Faculty of Science

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Advances in composite integer factorization
(International Knowledge Sharing Platiform, 2013-01-01) Wanambisi, Adrin W.; Aywa, Shem; Maende, Cleophas; Muketha, Geoffrey Muchiri
In this research we propose a new method of integer factorization. Prime numbers are the building blocks of arithmetic. At the moment there are no efficient methods (algorithms) known that will determine whether a given integer is prime or and its prime factors[1]. This fact is the basis behind many of the cryptosystems currently in use.
Algebraic approach to composite integer factorization
(European Centre for Research, Training and Development : IJMSS, 2013-03-01) Wanambisi, Adrin W.; Aywa, Shem; Maende, cleophas; Muketha, Geoffrey Muchiri
There various algorithms that can factor large integers but very few of these algorithms run in polynomial time. This fact makes them inefficient. The apparent difficulty of factoring large integers is the basis of some modern cryptographic algorithms. In this paper we propose an algebraic approach to factoring composite integer. This approach reduces the number of steps to a finite number of possible differences between two primes.
Characterization of codes of ideals of the polynomial ring for control in computer applicatons
(C o u n c i l f o r I n n o v a t i v e R e s e a r c h : J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s, 2016-06-01) Olege, Fanuel; Owino, M.O.; Aywa, Shem; Okaka, Colleta A.
The study of ideals in algebraic number system has contributed immensely in preserving the notion of unique factorization in rings of algebraic integers and in proving Fermat’s last Theorem. Recent research has revealed that ideals in Noetherian rings are closed in polynomial addition and multiplication.This property has been used to characterize the polynomial ring F_2^n[x] mod(xn-1) for error control. In this research we generate ideals of the polynomial ring using GAP software and characterize the polycodewords using Shannon’s Code region and Manin’s bound.
Equivalent Banach Operator Ideal Norms1
(HIKARI: International Journal of Mathematical Analysis, 2012-01-01) Musundi, Sammy; Aywa, Shem; Jan, Fourie
Let X, Y be Banach spaces and consider the w′-topology (the dualweak operator topology) on the space (L(X, Y),‖.‖) of bounded linearoperators from X into X with the uniform operator norm.Lw′(X,Y) is the space of all T∈L(X, Y) for which there exists a sequence ofcompact linear operators (Tn)⊂K (X, Y) such thatT=w′−limnTn Two equivalent norms, ‖|T‖|:=inf{█(sup@n)┤‖Tn‖:Tn∈K(X,Y),Tnw′→T}and ‖T‖u:=inf{█(sup@n)┤ {max{‖Tn‖,‖T−2Tn‖}}:‖:Tn∈K(X,Y),Tnw→T}on Lw′(X, Y), are considered. We show that (Lw′,|‖.‖|) and (Lw′,‖.‖u) are Banach operator ideals.
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.
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 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.
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.
Partition of measurable sets
(council for Innovative Research, 2015-06-18) Owino, Maurice Oduor; Otanga, Levis Olwamba; Aywa, Shem
The theory of vector measure has attracted much interest among researchers in the recent past. Available results show that measurability concepts of the Lebesgue measure have been used to partition subsets of the real line into disjoint sets of finite measure. In this paper we partition measurable sets in Rn for n ≥ 3 into disjoint sets of finite dimension.
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
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.
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.

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