Department of Mathematics
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Item Advances in composite integer factorization(2013) Wanambisi, Aldrin 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.Item Advances in composite integer factorization(International Knowledge Sharing Platiform, 2013-01-01) Wanambisi, Adrin W.; Aywa, Shem; Maende, Cleophas; Muketha, Geoffrey MuchiriIn 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.Item Advances in composite integer factorization(Mathematical theory and modelling, 2013) Wanambisi, Aldrin.; 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.Item Algebraic approach to composite integer factorization(International journal of mathematics and statistics, 2013) Wanambisi, W.; Aywa, Shem.; Maende, C.; Muketha, MG.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.Item 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 MuchiriThere 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.Item 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].Item 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 OtienoItem Characterization of codes of ideals of the polynomial ring f30 2 [x] mod ô€€€ x30 ô€€€ 1 for error control in computer applicatons(2016-06-30) Olege, Fanuel.; Oduor, Oduor M.; 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 Noethe-rian rings are closed in polynomial addition and multiplication.This property has been used to characterize the polynomial ring Fn 2 [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.Item 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.Item Conecture of banach space operator ideals in nuclear spaces(Asian journal of current engineering and maths, 2013) Musundi, W.; Ombaka, S.O.; Njogu, C.M.; Muthengi, S.F.; Mugambi, D.; Aywa, Shem.We apply the notion of Banach space operator ideals in nuclear spaces through topological vector spaces. The motivation for this study came from attempts to generalize the structure of nuclear spaces as a result of nuclear maps from functional analysis context. The compact closed structure associated with the category of relations results to nuclear ideals. Basic properties of Banach space operator ideals in relation to the structure of nuclear spaces will be demonstrated. We therefore establish a close correspondence between Banach space operator ideals and nuclear ideals through topological vector.Item Controllability of a quasi-linear heat equation(Asian Academic Research Associates : AARJMD, 2015-04-01) Andanje, Mulambula; Matuya, John WanyonyiIn this article we have examined the controllability of the quasi-linear heat equation. We considered distributed controls with support in a small sub-domain. We assumed that the quasi-linear terms are locally Lipschitz. We have proved that there exists a control that drives to desired state within a finite time. Our main objective was to determine the existence, uniquiness and regularity of solutions of controllability of a quasi-linear heat equationItem 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.Item 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.Item Distribution of spectrum in a direct sum decomposition of operators into normal and completely non-normal parts(Modern Scientific press : IJMMS, 2014-01-01) Mwenda, E.; Musundi, Sammy; Nzimbi, B.M; Marani, vincent Nyongesa; Loyford, NWe discuss the distribution of spectra of a direct sum decomposition of an arbitrary operator into normal and completely non normal parts. We utilize the fact that any given operator 𝑇∈𝐵(𝐻) can be decomposed into a direct summand 𝑇=𝑇1⊕𝑇2 with 𝑇1 and 𝑇2 are the normal and completely non normal parts respectively. This canonical decomposition is preferred to other forms of decomposition such as Polar and Cartesian decompositions because these two do not transfer certain properties (for instance the spectra, numerical range, and numerical radius) from the original /decomposed operator to the constituent parts. This is presumably done since these parts are simpler to deal with.Item 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.Item 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.Item 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 recommendedItem Elementary partial differential equations(Lambert, 2013-06-01) Wanjala, Charles C.; Manyonge, Alfred W.; Kwach, Boniface OtienoThe 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.Item Equivalent banach operator ideal norms(International journal of mathematical analysis, 2012) Musundi, S.; Aywa, Shem.; Fourie, J.We continue the investigation of coorbit spaces which can be attached to every integrable, irreducible, unitary representation of a locally compact groupG and every reasonable function space onG. Whereas Part I was devoted to atomic decompositions of such spaces, Part II deals with general properties of these spaces as Banach spaces. Among other things we show that inclusions, the quality of embeddings, reflexivity and minimality and maximality of coorbit spaces can be completely characterized by the same properties of the corresponding sequence spaces. In concrete examples (cf. Part III) one recovers several and often difficult theorems with ease.Item Equivalent Banach operator ideal norms1(Hikari: International Journal of mathematical analysis, 2012-12-01) Musundi, Sammy.; Aywa, Shem.; Jan, Fourie.We present some results concerning the general theory of Banach ideals of operators and give several applications to Banach space theory. We give, in Section 3, new proofs of several recent results, as well as new operator characterizations of the p-spaces of Lindenstrauss and Pelczynski. In Section 4 we prove that the space of absolutely summing operators from E to F is reflexive if both E and F are reflexive and E has the approximation property. Section 5 concerns Hilbert spaces. In particular, we compute the relative projection constant of Hilbert spaces in Lp(μ)-spaces.