Department of Mathematics

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On generation of measurable covers for measurable sets using multiple integral of functions
(200-09-20) Olwamba, Otanga Levi.; Aywa, Shem.; Oduor, Owino Maurice.
A fundamental notion in many areas of mathematics, including optimization, probability, variational problems, functional analysis and operator theory, is that of an integral functional. By this is meant an expression of the form If(x) = ~ f(s,x(s))p(ds), x E X, where X is a linear space of measurable functions defined on a measure space (S,A,~) and having values in a linear space E. The function f: S x E § R is the associated Inte~rand. Classically, only finite integrands on S x R n were studied, usually under the assumption that f(s,x) was continuous in x and measurable in s (the Carathgodory condition). However, from the modern point of view it is essential to admit possibly infinite values for f and If, since it is in this way that important kinds of constraints can most efficiently be represented. Such integrands require a distinctly new theoretical approach, where questions of measurability and the existence of measurable selections are prominent and are reflected in a concept of "normality". The purpose of these notes is to provide a relatively thorough treatment of the most common case in applications, that where E = R n. While many of the results have extensions in one way or another beyond this case, as indicated to some extent in the text, these are often more complicated technically and may require further restrictions. For example, it is only for R n that one presently knows how to develop a complete theory without assuming that the measurable space is complete, an assumption which appears to be awkward in some situations. In treating Inflnite-dimenslonal spaces E, there are the usual problems of the multiplicity of topologies and dualities which must be ironed out. It is desirable, therefore, to have available a full and consistent exposition of the details in the basic case of E = R n, freeing one from the need to search for auxilliary results through sequences of papers wlth varying frameworks.
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.
Sequence in the range of a vector- value measure
(Journal of school of computer, statistical and mathematical sciences, 1999) Aywa, Shem.
Liapounoff, in 1940, proved that the range of a countably additive bounded measure with values in a finite dimensional vector space is compact and, in the nonatomic case, is convex. Later, in 1945, Liapounoff showed, by counterexample, that neither the convexity nor compactness need hold in the infinite dimensional case. The next step was taken by Halmos who in 1948 gave simplified proofs of Liapounoff's results for the finite dimensional case. In 1951, Blackwell [I] considered the case of a measure represented by a finite dimen- sional vector integral and obtained results similar to those of Lia- pounoff for these measures. Various versions of Liapounoff's theorem appeared in the 1950's and 1960's, and in 1966, Lindenstrauss [8] gave a very elegant short proof of Liapounoff's earlier result. Finally, in 1968, Olech [9] considered the case of an unbounded measure with range in a finite dimensional vector space. The purpose of this note is to demonstrate that the closure of the range of a measure of bounded variation with values in a Banach space, which is either a reflexive space or a separable dual space, is compact and, in the non- atomic case, is convex. To this end, let Q be a point set and z be a a-field of subsets of U. If X is a Banach space, then an p-valued measure is a countably addi- tive function F defined on 2 with values in X. F is of bounded varia- tion if var(F)(Q) =sup E ll F(En)|| < a W n where the supremum is taken over all partitions r = { En }=C2 consisting of a finite collection of disjoint sets in z whose union is U. A set EE2 is an atom of F if F(E) $0 and E'C2, E'CE imply F(E') = 0 or F(E') = F(E). F is nonatomic if F has no atoms. The following theorem is the main result of this note. THEOREM 1. Let X be a Banach space which is either a reflexive space or a separable dual space. If F: 2;- X is a measure of bounded variation, then the range of F is a precompact set in the norm topology of t. More- over, if F is nonatomic, then the closure of the range of F is compact and convex.
On convergence of sections of sequences in Banach spaces
(Springer-Verlag, 2000-02-01) Aywa, Shem; Jan, Fourie
An elementary proof of the (known) fact that each element of the Banach spaceℓ w p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ w p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.
On convergence of sections of sequences in Banach spaces
(Springer-verlag, 2000-02-01) Aywa, Shem.; Fourie, Jan H.
An elementary proof of the (known) fact that each element of the Banach spaceℓ w p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ w p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.
On summing multipliers and applications
(Academic press, 2001-01-01) Aywa, Shem.; Fourie, Jan H.
A scalar sequence (αi) is said to be a p-summing multiplier of a Banach space E, if ∑∞i = 1‖αixi‖p < ∞ for all weakly p-summable sequences in E. We study some important properties of the space mp(E) of all p-summing multipliers of E, consider applications to E-valued operators on the sequence space lp, and extend this work to general “summing multipliers.” The case p = 1 shows close resemblance to the work of B. Marchena and C. Piñeiro (Quaestiones Math., to appear), where the results originated from the authors' interest in sequences in the ranges of vector measures.
On summing multipliers and applications
(Journal of Mathematical Analysis and Applications, 2001-01-01) Aywa, Shem; Jan, Fourie
A scalar sequence (αi) is said to be a p-summing multiplier of a Banach space E, if ∑∞i = 1‖αixi‖p < ∞ for all weakly p-summable sequences in E. We study some important properties of the space mp(E) of all p-summing multipliers of E, consider applications to E-valued operators on the sequence space lp, and extend this work to general “summing multipliers.” The case p = 1 shows close resemblance to the work of B. Marchena and C. Piñeiro (Quaestiones Math., to appear), where the results originated from the authors' interest in sequences in the ranges of vector measures.
Spaces of compact operators and their dual spaces
(Business mathematics and informatics, 2002) Aywa, Shem.; Fourie, HJ.
Theω′-topology on the spaceL(X, Y) of bounded linear operators from the Banach spaceX into the Banach spaceY is discussed in [10]. Let ℒw' (X, Y) denote the space of allT∈L(X, Y) for which there exists a sequence of compact linear operators (T n)⊂K(X, Y) such thatT=ω′−limnTn and let |||T|||:={supn||Tn||:Tn∈K(X,Y),Tn→w′T} . We show that (Lw′,|||⋅|||) is a Banach ideal of operators and that the continuous dual spaceK(X, Y)* is complemented in (Lw′(X,Y),|||⋅|||)∗ . This results in necessary and sufficient conditions forK(X, Y) to be reflexive, whereby the spacesX andY need not satisfy the approximation property. Similar results follow whenX andY are locally convex spaces.
Spaces of compact operators and their dual spaces
(Springer-Link, 2004-01-13) Aywa, Shem; Jan, Fourie
Theω′-topology on the spaceL(X, Y) of bounded linear operators from the Banach spaceX into the Banach spaceY is discussed in [10]. Let ℒw' (X, Y) denote the space of allT∈L(X, Y) for which there exists a sequence of compact linear operators (T n)⊂K(X, Y) such thatT=ω′−limnTn and let|||T|||:={supn||Tn||:Tn∈K(X,Y),Tn→w′T}. We show that(Lw′,|||⋅|||) is a Banach ideal of operators and that the continuous dual spaceK(X, Y)* is complemented in(Lw′(X,Y),|||⋅|||)∗. This results in necessary and sufficient conditions forK(X, Y) to be reflexive, whereby the spacesX andY need not satisfy the approximation property. Similar results follow whenX andY are locally convex spaces.
Effectiveness of a Language Based Program in School Mathematics on Students Understanding of Statistics
(2006-12-01) Wekesa, W.D.
. Mathematical knowledge and understanding is important not only for scientific progress and development but also for its day-to-day application in social sciences and arts, government, business and management studies and household chores. But the general performance in school mathematics in Kenya has been poor over the years. There is evidence that students have problems in understanding and interrelating the symbols and special language structure as used in mathematics. Nevertheless in a recent study, a program called Socialized Mathematical Language (SML) module was designed to enhance student’s learning outcomes in school mathematics. The study was carried out in a real classroom setting that involved comparisons between the treatment and control groups. A Solomon Four Group quasi-experimental design was employed to involve four high schools in Bungoma District. A total of 156 form two students enrolled in four intact classes from the selected schools were exposed to the same content in statistics for a period of two weeks. Three dependent measures namely the Mathematics Achievement Test (MAT), the Mathematics Skill Test (MST) and the Mathematics Classroom Environment Questionnaire (MCEQ) were used to assess the effectiveness of the program on students’ academic achievement in understanding of statistics, their skill performance and perceptions of the classroom environment during statistics lessons. The results affirm statistically significant learning gains in favour of the treatment groups. The study concludes that the use of SML program has a major implication for school mathematics instruction in the area of statistics.
Measurable feller semigroups on rn
(2007) Oduor, Maurice.; Owino; Aywa, Shem.
Fractional derivatives are used to model anomalous diffusion, which occurs when the particles spread in a different manner than the prediction of the classical diffusion equation ∂ ∂tu(x, t) = D ∂ 2 ∂x2 u(x, t), u(x, 0) = f(x). The solution u (x, t) depends on location x ∈ R and time t ≥ 0 and models the dispersion. A known model for an anomalous diffusion (see [6]) is the fractional diffusion equation, where the usual second derivative in space is replaced by a fractional derivative of order α, 0 < α < 2, ∂ ∂tu(x, t) = D ∂ α ∂xα u(x, t), u(x, 0) = f(x). We observe that ∂ α ∂xα is a pseudodifferential operator. Thus, we can extend this equation to ∂u ∂t (x, t) = (Au(· , t))(x), u(x, 0) = u0(x), where A is a pseudodifferential operator. A study of the solutions of a generalized reaction-diffusion equation of the form ∂u ∂t (x, t) = (Au (· , t)) (x) + f (x, u (x, t)), u (x, 0) = u0 (x), MATH. REPORTS 12(62), 2 (2010), 181–188 where A is a pseudodifferential operator which generates a Feller semigroup, was given in [9]. In this paper we consider the fractional Cauchy problem ∂ β ∂tβ u(x, t) = (Au(·, t))(x), u(x, 0) = f(x), where ∂ β ∂tβ u (x, t) is the Caputo fractional derivative in time and A is a pseudodifferential operator which generates a Feller semigroup. In [1] and [2] was shown that the solution of fractional Cauchy problem, where 0 < β < 1, t ≥ 0 and A is the generator of bounded continuous semigroup {T (t)}t≥0 on the Banach space X, can be expressed as an integral transform of the solution to the initial Cauchy problem ∂ ∂tu(x, t) = (Au(·, t))(x), u(x, 0) = f(x). Starting from this integral transform, we give a formula for the solution u(x, t) = S(t)f(x) of the fractional Cauchy problem. We show that {S(t)}t≥0 is a family of pseudodifferential operators. Their symbols are obtained by transformation of the symbols of the semigroup {T(t)}t≥0, where u(x, t) = T(t)f(x) is the solution to the initial Cauchy problem.
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
Aspects of (p.q) summing multipliers
(Vdm verlag Dr. Muller aktiengeselischaft & co. kg., 2010) Aywa, Shem.; Agure, J.O.; Rao, W.
A sequence (uj )j∈N of operators in L(X, Y ) is a (p, q)-summing multiplier (or (p, q)-summing sequence of operators), in short (uj ) ∈ `πp,q (X, Y ), if there exists a constant C > 0 such that, for any finite collection of vectors x1, x2, . . . xn in X, it holds that ³Xn j=1 kujxjk p ´1/p ≤ C sup n³Xn j=1 |x ∗ xj | q ´1/q ; x ∗ ∈ BX∗ o . Some examples of these operators, inclusions between the spaces and connections with spaces of multipliers are presented.∗ Mathematics Subject Classification (2000): 47B10. Key words: Summing operators, vector-valued multipliers. 1. Introduction. Let X and Y be two real or complex Banach spaces and let E(X) and F(Y ) be two Banach spaces whose elements are defined by sequences of vectors in X and Y (containing any eventually null sequence in X or Y ). A sequence of operators (un) ∈ L(X, Y ) is called a multiplier sequence from E(X) to F(Y ) if there exists a constant C > 0 such that ° °(ujxj ) n j=1 ° ° F (Y ) ≤ C ° °(xj ) n j=1 ° ° E(X) for all finite families x1, . . . , xn in X. The set of all of multiplier sequences is denoted by (E(X), F(Y )). For the study of such multipliers for the cases of E(X) and F(Y ) corresponding to vector-valued Hardy spaces, vector-valued Bergman spaces, vector-valued BMOA or spaces of vector valued Bloch functions the reader is referred to [AB1, Bl1, Bl2, Bl3, Bl4].
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.
Income source diversification and financial performance of commercial banks in Kenya
(Mku Journals, 2011-01-21) Teimet, Paul Rotich.; Ochieng, Damianus Okaka.; Aywa, Shem.
The profitability of commercial banks depends heavily on the net of income generating activities and the related activities’ expense. Due to the problem of profitability and stiff competition in the industry, commercial banks have changed their behavior of income sources, by increasingly diversifying into non-intermediation income generating activities as opposed to the traditional inter-mediation income generating activities. The objective of this paper was to establish the impact of income source diversification on financial performance of commercial banks in Kenya. This has been achieved through: establishing the level of income source diversification of commercial banks in Kenya and establish whether income source diversification improves financial position of commercial banks. This was a census study of all registered 44 commercial banks in Kenya and relied heavily on documentary secondary data for 5 year study period (2005-2009) and validated by primary data achieved through keyinformant method. Herfindahl-Hirschman Index, Correlations and Regression analysis were mainly used and revealed on aggregate that all commercial banks in Kenya are diversified with large banks in lead while Islamic banks trail. Further, diversification level has a positive influence on financial performance of commercial banks in Kenyan and the two main revenue streams are positively related.
Income Source Diversification and Financial Performance of Commercial Banks in Kenya
(Mount Kenya University Journals, 2011-04-01) Rotich, Paul T.; Okaka, Damianus Ochieng; Aywa, Shem
The profitability of commercial banks depends heavily on the net of income generating activities and the related activities’ expense. Due to the problem of profitability and stiff competition in the industry, commercial banks have changed their behavior of income sources, by increasingly diversifying into non-intermediation income generating activities as opposed to the traditional inter-mediation income generating activities. The objective of this paper was to establish the impact of income source diver-sification on financial performance of commercial banks in Kenya. This has been achieved through: establishing the level of income source diversification of commercial banks in Kenya and establish whether income source diversification improves financial position of commercial banks. This was a census study of all registered 44 commercial banks in Kenya and relied heavily on documentary sec-ondary data for 5 year study period (2005-2009) and validated by primary data achieved through key-informant method. Herfindahl-Hirschman Index, Correlations and Regression analysis were mainly used and revealed on aggregate that all commercial banks in Kenya are diversified with large banks in lead while Islamic banks trail. Further, diversification level has a positive influence on financial performance of commercial banks in Kenyan and the two main revenue streams are positively related.
Mathematical model for detecting diabetes in the blood
(Academia, 2011-07-01) Kwach, Boniface Otieno; Ongati, Omolo; Simwa, Richard
This study presents a new mathematical model for Blood Glucose Regulatory System(BGRS) which includes epinephrine as a third variable in the form, Ў= 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.
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
On characterization of u-ideals determined by sequences
(2012) Matuya, Wanyonyi John.; Makila, Patrick.; Achiles, Simiyu.; Aywa, Shem.; Sammy, Musundi.
The area of ideals is important in the study of Analysis, algebra, Geometry and Computer science. The various types of ideals have been studied, for example m ideals and h ideals. The m ideals defined on real Banach spaces are referred to as u - ideals. The natural examples of u - ideals with respect to their biduals, are order continuous Banach lattices. Using the approximation property, we shall study properties of u - ideals and their characterization. We define the set of compact operators K X( ) on X to be u - ideals given that X is a separable reflexive Banach space with approximation property if and only if there is a sequence (Tn ) of finite rank of operators with lim 2 1 n n →∞ I T − = and lim n n →∞Tx x = . We shall show that u -ideals containing no copies of sequences 1  are strict u - ideals.
On redemacher boundedness in operator – norm
(Journal of agriculture, pure and applied science and technology, 2012) Wanyonyi, W.; Simiyu, J.A.; Aywa, Shem.; Kirui, C.S.
In these lecture notes we report on recent breakthroughs in the functional analytic approach to maximal regularity for parabolic evolution equations, which set off a wave of activity in the last years and allowed to establish maximal L p -regularity for large classes of classical partial differential operators and systems. In the first chapter (Sections 2-8) we concentrate on the singular integral approach to maximal regularity. In particular we present effective Mihlin multiplier theorems for operator-valued multiplier functions in UMD-spaces as an interesting blend of ideas from the geometry of Banach spaces and harmonic analysis with R-boundedness at its center. As a corollary of this result we obtain a characterization of maximal regularity in terms of R-boundedness. We also show how the multiplier theorems “bootstrap” to give the R-boundedness of large classes of classical operators. Then we apply the theory to systems of elliptic differential operators on Rn or with some common boundary conditions and to elliptic operators in divergence form. In Chapter II (Sections 9-15) we construct the H∞ -calculus, give various characterizations for its boundedness, and explain its connection with the “operator-sum” method and R-boundedness. In particular, we extend McIntosh’s square function method form the Hilbert space to the Banach space setting. With this tool we prove, e.g., a theorem on the closedness of sums of operators which is general enough to yield the characterization theorem of maximal L p -regularity. We also prove perturbation theorems that allow us to show boundedness of the H∞ -calculus for various classes of differential operators we studied before. In an appendix we provide the necessary background on fractional powers of sectorial operators.

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