEV-CML burden due to coinfection compared to single infections of either HEV or CML.
Perturbation theory of differential operators
(Strathmore University, 2017) Nyamwala, Fredrick Oluoch
Suppose that M and S are differential operators and T is given by T = M-zS, where z is the spectral parameter. Here, we give the necessary and sufficient conditions for the deficiency indices and spectral properties of selfadjoint realisations of T and M to be equal on Hilbert spaces of square integrable functions.
Bayesian estimation of Multivariate Stochastic Volatility by applying state space models
(Strathmore University, 2017) Agasa, Lameck; Ombasa, Kiame
This work seeks to apply a Bayesian analysis in estimating multivariate stochastic volatility (MSV) using state space models. A multiplicative model based on inverted Wishart and multivariate singular beta distributions is proposed for the evolution of the volatility, and a flexible sequential volatility updating is employed. Being computationally fast, the resulting estimation procedure is particularly suitable for on-line forecasting. Bayesian MCMC is applied to estimate high dimensional problems. Three test are conducted on estimates: the log likelihood criterion, the mean of standardized one-step forecast errors, and sequential Bayes factors. The test and procedure are applied in real data set that will comprise ten exchange rate Kenyan shillings versus other currencies in Nairobi stock exchange.
On one nonclassical problem for the Laplace equation
(Strathmore University, 2017) Danilkina, Olga
Nonlocal problems received special attention over the past few decades due to efficient descriptions of various phenomena in physics, chemistry and engineering in cases when data from the boundary of the process or initial data are not available. In this paper, we study a nonclassical problem for the Laplace equation with nonlocal integral boundary conditions in the rectangular domain. We introduce a number of auxiliary problems, obtain a system of integral equations and then construct a recurrent relation based on the homotopy analysis method (HAM) to find a solution of the nonlocal problem. The existence and uniqueness theorem is proved.
Rational homotopy type of the component of inclusion in the nthspace of continuous mappings from Gr (k, n) to Gr(k, n + 1)
(Strathmore University, 2017) Otieno, P.A; Gatsinzi, J.B; Onyango, Otieno V
A complex manifold can be embedded in some complex projective space CP (N ), in particular, the Grassmann manifold Gr(n, k) of k dimensional subspaces in Cn can bembedded in some complex projective space CP (N ).Moreover G(k, n) ‹→ G(k, n + 1). For k = 1, we get a one dimensional vector space which is the complex projective plane and is an embedding of CP (n)in CP (n + 1).The Grassmanian admits a CW structure and any CW structure on a space provides a filtration relative to the empty space. To a simply connected topo-logical space, Sullivan associates a commutative differential graded algebra(∧V, d) which encodes the rational homotopy type of X. This is called aSullivan model of X. Given that H∗(CP (n), Q) is the truncated polynomialalgebra ∧x/(xn+1), one gets a a Sullivan model of the form () ∧ (x, y), d)where |x| = 2, |y| = 2n + 1 and dx = 0, dy = xn+1. For k ≥ 1, one might usethe homeomorphism G(k, n) = U (n)/(U (k) × U (n − k)) to find a Sullivanmodel. In this paper, we use a Sullivan model of the inclusion Gr(k, n) −→ Gr(k, n+1) to compute the rational homotopy type of the component of the inclusionin the space of mappings from Gr(k, n) to Gr(k, n + 1).
Identifying important Mathematics underlying student thinking
(Strathmore University, 2017) Ochieng, Mary
Research in mathematics teacher education suggests the benefits of instructional practices that use student mathematical thinking (e.g., Fennema, et al., 1996; Stein & Lane, 1996), but such practices are complex and difficult both to understand and to enact (Ball & Cohen, 1999; Feiman-Nemser, 2001; Sherin, 2002; Silver, Ghousseini, Gosen, Charalambous, & Font Strawhun, 2005). Preliminary efforts to investigate the practice of using student thinking suggest that productive use requires engaging the class in making sense of mathematical ideas that have originated with students (Leatham, Peterson, Stockero & Van Zoest, 2015). In order to facilitate students’ engagement in making sense of these mathematical ideas, the teacher needs to be able to identify the mathematics to which the idea relates. In this session we will talk about the teaching practice of identifying mathematical points in student thinking, including how this practice may be used to foster understanding of important mathematical ideas and how to support teachers to develop the practice.
The Squares of numbers using Komozian series
(Strathmore University, 2017) Omoze, Hillary Ongoyo; Matuya, John Wanyonyi
In this paper we have determined the squares of numbers using the Komozian series. The method has established series which analyses the digits present in a given number and sets priority for every digit depending on its place value. The number of digits present are represented by k 1: kis the number of digits present other than the last digit which is taken to be  and has a place value of ones while the place values of the other digits is given by 10 j and the total value of the digits is given by 10 j m wherejj  k, k 1, k  2,..., 3, 2,1, 0 . The number x would be given byx 10k m10k 1 mk 110k 2 mk 2 ... 103 m102 m101 m100 mThe formula is given in series depending on the value of k , it is assumed that when square is known and therefore needs not to be calculated.k  0 the The presentation spans k  1,k  2,k  3 and then generalizes for k  nk 2.
The reproductive ratio: at macro and micro levels
(Strathmore University, 2017) Mbogo, Rachel Waema
The basic reproduction ratio, R0, is one of the fundamental concepts in mathematical biology defined as “the average number of secondary infections caused by a single infectious individual during their entire infectious lifetime”, however, “secondary” depends on context For both in vivo and epidemiological models of pathogen dynamics, the basic reproductive ratio, R0, is a powerful tool for gauging the risk associated with an emerging pathogen, or for estimating the magnitude of required control measures. Techniques for estimating R0, either from incidence data or in-host clinical measures, often rely on estimates of mean transition times, that is, the mean time before recovery, death or quarantine occurs. In many cases, however, either data or intuition may provide additional information about the dispersal of these transition times about the mean, even if the precise form of the underlying probability distribution remains unknown. In this paper we elucidate common situations in which R0 is sensitive to the dispersal of transition times about their respective means both at macro and micro levels. We then provide simple correction factors that may be applied to improve estimates of R0 when not only the mean but also the standard deviation of transition times out of the infectious state can be estimated.
Correlation between Malaria and Pneumonia in Kwale County, Kenya
(Strathmore University, 2017) Mungai, Esther
Malaria and Pneumonia have been the major causes of morbidities and mortilities in Kwale county in the past years. Majority of the people in the county who get pneumonia have a higher incidence of Malaria. Thus, even though Malaria is prevalent in Kwale, individuals whose Malaria incidence is lower rarely get Pneumonia. In this project we try to find whether there is any correlation between the two diseases. We investigate the factors that may contribute to this correlation. We derive that control of these factors would lead to the reduction of both diseases.
Initial reflections on teaching and learning Mathematics using tablet in Aprison Education Centre
(Strathmore University, 2017) Njurai, Evelyn; Stefansson, Gunnar; ónsdóttir, Anna H. J; Mwenda, Patrick; Obonyo, Mose; Kariuki, Joseph
The reflections reported in this paper involve an intervention using a tablet as a tool for enhancing teaching and learning of mathematics in a prison education centre in Kenya. The tablet is connected wirelessly to a server running the tutor-web mathematics software, installed at the Centre. The first duty of prison staff-teachers is security and then teaching. With these priorities, the teachers sometimes do not attend to their mathematics lessons for hours or days and hence inmate-students miss continued mathematical instructions. To address the challenge, the use of tablet was implemented as a tool to facilitate continued instructions times and subsequent learning. The participants were Form 1 students in the secondary section of the Centre. Findings show that the use of technology did not only facilitate continued learning, but also changed the inmate-students’ negative attitude towards mathematics, motivated them and improved their mathematics performance. Challenges of using the tablet as a tool were noted. In general, the project continues to facilitate achievement of the mission of Kenya Prison Service (KPS) of containment and rehabilitation of offenders, responsive administration of justice, social reintegration and community protection. Future improvements on the project have been suggested. The findings inform the government’s implementation of digital learning of mathematics which is key for the development of the country.
A Latent profile analysis of the Science teacher-trainees’ self-regulated learning strategies
(Strathmore University, 2017) Muwonge, Charles Magoba; Ssenyonga, Joseph
In the present study, we conducted a latent profile analysis to identify three clusters of Science teacher-trainees based on their self-regulated learning strategies. The sample comprised of 527 undergraduate teacher-trainees (Mean age = 22 years; SD = 2.20) randomly selected from seven universities in Uganda. Data was collected using the modified Motivated Strategies for Learning Questionnaire (MSLQ). These profiles including the (a) competent/expert self-regulated learners (259 students; 49.2%), (b) average self-regulated learners (193 students; 36.6%), and (c) novice self-regulated learners (75 students; 14.2%) differed significantly with respect to their motivational beliefs and academic performance, with the expert and average self-regulated learning profiles having better grade point average (GPA) scores and higher motivation compared to students in the novice self regulated learning profile. All profiles were similar with respect to age, gender and year of study. Additionally, task value and self-efficacy (but not demographic characteristics) significantly predicted latent profile membership. It is therefore important to understand such individual differences among science teacher-trainees in order to improve on their self regulated learning skills. Further implications are discussed in the paper.
Motivated strategies for learning questionnaire - modeling the relationship between motivational beliefs, cognitive learning strategies and academic performance of Mathematics teacher-trainees
(Strathmore University, 2017) Muwonge, Charles Magoba
Self-regulated learning refers to the conscious planning, monitoring, and evaluation of one’s own learning through use of appropriate behavioral, meta-cognitive and cognitive strategies to enhance academic performance. Self-regulated learners are highly motivated, use a variety of learning strategies in achieving their academic goals and as such are high achievers. Although self-regulated learning has received much attention over the past decades, research on how teacher-trainees regulate their own learning has been scarce and especially in third world countries. Lack of research in this aspect has not only created a knowledge gap but also compromised efforts to tackle to challenge of poor academic performance among teacher trainees. In this study, we examined the relationship between motivational beliefs, cognitive learning strategies and academic performance among Mathematics teacher-trainees in Uganda. The sample comprised of 689 undergraduate students randomly selected from eight universities across Uganda. Data was collected using the Motivated Strategies for Learning Questionnaire (MSLQ) and analysed using structural equation modeling. A structural equation model indicated that cognitive learning strategies mediated the relationship between motivational beliefs and academic performance of these undergraduate teacher-trainees. Motivational beliefs influenced student’s performance through enhancing their critical thinking and organization learning strategies. Therefore, there is need to enhance leaner’s ability to adopt adaptive learning strategies in-order to enhance their academic achievement at the university.
Modelling the role of treatment and vaccination in the control of transmission dynamics of pneumonia among children in Uganda
(Strathmore University, 2017) Kizito, Mohammed; Tumwiine, Julius
Pneumonia is one of the leading causes of serious illness and deaths among children around the world. Efforts to effectively treat and control the spread of pneumonia is possible if its dynamics is well understood. In this paper, a mathematical model for the transmission dynamics of pneumonia is studied. The population is divided into five epidemiological classes to evaluate the role of treatment and vaccination in mitigating the spread of the disease. A system of differential equations is used to study the disease dynamics. Model analysis is carried out to establish the existence and stability of the steady states. It is revealed that the disease-free equilibrium point is globally stable if and only if the basic reproduction number R0<1 and the disease will be wiped out of the community. If R0>1, the endemic equilibrium point is globally stable and the disease persists at the endemic steady state. We infer the impact of control strategies on the dynamics of the disease through sensitivity analysis of the effective reproduction number Re from which the results showed that the combination of treatment and vaccination can eradicate the pneumonia infection.
On a-self-adjoint, a-unitary opeators and quasiaffinities
(Strathmore University, 2017) Sitati, Isaiah
In this paper, we investigate properties of A-self adjoint operators and other relations on Hilbert spaces. In this context, A is a self-adjoint and an invertible operator. More results on operator equivalences including similarity, unitary and metric equivalences are discussed. We also investigate conditions under which these classes of operators are self- adjoint and unitary. We finally locate their spectra.
Metric equivalence as an almost similarity property
(Strathmore University, 2017) Gitonga, Eric M.; Musundi, Sammy W.; Nzimbi, Benerd M.
Various results that relate to almost similarity and other classes of operators such as isometry, normal, unitary and compact operators have been extensively discussed. It has been shown that if operators S and T are unitarily equivalent, then S is almost similar to T. Similarly, it has been shown that operators A and B are such that A is almost similar to B and if A is Hermitian, thenA and B are said to be unitarily equivalent. Metric equivalence property which is a new relation in operator theory has drawn much attention from mathematicians in the recent past. Two operators S and T are unitarily equivalent if they are metrically equivalent projections. It has been shown that if operators S and T are unitarily equivalent, then S is metrically equivalent to T. However, there is no literature that has been shown for the conditions under which metric equivalence and almost similarity coincide. In this paper we will therefore strive to establish the equivalence relation between metric equivalence property and almost similarity relation. To achieve this, properties of invertible operators, normal operators, similar operators, unitarily operators as well as projection and self-adjoint operators will be employed.
Evidence-based technology to enhance mathematics education from Iceland to Kenya
(Strathmore University, 2017) Stefansson, Gunnar; Stern, David; Lentin, Jamie; Jonsdottir, Anna Helga
Enhancing the mathematics classroom using computerised drills for homework is shown to provide considerable benefits for student learning in diverse regions, from high-tech westerns classrooms to those with neither Internet nor stable electricity. This paper describes results from several experiments using the tutor-web educational system for teaching mathematics with student groups ranging from upper primary school through graduate school. When existing teaching methods are augmented using this technology, quantitative and qualitative results across all regions and age groups indicate improvements in learner performance. Although it has a wide range of functions, the tutor-web system is most importantly a drilling system, which would normally be classified as an Adaptive and Intelligent Web-based Educational System (AIWBES) providing quite personalised learning, tailoring a drill sequence for each student. In addition to personalised grading schemes aimed for enticing the student to continue, a different reward scheme is implemented by giving the students a cryptocurrency whenever they complete a topic with excellence. Initial results of the effects of these various reward schemes are also presented.
On the Banach algebra numerical range
(Strathmore University, 2017) Kaubya, Francis; Agure, John Ogonji; Emenyu, John
Let A denote a unital Banach algebra and SA denote its unit sphere. It was proved by F. F. Bonsalland J. Duncan that when the Banach algebra is unital, the algebra numerical rangeis identical to a subset of itself. Basing on this fact, we have established that the union overall elements of SA of the sets of support functionals for the unit ball at each x ∈ SA i.e.Sx∈SD(A, x) is equal to the set of normalized states i.e. D(A, eˆwhen the Banach algebraAˆ).ˆeeeeA is unital (eˆ is the unit element). The implication is that even when the algebra is smooth, theset of normalized states i.e. D(A, e) is not a singleton. Hence algebra numerical range is notsingleton except when the element is a scalar multiple of the unit. Also, consider the statementsP1 and P2;(P1) The union over all elements of SA of the sets of support functionals for the unit ball ateach x ∈ SA i.e. Sx∈S D(A, x) is equal to the set of normalized states i.e. D(A, eˆ-component algebra numerical(P2) The algebra numerical range i.e. V (A, a) is equal to the eˆrange i.e. V (A, a, e).We have proved that the statements P1 and P2 are equivalent under a suitable condition. Further,F.F Bonsall and J. Duncan proved that for a unital Banach algebra, the numerical radius is anequivalent algebra norm. In that proof, the inequalities k > v(a) and k ≥ 1 "a" were used toconclude that 1 "a" ≤ v(a) (e is the irrational number 2.718...). However, these two inequalities lead to the undesired inequality 1 "a" ≥ v(a). In this dissertation we improve on this proof bymerging the property of the complex roots of unity used by Bonsall and Duncan together withthe geometric series argument and making a different choice of the arbitrary element b ∈ A ofnorm less than one to derive the desired inequality 1 "a" ≤ v(a).andˆD(A, eˆe) = 1}.) = {f ∈ SAt : f (ˆ Elements of D(A, x) are called support functionals for the unit ball at x ∈ SA while elements of D(A, e) are called normalized states. For each a ∈ A and x ∈ SA defineV (A, a, x) = {f (ax) : f ∈ D(A, x)}

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