Department of Mathematics

Permanent URI for this collectionhttp://erepository.kibu.ac.ke/handle/123456789/112

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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.
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 Otieno
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.
Controllability of a quasi-linear heat equation
(Asian Academic Research Associates : AARJMD, 2015-04-01) Andanje, Mulambula; Matuya, John Wanyonyi
In 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 equation
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.
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, N
We 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.
Elementary partial differential equations
(Lambert, 2013-06-01) Wanjala, Charles C.; Manyonge, Alfred W.; Kwach, Boniface Otieno
The 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.
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.
Estimating Non-Smooth Functional Using Non-Parametric Procedure in the Hilbert Sample Space
(International Journal of Engineering & Mathematical Sciences., 2017-06-01) Kololi, Moses Mukhwana; Orwa, George O.; Odhiambo, Romanus O.
One of the problems in statistical inference is that of estimating functional. A functional is a mathematical relation that maps two or more functions in one number. They are either smooth or non-smooth. The smoothness properties of functional determine the quality of estimation. However, non-smooth functional lack some degree of properties traditionally relied upon in estimation. Lack of these properties highlights the reason why standard techniques fail to give sharp results. In this paper, an estimator for an arbitrary non-smooth functional is proposed in the nonparametric set-up using robust stochastic Hilbert sample spaces. The estimator is based on the Mini Max criterion where lower and upper bounds are constructed. However, while working in the context of Mini Max estimation, the lower bounds are most important. The approximation theory is used to construct an estimator that is asymptotically sharp Mini Max when the means are bounded. The procedure used allows analysis and presentation data at hand without making any assumption about the underlying distribution. Therefore, the predictions do not depend on whether or not the underlying distribution is normal
Forensic estimation of time of death: a mathematical model
(International Journal of Management,IT and Engineering (IJMIE), 2013-07-01) Kwach, Boniface Otieno; Ongati, Omolo; Nyakinda, J.O.; 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.
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.
Generalized estimation of missing observations in nonlinear time series model using state space representation
(Science Publishing Group, 2013-01-01) Biwott, Daniel Kiprotich; Odongo, Leo O.
The aim of the study was to formulate a Time Series Model to be used in obtaining optimal estimates of missing observations. State space models and Kalman filter were used to handle irregularly spaced data. A non-Bayesian approach where the missing values were treated as fixed parameters. Simulated AR (1) data and corresponding estimated missing values were generated using a computer programme. Values were withheld and then estimated as though they were missing. The results revealed that simple exposition of state space representation for commonly used Time Series Models can be formulated.
A HIV/AIDS model in a homogeneous population with time lags in the transmission
(HIKARI Ltd: Applied Mathematical Sciences, 2014-01-01) Wasike, Adu A.M.; Kibwott, Frankline T; Lawi, GO; Mulati, Nyukuri Omukoba
We investigate a simple HIV/AIDS epidemic model in a heterosexual population by by modifying a Susceptible-Infective-Removed (SIR) model. The model considers sexual transmission of HIV where individuals are being recruited into sexually matured age group at a constant rate and incorporates time lags for one to become infective and the other to become fully blown. A complete qualitative analysis of the model, including the boundedness and positivity of the solutions, local stability of the equilibrium points is done. We applly the next generation matrix to determine the disease reproduction number R0. The Model is shown to be completely determined by the reproduction number. The model is numerically analysed to asses the effects of the delay on the dynamics of HIV/AIDS and the demographic impact of the epidemic using the demographic and epidemiological parameters for Kenya.
Hybrid single particle simulation of atmospheric pollutants over Bungoma County in Kenya
(Ethiopian Journal of Environmental Studies and Management (EJESM), 2018-12-20) Shem, Godfrey Juma; Makokha, John Wanjala; Kelonye, Festus Beru
Microscale dispersion models with different levels of complexity may be used to assess urban air quality and support decision making for pollution control. Mathematical models calculate pollutant concentrations by solving either analytically a simplified set of parametric equations or numerically a set of differential equations that describe in detail wind flow and pollutant dispersion. Air pollution has been evident at Webuye town in Bungoma Kenya and this may be attributable to the current and residual effect of the pan paper and heavy chemical processing plants. A Lagrangian trajectory modeling system for urban air pollution was used to model the flow of atmospheric pollutants in Bungoma County with emphasis to Webuye, an industrial town within the County. Results of the study reveal that pollutants from the area of study exhibit a seasonal dispersion trend over Lake Victoria and surroundings to the Eastern part of Uganda. An investigation of factors that influence this trend is recommended.
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.
Long term assessment of aerosol radiative forcing over selected sites of East Africa
(Scientific Research Publishing Inc., 2018-04-02) Makokha, John W.; Odhiambo, Jared O.; Shem, Godfrey Juma
Atmospheric aerosols have contributed to radiative forcing through direct and indirect mechanisms. Aerosol effects are important in computing radiative forcing estimates for the past, current and future climate. In this study, a comprehensive assessment of regional aerosol radiative forcing, Optical Properties of Aerosol and Clouds (OPAC) model (wavelength range of 0.25 - 4.0 μm) over selected sites in East Africa was done. Aerosol optical properties constituted the inputs of a Radiative Transfer Model (RTM). Optical properties investigated included Aerosol Optical Depth (AOD), Single Scattering Albedo (SSA) and Asymmetry Parameter (AP). Aerosol Radiative Forcing (ARF) during the study period at the surface (SFC), top of the atmosphere (TOA) and the atmosphere (ATM) was estimated to be –18.4 ± 1.4 W∙m−2 , +1.1 ± 0.3 W∙m−2 and +19.5 ± 2.5 W∙m−2 , respectively. This corresponds to an increment in net atmospheric forcing at a heating rate of about 0.55 ± 0.05 K/day (0.41 ± 0.03 to 0.78 ± 0.03 K/day) in the lower troposphere. The study points out the significant role played by atmospheric aerosols in climate modification over the area of study. It is recommended that a further assessment be done in view of uncertainties that may impact on the findings and which were not within the scope of this research.
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.
Mathematical model for drug therapy in patients with diabetes mellitus
(International Journal of Engineering, Science and Mathematics, 2013-03-01) Kwach, Boniface Otieno; Ongati, Omolo; Okoya, Oduori; Otedo, Amos
This study presents a new mathematical model for Drug Therapy in Patients with Diabetes Mellitus which includes external rate at which blood glucose, insulin and epinephrine is being increased in the form Y= fi (g,h,e)+ri (t) . The system has been analyzed and solved to provide the systems natural frequency, ω0, which is the basic descriptor of saturation level of the drug. We establish that the resonance period for the final model, that is, T0=3.76912 hrs, agrees well with the data for the existing insulin therapy, showing that the peak, which is the time period for insulin to be most effective in lowering blood sugar, is in the acceptable therapeutic range
Mathematical modeling of insulin therapy in patients with diabetes mellitus
(International Journal of multidisciplinary sciences and Engineering, 2015-06-01) Kwach, Boniface Otieno; Ongati, Omolo; Okoya, Oduori; 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)) ̇ andwhose solution was analyzed to provide the systems natural frequency, ω_o which is the basic descriptor of saturation level of the drug. It was established that the resonance period for the final model, that is, T0=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 T0= 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.

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