maths conference programme
TRANSCRIPT
TENTATIVE PROGRAMME
ORGANIZING COMMITTEE
REGIONAL COMMITTEE
Dr. Leo Odongo, Kenyatta University. ‐ Chairman
Ms Mary Opondo, Kenyatta University. ‐ Secretary
Dr. Isaac Chepkwony, Kenyatta University. ‐ Treasurer
Dr. David Malonza, Kenyatta University.
Dr. Ireri Kamuti, Kenyatta University.
Mrs. Eunice Muchai, Kenyatta University.
Dr. Jeconia A. Okello, Jomo Kenyatta University.
Prof. Patrick G. O. Weke, University of Nairobi.
Dr. Samuel Karoki, Catholic University of Eastern Africa.
Dr. Benard Kivunge, Kenyatta University
Dr. Edward Njenga, Kenyatta University
Mr. Christopher Ouma, Kenyatta University
Ms. Lydia Njuguna, Kenyatta University
SCIENTIFIC COMMITTEE
Prof. Enrico Rogora, University of Rome “La Sapienza”, Italy
Prof. John O. Agure, Maseno University, Kenya
Prof. Joseph Y.T. Mugisha, Makerere University, Uganda
Prof. Moses M. Manene, University of Nairobi, Kenya.
CONFERENCE ADDRESS:
The Chairman, Regional Committee
Department of Mathematics
P.O BOX 43844 – 00100, Nairobi, Kenya
Email: ku‐[email protected], [email protected]
Website: www.ku.ac.ke/mathsconference
TENTATIVE PROGRAMME
DAY 1: Wednesday 8th June 2011
08.00 – 08.30 Registration
08.30 – 09.30 Opening Session:
Chairman /MC Dr. Benard Kivunge
Dr. David Malonza Chairman Mathematics
Dr. Leo Odongo, Chairman Organizing Committee
Dr. Eunice Kairu Dean School of Pure & Applied Sciences
Prof. John Okumu DVC Academics
Prof. Olive Mugenda Vice Chancellor, Kenyatta University
09.30 – 10.10 Keynote Speaker: Prof. Oluwole Daniel Makinde.
Title: Impact of Mathematical Sciences Research on National Development.
Cape Peninsula University of Technology, South Africa.
10.10 – 10.40 Photo Session and Tea Break
SESSION I
Chairman : Dr. James Kahiri
Rapporteur : Mr. Christopher Ouma
11.00 – 11.20 Rongoro Enrico
11.20 – 11.40 Views on the role of mathematics in development. Jan Persens, Western Cape University, South Africa.
11.40 – 12.00 On the characterization of class R1 of non‐normal operators in a Hilbert space. Mile Justus Kitheka, University of Nairobi.
12.00 – 12.20 A Mathematical model for the spread and control of tuberculosis on a population with temporally immunity. Cyprian Turyatemba, Julius Tumwiine, Mbarara Universty , Uganda.
12.20 – 12.40 To weight or not to weight.
Edward Njenga, Kenyatta University, Kenya.
12.40 – 13.00 An Iterative Procedure for Solving a Cauchy Problem for the Helmholtz Equation. Lydie Mpinganzima, Link¨oping University, Sweden.
13.00 – 14.00 Lunch Break
SESSION II
Chairman : Dr. Ireri Kamuti
Rapporteur : Ms. Mary Opondo
14.00 – 14.20 Copula based risks classification: A simulation study
J. K. Mung’atu, S. M. Mwalili, JKUAT and P. G. O. Weke University of Nairobi.
14.20 – 14.40 Formulas for the computation of the Tutte polynomial of a graph with parallel classes, Eunice Mphako‐Banda, University of Witwatersrand, South Africa.
14.40 – 15.00 Model‐based nonparametric estimation of the finite population total under two‐stage
cluster sampling. Karoki Samuel, Catholic University, Odongo Leo and Kahiri James,
Kenyatta University.
15.00 – 15.20 Split operator method for parabolic partial differential equation.
Nthiiri Joyce Kagendo, Mary Okombo, Shem Aywa, Masinde Muliro University and Michael Oduor Okoya, Maseno University.
15.20 – 15.40 Tea Break
15.40 – 16.00 A numerical simulator to estimate hydrologic parameters of a fracture using fluoresce in thermal decay correction. Benson Wang’ombe, Willis Ambusso and Peter Omenda, Kenyatta University
16.00 – 16.20 Necessary conditions for existence of a friend of 38.
R. K. Gachimu, C. W. Mwathi, JKUAT, Kenya and I. Kamuti, Kenyatta University.
16.20 – 16.40 Applications of u‐ideals in Banach spaces. Kayiita Z. Kaunda, Baraton University, Kenya.
16.40 – 17.00 A model for misclassification errors in Single Stage cluster sampling. Kahiri James, Odongo Leo, Kenyatta University and Karoki Samuel, Catholic University.
DAY 2: Thursday 9th June 2011
SESSION III A MAIN HALL
Chairman : Dr. Eunice Mphako ‐ Banda
Rapporteur : Mr. Kennedy Awuor
08.00 – 08.40 Keynote Speaker: Prof. Cecilia Mwathi
Title: Pure Mathematics as a cornerstone for technological development.
Jomo Kenyatta University of Agriculture and Technology(JKUAT), Kenya.
08.40 – 09.00 Exact Minimizer and Duality. Japhet Niyobuhungiro, Link¨oping University, Sweden.
09.00 – 09.20 MAC model for an elastic plate.
Igor Neygebauer, University of Dodoma, Tanzania
09.20 – 09.40 Sign matrices for frames of 2n‐ons under Smith Conway and Cayley Dickson multiplications. Benard Kivunge, Kenyatta University, Kenya.
09.40 – 10.00 A Numerical Method for Improving Shielded Thermocouple Accuracy.
Fredrik Berntsson, Linkoping University, Sweden.
10.00 – 10.20 A mathematical model for transmission and control of bovine brucellosis in cattle populations. Godwin Robert, Mbarara University, Uganda.
10.20 – 10.40 Tea Break
SESSION IV A MAIN HALL
Chairman : Dr. Isaac Chepkwony
Rapporteur : Ms. Lydia Njuguna
10.40 – 11.00 On the existence of prime numbers in polynomial sequences. Acquaah Peter, University of Ghana, Ghana.
11.00 – 11.20 Mathematical model for malaria and meningitis co‐infection among children. Lawi G.Owuor , Omolo – Ongati, Maseno University, Kenya and Mugisha J.Y.T., N., Makerere University, Uganda.
11.20 – 11.40 A Secured Hybrid Architecture Model for Mobile Financial Services: A Case Study of Zimbabwe. Weston D. Govere and Dumisani Sibanda, Midlands State University, Zimbabwe.
11.40 – 12.00 Analysis of unsteady hydromagnetic thermal boundary layer over a flat plate with Navier
slip and a convective surface boundary condition.
Oluwole Daniel Makinde, Cape Peninsula University of Technology, South Africa.
12.00 – 12.20 Application of fourier series to 1‐, 2‐, 3‐ dimensional signals with examples in matlab. Eyram Schwinger, University of Ghana, Ghana.
12.20 – 12.40 Effects of internal heat generation, thermal radiation, and buoyancy force on boundary
layer over a vertical plate with a convective surface boundary condition.
Olanrewaju Philip Oladapo, Covenant University, Nigeria.
12.40 – 13.00 Principal component and Cluster analysis an approach to goat market segmentation in Matebele land Province‐Zimbabwe.
Govere W., Chinofunga P.T., Charumbira W.F., Chikudo E, Midland State University, Zimbabwe.
DAY 2: Thursday 9th June 2011
SESSION III B HALL 2
08.40 – 09.00 Implicative algebras. K.Venkateswarlu , Berhanu Bekele, Adis Ababa University, Ethiopia.
09.00 – 09.20 On N‐curvature inheritance in Finsler spaces.
J.K.Gatoto, Kenyatta University, Kenya.
09.20 – 09.40 Making Markov chain modeling of rainfall data accessible.
J.O. Ong’ala, The Great Lakes University of Kisumu,Kenya. D.A. Stern, Maseno
University, Kenya and R.D. Stern, University of Readings, UK.
09.40 – 10.00 A Mathematical comparative model of HIV/AIDS, poverty and development.
Cecilia Musyoka, South Eastern University College, Kenya.
10.00 – 10.20 The role of Mathematics in development of Technical and Vocational Education and
Training (TVET) in the 21st century.
Samuel M. Mwangi, Gusii Institute of Technology, Kenya.
10.20 – 10.40 Tea Break
SESSION IV B HALL 2
Chairman :
Rapporteur :
10.40 – 11.00 On a more generalized class of bounded starlike functions of complex order. Biodun Oladipupo, Ladoke Ankintola University of Technology, Nigeria.
11.00 – 11.20 The Limitations in the Convectional ISBN‐10 Code. Peter Waweru, Cecilia Mwathi, JKUAT, Kenya and Benard Kivunge, Kenyatta University.
11.20 – 11.40 Accelerated failure time frailty model for CD4 cells count.
Nyakundi Wycliffe and Mwalili Samwuel JKUAT, Kenya.
11.40 – 12.00 Demystifying mathematics: Handling learning difficulties in mathematics among low
achievers in Kenyan schools.
Gladys Gakenia Njoroge, Kenyatta University, Kenya.
12.00 – 12.20 Completely prime modules. Ssevviiri David and N. J. Groenewald , Nelson Mandela Metropolitan University, South Africa.
12.20 – 12.40 Convergence of Ishikawas's Iteration Method for Finite Family of Pseudocontractive Mappings.
Habtu Zegeye, University of Botswana, Gaborone, Botswana.
12.40 – 13.00 Solution of the third order viscous wave equation using Finite Difference Method. Ogang a Duncan O, Shem Away, Masinde Muliro University and Michael O. Okoya, Maseno University.
13.00 – 14.00 Lunch Break
14.00 – 17.00 Round table discussion for MARM participants HALL 2
SESSION V MAIN HALL
Chairman : Dr. Titus Kibua
Rapporteur : Mr. Mutie Kavila
14.00 – 14.20 A model based inference of finite population total in two stage cluster sampling.
Ouma C. Onyango, Kenyatta University, Kenya.
14.20 – 14.40 Moving teaching beyond calculations towards mathematics through eLearning; a case study of e‐Statistics Made Simple. James Musyoka, Thomas Mawora and David Stern, Maseno University, Kenya.
14.40 – 15.00 Modelling stray dogs in Gweru urban.
Charumbira W.F., Govere W., Chinofunga P.T. and Borerwe T, Midland State University,
Zimbambwe
15.00 – 15.20 Geospatial modeling on prevalence of Buruli Ulcer in Amansie district in Ghana .
Dontwi I.K, Frempong, N.K. and Bonyah, E, Kwame Nkrumah University of Science and
Technology, Ghana
15.20 – 15.40 Tea Break
15.40 – 16.00 Computational Group‐Testing Strategy. K.L. Nyongesa, Masinde Muliro University and J.W. Mwangi, Egerton University.
16.00 – 16.20 Mathematics as the key instrument in modern technology.
Josephat Itambu, University of Dodoma, Tanzania.
16.20 – 16.40 A mathematical model for the control of banana bacterial wilt disease.
Kweyunga Eliab Horub and Tumwiine Julius, Mbarara University, Uganda.
16.40 – 17.00 Automated Teller Machine Reliability, Problems and Risks.
Chinofunga P.T., Charumbira W.F., Govere W.D., Dzvuke G, Midlands State University.
DAY 3: Friday 10th June 2011
SESSION VI A MAIN HALL
Chairman : Dr. Fredrik Berntsson
Rapporteur : Mrs. Winnie Mutuku‐Njogu
08.00 – 08.20 Fuzzy Random variables. Eunice Muchai, Kenyatta University, Kenya.
08.20 – 08.40 Adjusting rainfall for climate change scenarios as inputs for crop simulation models.
Mawora Thomas Mwakudisa and David Stern, Maseno University.
08.40 – 09.00 Rank and subdegrees of the symmetric group nS acting on ordered r‐element subsets .
Rimberia Kagwiria Jane, Kenyatta University, Kenya.
09.00 – 09.20 Sequential Bayesian analysis of binomial opinion polls. Kiingati Jeremiah, Samuel Mwalili and Anthony Waititu, JKUAT, Kenya
09.20 – 09.40 Pricing Asian currency options in a developing financial market: A modified lattice approach. Ogutu Carolyne Adhiambo, University of Nairobi.
09.40 – 10.00 Ideals of the polynomial ring xF 1mod nx for error control in computer
applications. Olege Fanuel & Aywa Shem, Masinde Muliro University, Kenya.
10.00 – 10.20 Investigating the maximal determinant of an nn matrix.
Irene Okello, Cecilia Mwathi, JKUAT, Kenya and Kivunge B., Kenyatta University.
10.20 – 10.40 Tea Break
SESSION VII A MAIN HALL
Chairman : Dr. Benard Kivunge
Rapporteur : Ms. Jane Rimberia
10.40 – 11.00 On the unit groups of completely primary finite rings of characteristic p. Owino Maurice Oduor, Maseno University, Kenya.
11.00 – 11.20 Same and equivalent linear codes in relation to categorising Goppa codes.
Govere W and Sibanda D, Midland State University, Zimbabwe.
11.20 – 11.40 Mathematical modeling for road design in Kenya. The mechanistic empirical option. S.A. Nyakiti, D. A. Stern and O. Ongati, Maseno University, Kenya.
11.40 – 12.00 Action of on .
I. N. Kamuti, J. K. Rimberia, Kenyatta University and E. B. Inyangala
12.00 – 12.20 Transforming attitudes and performance of management students’ statistical training.
Bernard Manyalla, Siaya Institute of Technology, David Stern, Maseno University.
12.20 – 12.40 On characterization of u ‐ ideals determined by sequences.
Matuya John Wanyonyi, Narok University, Shem Aywa and Achiles Simiyu, Masinde
Muliro University, Kenya.
12.40 – 13.00 The M0‐ matrix completion problem for selected 4*4 matrices, p=4 and q =4; Digraphs without completion.
Peter Waweru, Cecilia Mwathi, JKUAT, Kenya and Bernard Kivunge, Kenyatta University.
DAY 3: Friday 10th June 2011
SESSION VI B HALL 2
Chairman : Dr. Leo Odongo
Rapporteur : Mr. Ananda Kube
08.00 – 08.20 08.20 – 08.40 Generalized equation of viscous flow through non‐circular flow channels.
George Xyplagus Stower, Kenyatta University, Kenya.
08.40 – 09.00 Mathematics is the key to sustainable development. Taliba Caroline and Cyrus Ssebugenyi, Makere University, Uganda.
09.00 – 09.20 On decomposition of curvature tensors in conformal Finsler spaces.
J.K.Gatoto and S.P.Singh, Kenyatta University.
09.20 – 09.40 Moving teaching beyond calculations towards mathematics through eLearning; a case study of e‐Statistics made simple. Kaleli James and Mawora Thomas, Maseno University, Kenya.
09.40 – 10.00 Sedenion Extension Loops and Frames of General Hypercomplex Numbers. Njuguna Lydia N., Kenyatta University, Kenya.
10.00 – 10.20 On banach space ideal properties. Musundi Sammy W, Chuka University College, Kenya.
10.20 – 10.40 Tea Break
SESSION VII B HALL 2
Chairman : Dr. Edward Njenga
Rapporteur : Mr. Wahome
10.40 – 11.00 Laser technology as a tool for development. Opondo Mary, Kenyatta University, Kenya.
11.00 – 11.20 Solutions of the third order viscous wave equation using finite difference method.
Otieno Duncan, Masinde Muliro University, Kenya. 11.20 – 11.40 On the characterization of class R1 of non‐normal operators in a Hilbert space.
Mile Justus Kitheka, University of Nairobi, Kenya.
11.40 – 12.00 Distributed controllability of Cochlea model.
Chepkwony Isaac, Kenyatta University, Kenya.
12.00 – 12.20 On commutants & spectral properties of ‐commuting operators.
Mutie Kavila, Kenyatta University , Kenya.
12.20 – 12.40 Encouraging dialogic learning in mathematics’ class discourse.
Kahenya Paul, African Nazarene University, Kenya.
12.40 – 13.00 Banach space ideal structures in operator ideals. Makila Patrick W., Aywa Shem & Simiyu Achile, Masinde Muliro University.
13.00 – 13.30 Closing Session MAIN HALL
Dr. Leo Odongo
Dr. David Malonza
13.30 Lunch Break
14.30 Excavation, Visit to the Nairobi National Park.
ABSTRACTS KEY NOTE SPEACH
Impact of Mathematical Sciences Research on National Development. Oluwole Daniel Makinde, Cape Peninsula University of Technology, South Africa.
Abstract: Mathematics plays a rapidly increasing role as a universal language for science. Without it, science and technology cannot address the complex issues facing the modern world. It is the level of research in mathematical sciences that determines the level of the science and technological advancement of any nation. The foundation of science and technology, which is the basic requirement for development of a nation, is mathematics. Unless a country or a group of countries are well equipped with the necessary mathematical skills and knowledge to unlock its enormous scientific and technological potentials, they will be lagging behind in the race of development. This talk is aimed at promoting greater awareness of the impact and utilization of mathematical science research for national developments.
To weight or not to weight.
Edward Njenga, Kenyatta University, Kenya.
Abstract:
Weighting by the inverse of unit selection probability is the basis of randomization inference. In model
based frame work , probability designs are ignorable and so probability Weights have no obvious role.
Using results from size biased sampling and penalized Spline regression, it is shown that randomization
estimators can be justified. The problem addressed here is whether probability Weights have a role in
model based inference for sample surveys or not.
Copula based risks classification: A simulation study
J. K. Mung’atu, S. M. Mwalili, JKUAT and P. G. O. Weke University of Nairobi.
Abstract:
Risks’ classification generally forms the basis of rate making in practically all branches of insurance.
Similar risks should be assigned to the same class with respect to each variable. Modern portfolio theory
is based on correlation as a measure of dependence while the criterion presented here is based on the
copula theory which is more superior to the correlation as a measure of the intrinsic relatedness of
different risks under certain conditions. Dependence between risks reduces the benefits of
diversification. Data were simulated following four loss distributions. Dependencies are examined by
fitting copulas, estimating the dependence parameters and lastly using distance matrices to cluster the
risks together by the minimum distance criterion. Cophenetic correlation coefficient is employed to
choose between the best distances to use. The Manhattan distance performs better than the Euclidean
distances and so we base our clustering on the Manhattan distances.
Key words: Copula, Measures of dependence, Cluster, Lines of business, Distances.
Fuzzy Random Variables.(An overview paper)
Eunice Muchai, Kenyatta University, Kenya.
Abstract:
The concept of fuzzy set was introduced by Zadeh in 1965. Since then the fuzzy concept has grown.
However it is notable that many fields which were expected to apply the fuzzy concept have not applied
it. The reason for this is may be that potential users are not sufficiently familiar with fuzzy random
variables. This article introduces fuzzy random variables to statisticians who may not be exposed to the
fuzzy concepts. It is hoped that by doing this then they can apply these concepts in their areas of
specialization.
Accelerated failure time frailty model for CD4 cells count.
Nyakundi Wycliffe and Mwalili Samwuel JKUAT, Kenya.
Abstarct:
The accelerated failure time frailty model will be studied and in particular we shall look at the various
patients with HIV AIDS co‐infections during the surveillance period. Firstly, we assess the time impact of
the co‐infection type on the surrogate endpoint of the HIV, prior to the development of AIDS and the
development of the AIDS condition, in this case, the clinical endpoint. Longitudinal multivariate data
with the CD4 counts was collected from the Westlands District Health Facilities with an assumption of
the entry level being the time of infection of the HIV condition. Surrogate endpoints in this case
serve to generalize the whole health of a patient in a medical trial. Random purposive sampling
technique with consideration of all intended study variables were used to select the sample of 95
patients.
To determine the impact of time on the HIV/AIDS and its co‐infections the Kaplain Meier survival curves
will be used. To model the data using the AFT model given as, UBX
oitt )()( the covariates which
affect the β are introduced. The accelerated failure time frailty model will analyze the CD4 cell counts
to replica the longitudinal data observations at the surrogate endpoint in consideration of the clinical
endpoint.
Model based inference of finite population total in two stage cluster sampling.
Ouma C. Onyango, Kenyatta University, Kenya.
Abstract.
The bootstrap approach to model based inference was first proposed by Chambers and Dorfman [2002].
Ouma and Wafula [2005] re‐looked at the conditions and extended this work. Both cases focused on
simple random sampling in cases where the auxiliary variables are known for the entire population.
Recently, Ouma et al (2010) re‐looked at the same kind of inference in two stage cluster sampling with
unequal cluster sizes. However in all these previous work, they assumed that the sampling distribution
of the parameters were normal. This may not necessarily be the case especially when the characteristics
of sampling units are heterogeneous.
This paper considers the same approach to constructing bootstrap confidence intervals for the
population total but unlike the previous work the sampling procedure takes into account the
heterogeneity of the sampling distributions of different clusters. This consideration enables us to choose
different sampling weights in different clusters within the sampling frame. This improves the efficiency
of the estimator of the population total.
Keywords: Model Based Surveys, Bootstrapping, Two Stage cluster sampling
Geospatial modeling on prevalence of Buruli Ulcer in Amansie district in Ghana .
Dontwi I.K, Frempong, N.K. and Bonyah, E. Kwame Nkrumah University of Science and Technology,
Ghana
Abstract:
Buruli ulcer (BU) is a disease caused by mycobacterium ulcerans (MU). MU is a pathogenic bacterium
that causes dermal ulcers known as “Buruli ulcer” (BU) is fast becoming a debilitating affliction in many
countries worldwide. Buruli ulcer has been reported in over 30 countries where it has not yet been
recognized. If left untreated BU may lead to extensive soft tissue destruction, with inflammation
extending to deep fascia if patient do not report early for treatment. The large number of cases and the
complications currently associated with the disease as well as its long‐term socio‐economic impact could
have a substantial effect on the rural economy. Knowledge gaps about the exact mode of transmission
and factors that pre‐dispose to infection motivate this study. This study employed geographical
information systems (GIS) and geostatistics to establish relationship between BU and postulated risk
factors in a subdivision of Ghana. Semivariograms were computed to determine the strength and spatial
dependency of the pattern of disease as well as summarize the variation. The risk of developing the
disease was estimated by kriging. Ordinary kriging was chosen in the variogram modeling.
The BU data sets exhibited a highly positively skewed histogram with possible outliers. The length of
spatial autocorrelation (practical range) was much longer than sampling interval (lag size). The kriged
map showed that there are large patches of BU disease in the southern part of the study area with few
isolated cases in the other parts. The large patches were greater with towns closer to river Offin than
that of river Oda in the southern of the study area. This study has identified the presence of spatial
pattern in the distribution of BU in the study area. Through geostatistics procedures, non‐randomness in
the distribution of BU and the identification of unusual concentration of BU incidence has been
identified and arsenic may be implicated in the Buruli ulcer disease.
Key words: Variogram, kriging, geographical information systems, spatial patterns, semivariograms
Computational Group‐Testing Strategy
K.L. Nyongesa, Masinde Muliro University & J.W. Mwangi, Egerton University.
Abstract:
Screening of pooled urine sample was suggested during the Second World War as a method for reducing
the cost of detecting syphilis in U.S. soldiers. Recently, pooling has been used in epidemiological studies
for screening for human immunodeficiency virus HIV/AIDS antibody to help curb the spread of the virus.
Pooling reduces the cost but also – and more importantly – offers a feasible way to lower the
misclassifications associated with labeling samples when imperfects tests are used. Furthermore,
misclassifications can be reduced by re‐testing in the pool testing procedure. This study will develop a
computational group‐testing model; statistical measures will be computed. The efficiency of the design
will be compared with individual testing.
Principal component and Cluster analysis an approach to goat market segmentation in Matebele land Province‐Zimbabwe. Govere W., Chinofunga P.T., Charumbira W.F., Chikudo E, Midlands State University, Zimbabwe.
Abstract: Zimbabwe’s goat population is estimated to be 3 million (Small‐Ruminant‐Breeds‐J.L.Sikosana). Of these only 2% are found in the commercial sector and 98% in the small holder sector (Sabine Homann‐(2007) Goat production and marketing). This reflects that marketing constraints are eradicating commercial mind set of goat farmers. Scant attention has been paid to date to these marketing constraints and therefore are poorly understood and incompetently documented. This paper seeks to unveil key marketing constraints that are being faced by goat farmers across six districts of Southern Zimbabwe. This will be done by determining principal factors in terms of market variables of six districts through factor reduction method. Thereafter similarities and dissimilarities that exist in existing goat marketing infrastructure and strategies among districts will be determined. Key words: Principal component, Cluster analysis, market segmentation
Model‐based nonparametric estimation of the finite population total under two‐stage cluster
sampling.
Karoki Samuel, Catholic University, Odongo Leo and Kahiri James, Kenyatta University.
Abstract:
A design‐free, model‐based non parametric regression estimator for the finite population total under
two‐stage cluster sampling is proposed. Asymptotic properties of the estimator are investigated by
deriving its Asymptotic Mean Integrated Squared Error( AMISE ). In particular the applicability of the
AMISE to the choice of bandwidth is explored. It is shown that plug‐in methods for estimating
bandwidth based on the AMISE will be ineffective. The results of an empirical study are used to compare
the performance of the proposed estimator with that of the linear regression estimator and two local
polynomial regression estimators of the total in current use. It is observed that the estimator performs
better ( in terms of efficiency ) than the linear regression estimator if there is a model misspecification
and compares favourably with the other non parametric estimators ( that use design properties ) in
most situations. It is also noted that the proposed estimator does not require the inclusion probabilities
to implement as is the case with the local polynomial estimators.
Key words: Cluster Sampling: Nonparametric Estimation: Design‐Free: Asymptotic Mean Integrated
Squared Error
Modelling stray dogs in Gweru urban. Charumbira W.F., Govere W., Chinofunga P.T. and Borerwe T , Midland State University, Zimbabwe
Abstract: This paper seeks to model stray dogs in Gweru urban using the logistic models. The data for stray dogs was supplied by Zimbabwe Society for the Prevention of Cruelity to Animals (SPCA). Two control measures were considered; euthanasia and sterilization. Models for each of these were formulated. The
differential equations were solved using the MATLAB. It was realized that euthnasia is more effective in a period of less than 8 years than sterilization. Key words: stray dogs, sterilization, euthanasia. A model for misclassification errors in Single Stage cluster sampling . Kahiri James, Odongo L. O, Kenyatta University , Karoki Samuel, Catholic University of East Africa, Kenya.
Abstract: Single stage cluster sampling is considered where the study variable is binary and affected by misclassification errors. In studies carried out before, it has been found that population units can consistently be misclassified from one category to another, resulting in asymmetrical errors. In the our model, correlated misclassification errors are assumed. An empirical study is carried out to investigate the effect of misclassification errors on the estimate of the population proportion. It is observed that the bias of the estimate of the population proportion can be relatively large for small population proportions.
Key words: Misclassification errors, simple response errors, correlated errors.
Sequential Bayesian analysis of binomial opinion polls
Kiingati Jeremiah, Dr. Samuel Mwalili, Dr. Anthony Waititu JKUAT, Kenya
Abstract: Bayesian statistics takes a prior probability statement about a population parameter and then modify these prior beliefs in the light of current and relevant data in order to arrive at a posterior knowledge with less uncertainty. Opinion poll data change from time to time therefore a Bayesian model is the best option since the previous measure can be used as the prior of the current measure. In this paper our concern is to model the proportion of votes between two candidates, incumbent and Challenger. A Bayesian model of our binomial variable of interest will be applied sequentially to the K opinion poll data sets in order to arrive at a posterior probability statement.
Automated Teller Machine Reliability, Problems and Risks
Chinofunga P.T., Charumbira W.F., Govere W.D., Dzvuke G, Midlands State University, Zimbabwe.
Abstract:
As machines and devices are manufactured, engineers and designers try as hard as possible to be
perfect and produce devices that perform to optimal levels. However, no matter how good the design
is, it is prone to failure at some stage in its life time. This is because devices are mostly made of moving
parts, which may wear out due to friction, or electronic components which may be affected by the
environment they function in. This brings about the problem of device reliability. This paper aims at
coming up with reliability models for Automated Teller Machines (ATMs) that can be extended to other
computer hardware. Existing reliability growth models like the Duane reliability growth model, Weibull
distributed models and Exponential models, were used as foundation for model formulation.
Comparisons were made between simulated data and actual data collected over time at a local Building
society Bank in Zimbabwe. The results indicate that the value of the hazard rate function is 0.02. This
means that the probability of failure in a very small interval of time is very low and we are unlikely to get
a failure in such an interval. The mean time before failure (MTBF) is 28.704 which translates to 28 days
before a failure at the stated values of k (the average failure rate of the system) and , the initial number
of failures at time . This is a fairly large period that provides some confidence that the system is reliable.
Key words: Reliability, Failure
A numerical simulator to estimate hydrologic parameters of a fracture using fluorescein thermal
decay correction.
Benson Wang’ombe, Willis Ambusso and Peter Omenda, Kenyatta University
Abstract:
A numerical solution to model the flow of fluoresce in tracer in a fractured high temperature
geothermal system is presented. Results of this study show that correcting for fluoresce in decay at
elevated temperatures can be used to yield reservoir hydrologic parameters and improve the methods
of evaluating effects of injected fluids on reservoir temperature. The hydrologic parameters for this
study are better than those obtained using other methods. This outcome was obtained by solving the
material and tracer balance equations that were fully discretized using integral finite difference and
solved using Gauss‐Seidal recursive methods. This proves that computational methods such as those
used here can be used for industrial application.
On decomposition of curvature tensors in conformal Finsler spaces.
J.K.Gatoto and S.P.Singh, Kenyatta University.
Abstract:
M.S Knebelman (1929) has developed conformal geometry of generalized spaces. R.B Misra (1967) has
obtained Bianchi identities satisfied by curvature tensors in a conformal Finsler space. The
decomposition of recurrent curvature tensors in Finsler space was studied by B.B. Sinha and S.P.Singh
(1972). The present authors (2001), C.K.Mishra and Gautam Lodhi (2008) have decomposed curvature
tensors in recurrent Finsler space . The object of the present paper is to study the decomposition of
curvature tensor fields irjk
irjk KH , in conformal Finsler space. In the paper several important theorems
are established.
A Mathematical model for the spread and control of tuberculosis on a population with temporally immunity. Cyprian Turyatemba, Julius Tumwiine, Mbarara Universty , Uganda
Abstract: The SEIR epidemic model for the spread and control of tuberculosis studied includes constant inflows of new susceptible, exposeds and infectives. This model also incorporates a contact rate and a disease‐ related death rate. As the infected fraction of the exposed and infectious sub‐ groups cannot be eliminated from the population, this kind of the model has only the unique endemic equilibrium that is globally asymptotically stable. However, in the absence of the infected inflows, the model will have a disease free equilibrium point. We prove that the endemic equilibrium point is globally asymptotically stable when the basic reproductive number, Ro, is greater than unity. While the disease ‐free equilibrium point is globally asymptotically stable when Ro is less than unity. Numerical simulations are also provided to illustrate the analytical results.
Application of fourier series to 1‐, 2‐, 3‐ dimensional signals with examples in matlab. Eyram Schwinger, University of Ghana, Ghana. Abstract:
When Joseph Fourier derived the concept of the Fourier series, he was only considering the solution of the heat equation. Today, the Fourier Series and its generalization, the Fourier Transform have become indispensable in the area of signal and image processing. The general solution of the heat equation is the convolution of the dirichlet kernel and the initial condition. In general, signal diffusion is achieved by convolving one signal with an impulse response function. The problem with convolution is that it is computationally expensive. Fourier transforms however give us a way out. With the Fast Fourier Transform(FFT), we can perform Fourier Transform conversion as a O(N logN) problem and perform element‐wise multiplication which is faster. In this talk we apply Fourier transforms to 1‐dimensional signals (sound), 2‐dimensional signals (images) and 3‐dimensional signal (3‐D images / volumes). We look at how much information we can retrieve from sound _les in the frequency domain and the time‐frequency representation of a 1‐dimensional signal. We look at such low‐pass filtering as a convolution with the gaussian and the 1‐ and 2‐ dimensional pulse signals and their corresponding operations in the frequency domain as element wise multiplication with the Fourier transform representation of the pulse (sinc) and gaussian signals, high‐pass filtering as discrete differentiation and as frequency space filtering. We will also look at image restoration and enhancement in the frequency domain, removal of periodic noise in signals using Fourier transforms and the application of the projection slice theorem to the visualization of three dimensional images. A mathematical model for transmission and control of bovine brucellosis in cattle populations. Godwin Robert, Mbarara University, Uganda.
Abstract: Brucellosis is an infectious, contagious disease of animals and man caused by a bacterium of genus Brucella. The disease causing organism is found in blood and urine of infected animals and is abundant in fluids and membranes of infected aborting and delivering animals.It is communicable to man by infected animal products like unpasteurized milk and cheese and poorly prepared meat. A mathematical model for transmission and control of bovine brucellosis in cattle populations is constructed to provide a better understanding of the disease and its impact towards the animal populations. The model developed is based on SIRS model .Model analysis is carried out to establish the stability of the equilibrium points .The thresh hold parameter which is the basic reproductive number Ro , is calculated. The analysis reveals conditions under which brucellosis can clear, when Ro<1 or persist in animal populations when Ro>1. Some numerical simulations of the model are presented and discussed.
MHD boundary layer flow of a chemically reactive fluid over a horizontal flat plate with suction, Ibrahim Yakubu Seini
Abstract: An analysis is presented for a magnetohydrodynamic boundary layer flow of a chemically reacting fluid over a moving horizontal flat plate with suction. The effect of uniform suction on the velocity, thermal and concentration boundary layers are presented using the newton raphson algorism alongside the runge‐kutta integration scheme. The skin friction coefficient, the rate of heat and mass transfer rates are presented with respect to various varying parameters and discussed quantitatively.
MAC model for an elastic plate.
Igor Neygebauer, University of Dodoma, Tanzania
Abstract:
The method of additional conditions or MAC is applied to the boundary value problems of mathematical
physics, where the classical solution does not exist or a nonphysical generalized solution is obtained. The
elastic plate under a transversal force applied at some point inside the plane domain of the plate creates
the solution with singularity in bending moments and corresponding stresses. The MAC model is
introduced to avoid the above nonphysical singularities. The beam model is a basis to create the MAC
model for an elastic plate. Using the principle of superposition the integro‐differential equation of the
MAC model is obtained. The boundary conditions at the point inside domain could be arbitrary that is
not true in the classical plate problem.
Split operator method for parabolic partial differential equations.
Nthiiri Joyce Kagendo, Mary Okombo, Shem Away, Masinde Muliro University and Michael Oduor Okoya, Maseno University.
Abstract: In many mathematical models differential equations of different types such as parabolic partial differential equations are integrated in time. This is done with the help of operator splitting methods. The splitting applied is simple and straightforward to apply, independent of the order of the split operators, since the spatial variables in a given equation can be easily identified. In this paper we use operator splitting method to obtain numerical solutions of seepage parabolic partial differential equations. The numerical scheme developed using the operator splitting technique has additional mesh points as compared to the known standard finite difference schemes. The developed scheme that is implicit and thus stable for all values of mesh ratio, has an increased order of accuracy. Numerical results were obtained using Mathematica, tabulated graphical presentations made.
Mathematical model for malaria and meningitis co‐infection among children. Lawi G.Owuor , Omolo – Ongati, Maseno University, Kenya and Mugisha J.Y.T., N., Makerere University
Abstract: Diseases such as HIV/AIDS, malaria and tuberculosis are a hindrance to economic development, especially in developing countries where huge resources are spent in treatment and prevention. Coupled with poverty, many preventable infectious diseases are a threat to child survival in the developing world, where access to good nutrition, sanitation and health care is poor. In this paper, a mathematical model for the dynamics of malaria and meningitis co‐infection among children under five years of age is developed and analysed. We establish the basic reproduction number Rmm for the model, which is a measure of the course of the co‐infection. The analysis shows that the disease‐free equilibrium of the model may not be globally asymptotically stable whenever Rmm is less than unity. The Centre Manifold theorem is used to show that the model has a unique endemic equilibrium which is locally asymptotically stable when Rmm < 1 and unstable when Rmm > 1. We deduce, from the numerical simulations, that a reduction in malaria infection cases either through protection or prompt effective treatment, which is dependent on the socio‐economic status of a community, would reduce the number of new co‐infection cases. The public health implication is that the fight against malaria and re‐emerging childhood diseases such as meningitis may be won by comprehensive laboratory testing to rule out or confirm co‐infection cases.
An Iterative Procedure for Solving a Cauchy Problem for the Helmholtz Equation.
Lydie Mpinganzima, Link¨oping University, Sweden.
Abstract:
An iterative procedure for solving a Cauchy problem for the Helmholtz equation is proposed. In each
iteration of the procedure well‐posed mixed boundary value problems are solved. The problem is
severely ill‐posed in the sense that small errors in the data may cause very large errors in the computed
solution. When our proposed method is used together with a suitable stopping criterion we obtain a
regularization of the original ill‐posed problem. Both analytical and experimental results demonstrate
that the method works well.
This is joint work with Vladimir Kozlov, Bengt‐Ove Turesson, and Fredrik Berntsson. Link¨oping
University, Sweden.
Analysis of unsteady hydromagnetic thermal boundary layer over a flat plate with Navier slip and a convective surface boundary condition. Oluwole Daniel Makinde, South Africa.
Abstract: The hydromagnetic thermal boundary layer over a flat surface is of interest in several engineering and geophysical applications such as geothermal reservoirs, thermal insulation, enhanced oil recovery, packed‐bed catalytic reactors, cooling of nuclear reactors. Moreover, many experimental results have provided evidences to support the possibility of slip condition at the fluid‐solid interface. In this talk, the combined effects of Navier slip, flow unsteadiness and magnetic field on boundary layer flow of an electrically‐conducting Newtonian fluid over a flat plate with a convective surface boundary condition is examined. By taking suitable similarity variables, the governing boundary layer equations are transformed into a set of non‐linear coupled ordinary differential equations and then solved numerically using shooting algorithm with Runge‐Kutta Fehlberg integration scheme over the entire range of physical parameters. The effects of key parameters on the fluid velocity, temperature, local skin friction and Nusselt number in the flow regime are depicted graphically and analyzed in detail.
Effects of internal heat generation, thermal radiation, and buoyancy force on boundary layer over a
vertical plate with a convective surface boundary condition.
Olanrewaju Philip Oladapo, Covenant University, Nigeria.
Abstract:
In this paper we analyze the effects of internal heat generation, thermal radiation, and buoyancy force
on the laminar boundary layer about a vertical plate in a uniform stream of fluid under a convective
surface boundary condition. In the analysis, we assumed that the left surface of the plate is in contact
with a hot fluid while a stream of cold fluid flows steadily over the right surface with a heat source that
decays exponentially. Similarity variable method is applied to the steady state governing non‐linear
partial differential equations and was transformed into a set of coupled non‐linear ordinary differential
equations, which are solved numerically by applying shooting iteration technique together with sixth
order Runge‐Kutta integration scheme for better accuracy. The effects of Prandtl number, local Biot
number, the internal heat generation parameter, thermal radiation, and the local Grashof number on
the velocity and temperature profiles are illustrated and interpreted in physical terms. A comparison
with previously published results on special case of the problem shows excellent agreement.
On N‐curvature inheritance in Finsler spaces.
J.K.Gatoto, Kenyatta University, Kenya.
Abstract:
S.B Misra and A.K.Misra (1984) have studied affine motion in RNP Finsler space. U.P. Singh and
A.K.Singh (1981) have defined N‐curvature collineation and discussed the existence of N‐curvature of
different types.in Finsler space. Recently S.P.Singh (2003) has introduced and studied projective
curvature inheritance in Finsler space. J.K.Gatoto and S.P.Singh (2008) have investigated and discussed
projective curvature inheritance in finsler space for the relative curvature tensor. C.K.Mishra and
D.D.Yadav (2007) have investigated and discussed projective curvature inheritance in normal projective
Finsler space. In the present paper, the author has studied N‐curvature inheritance in Finsler spaces.
Some special cases of N‐curvature inheritance are also discussed. Several important Theorems are also
established and formulated.
A Numerical Method for Improving Shielded Thermocouple Accuracy.
Fredrik Berntsson, Linkoping University, Sweden.
Abstract: A shielded thermocouple is a measurement device used for monitoring the temperature in chemically,
or mechanically, hostile environments. The sensitive parts of the thermocouple are protected by a
shielding layer. In order to improve the accuracy of the measurement device we study an inverse heat
conduction problem where the temperature on the surface of the shielding layer is sought; given
measured temperatures in the interior of the thermocouple.
Mathematically we can formulate the problem as a Cauchy problem for the heat equation, in cylindrical
coordinates, where data is given along the line r = r1 and the solution is sought at r1 < r _ r2. The
problem is ill–posed, in the sense that the solution (if it exists) does not depend continuously on the
data. Thus regularization techniques
are needed. The ill–posedness of the problem is analyzed and a numerical method is proposed.
Numerical experiments demonstrate that the proposed method works well.
This is joint work with Yves Nyalihama and Fidele Ndahayo, National University of Rwanda, Butare,
Rwanda.
Generalized equation of viscous flow through non‐circular flow channels.
George Xyplagus Stower, Kenyatta University, Kenya.
Abstract:
In this paper the length scale for a flow passage whose cross‐section is non‐circular is obtained. An
attempt is made to generalize the estimate of the discharge through irregular cross‐sections by
introducing the concept of shape factor, F a constant for a given cross‐section, and the derived form of
length‐scale. Poiseulle’s equation is found to be a special case of this generalized discharge equation
with F = 1. The problem of slow viscous flow through a pipe of rectangular cross‐ section is solved and
the discharge equation obtained gives a shape factor value of F= 1.111.
A mathematical model for the control of banana bacterial wilt disease. Kweyunga Eliab Horub & Tumwiine Julius, Mbarara University, Uganda.
Abstract: Banana bacterial wilt, caused by xanthomonas campestris pv.musacearum is a very destructive and rapidly spreading disease which affects all known grown banana varieties. Presently there is no cure and
no resistant banana cultivars. In this paper, a deterministic SI plant epidemic model has been proposed and analyzed. The model is based on the integrated disease management component of cultural control strategies. Cultural control involves continuous (or periodic) inspections followed by removal of diseased plants (called rouging) and replanting of healthy plants. The existence and stability of equilibria are determined and analyzed. Key in the analysis is the basic reproductive number R0, the threshold that determines whether the disease will persist or die out in the population. Parameter estimations and computer simulation have been carried out.
Key words: banana bacterial wilt,disease management, basic reproductive number , endemic equilibrium and stability.
Action of on .
I. N. Kamuti, E. B. Inyangala and J. K. Rimberia
Abstract:
Let ,2PSL ; the modular group. The action of on the rational projective line
has been studied by several authors. Jones, Singerman and Wicks have shown that acts transitively
on
Q and the stabilizer of a point is an infinite cyclic group. In this paper some properties of
(stabilizer of ) on have been investigated. It has been shown that the action is simply transitive and imprimitive. Some properties of the suborbital graphs corresponding to this action have also been determined.
Applications of u‐ideals in Banach spaces
Kayiita Z. Kaunda, Baraton University, Kenya
Abstract: First introduced by Casazza and Kalton, u‐ideals are generalizations of M‐ideals. Suppose Y is a real
Banach space. We shall define that on the optimal condition when the canonical decomposition X
�X*is unconditional, a subspace X of a Banach space Y to be a u‐ideal if there is an hermitian projection
P (respectively a projection P with _I‐2P_=1) on Y with kernel X. In this paper we undertake a general
study of the notion of ideals which was introduced by Godefroy, Kalton and Saphar, extend this concept
to u‐ideals, we further characterize the strict u‐ideals and then consider the applications of u‐ideals in
Banach Spaces. For the applications of u‐ideals in Banach spaces, in any Banach space we shall consider
the spaces X embedded in their biduals that is Banach spaces X which are u‐ideals in X.
Sign matrices for frames of 2n‐ons under Smith Conway and Cayley Dickson multiplications. Benard Kivunge, Kenyatta University, Kenya.
Abstract:
There has been a great desire to develop doubling formulas that give better algebraic structures as the dimensions of the algebras so formed increase. Whenever these doubling formulas are applied, several interesting loop and algebraic properties are observed on the
structures so formed. The Cayley‐Dickson formula is given by cbdabdacdcba ,),)(,( while
the Smith‐Conway doubling formula is
0if)(,
0if),(),)(,( 1 bdbabcbdbac
bdaacdcba .
A Hadamard matrix of order n is a nn matrix H with entries 1 such that nT nIHH where nI
is the identity nn matrix. It is shown that the sign matrices for the frame multiplication under the Smith‐Conway and Cayley‐Dickson multiplications are skew Hadamard matrices.
Kronecker products are also introduced, and it is shown that the sign matrices for the quaternion and octonion frames are equivalent to Kronecker products.
Rank and subdegrees of the symmetric group nS acting on ordered r‐element subsets .
Rimberia Kagwiria Jane, Kenyatta University, Kenya.
Abstract:
The action of the symmetric group nS on ordered subsets from the set 1, 2, ,X n is an aspect
that seems to have received little attention for a long time. Most studies have focused on the action of
nS on unordered subsets leaving many properties about its action on ordered subsets unknown. In this
paper, we determine the rank and subdegrees of nS acting on rX , the set of all ordered r‐element
subsets from X. Particular cases when 2, 3r and 4 will be considered first and then a generalization
will be made for any value of r and n. In the action of nS on 2 3, X X and 4X , the rank is shown to
be 7, 34 and 209 respectively. By generalizing these results, we have come up with the formulas for the
rank and subdegrees of nS acting on rX . The results show that the rank is a function of r alone
provided 2n r while the subdegrees are functions of both r and n. The action of nS on rX is also
shown to be transitive but imprimitive. We have also formulated the conditions for a suborbit of nS
corresponding to this action to be either self‐paired or paired with another. A formula for computing the number of self‐paired suborbits has also been derived using character theory.
Exact Minimizer and Duality. Japhet Niyobuhungiro, Link¨oping University, Sweden.
Abstract: Denoising is one of the most fundamental problems in image processing. The problem of image
denoising is to find a clear image nS from a noisy image . The Rudin‐Osher‐Fatemi (ROF)'s TV denois‐
ing model is one of the most successful methods for denoising and suggests taking as an approximation
to u a ut which minimizes the functional
BVlututuu
BVuBVlL
22
,, 22
1inf),(
where BV
denotes the space of functions of bounded variation on a rectangular domain _ R2. In this paper we will show what happens with the exact minimizer of Lt;u_ (l2;X) under the duality, where X is in general a
Banach space. With examples in concrete situations for 1lX and BVX we show why it is
important to construct exact minimizer. Results connected with characteristics of exact minimizer are also derived. Keywords: Exact
On characterization of u ‐ ideals determined by sequences.
John Wanyonyi Matuya, Narok University, Shem Aywa and Achiles Simiyu, Masinde Muliro University,
Kenya.
Abstract:
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 ‐ ideal given that X is a separable reflexive Banach space with approximation property if
and only if there is a sequence nT of finite rank of operators with lim 2 1n nI T and limn nT x x .
We shall show that u ‐ideals containing no copies of sequences 1 are strict u ‐ ideals.
Mathematics as the key instrument in modern technology.
Josephat Itambu, University of Dodoma, Tanzania.
Abstract: Mathematics being a key instrument in sciences plays a big role in development of any society. Early
mathematicians used to relate mathematical concepts in solving problems which are directly facing the
society. Nowadays, science and technology is rapidly growing and all research relating sciences and
mathematics deal with the determination of efficient methods in solving such problems. The role of
mathematics as a tool for development can be observed if and only if the positive impact on problems
facing the society. In Africa for instance, main problems involve economic and development
planning, diseases, poor technology and communication skills.
Using mathematical concepts and by the aid computer programs mathematical models can logically be
created. These can be used for different uses such as solving problems in economy, finance, industrial
production, optimization in business and planning. Similarly using mathematics particularly in codes and
cryptography, we can se up security system. Moreover, in order to deal with ecological, health and
environmental problems, mathematics plays a fundamental role.
Most of developing countries are with poor technology that leads them not to develop. Poverty,
diseases and ignorance still are crucial enemies of these countries. Naturally Africa is very rich but her
people are very poor. African land is fertile and hence it is good for agricultural activities. Furthermore,
activities such as fishing industry, mining and tourism activities might have a great chance in the
development of our countries. The main problem is the lack of facilities as well as experts.
Keyword: programming, simulation, cryptography, portfolio, optimization, biostatistics.
Sedenion Extension Loops and Frames of General Hypercomplex Numbers. Njuguna Lydia Nyambura, Kenyatta University, Kenya.
Abstract: This research is on hypercomplex numbers. These are numbers which are obtained by extending complex numbers using various doubling formulae. They include 4‐dimensional quaternions, 8‐
dimensional octonions, 16‐dimensional sedenions and the general 2n ‐ dimensional 2n ‐ons.
Consider the non‐negative numbers Z = {0, 1, 2, 3, …}. Nim addition gives a way of defining addition in Z to make it a field of characteristic 2. The Nim‐sum of a number of distinct powers of 2 is the
ordinary sum while the Nim‐sum of two equal numbers is to 0. In this paper we perform the multiplication of basis elements of complex, quaternion, octonion and sedenion split extensions using the Jonathan Smith formula. In each case we show that the multiplication is related to Nim addition. We
also show that the multiplication of split extensions for general n2 ‐ons can be viewed in terms of Nim addition.
Banach space ideal structures in operator ideals.
Makila Patrick Wanjala, Aywa Shem & Simiyu Achile, Masinde Muliro University.
Abstract: The theory of operator Ideals is playing an important role in the study of the structures of locally convex
spaces such as Banach spaces.We in our research aim to focus on Ideals of operators in which the Ideals
of compact operators(or finite rank operators or nuclear operatos or Integral operators ), will be
Ideals,u‐Ideals and h‐Ideals.This will be in the sense that they hold for every pair X,Y of Banach space.Of
further interest to any Banach space specialist is our demonstration that sometimes the intrinsic
characteristics of the class of operators considered rather than the stracture of the underlying spaces
determine the Ideal properties.
On the characterization of class R1 of non‐normal operators in a Hilbert space. Mile Justus Kitheka, University of Nairobi
Abstract: The study of normal operators is so successful that much of the theory of non‐normal operators is
modelled after it. In this paper, we give a characterization of class R1 of operators which was introduced
by Paul R. Halmos in the course of studying reducible operators. Halmos showed that R1 contains the
normal and isometric operators. In particular we show that if a bounded operator T is normaloid,
spectraloid, paranormal,or hyponormal then T is in R1. Also, if T is hyponormal, or is with the sequential
G1‐property, or has the G1‐property; then for any compact operator K, T+K is in R1.
Key words: Class R1,normaloid, spectraloid, paranormal, hyponormal, sequential G1‐property.
Ideals of the polynomial ring xF 1mod nx for error control in computer applications.
Olege Fanuel & Aywa Shem, Masinde Muliro University, Kenya.
Abstract: A nonempty subset B of a ring A is called an ideal of A if B is closed with respect to addition and negatives and B absorbs products in A. The concept of ideals was first introduced by Ernst Kummer with the aim of preserving the notion of unique factorization in certain rings of algebraic integers. This
research provides ideals of the polynomials ring xF 1mod nx associated with the
codewords of a cyclic code C. If the set of polynomials corresponding to codewords is given by I(c), an
ideal of xF 1mod nx , we are able to show that C is a cyclic code. Principal ideals of cyclic codes
are defined from a new view point involving polynomials. The potentialities of these codes for error control in computer applications are described in detail.
Same and equivalent linear codes in relation to categorising Goppa codes.
Govere W and Sibanda D, Midlands State University, Zimbabwe
Abstract: In recent research it has become clear that the numbers of extended Goppa codes is far fewer that the
number of non extended Goppa codes. For example there are 56 Goppa codes of length 32 (degree 4)
but these 56 codes collapse to only 4 codes when extended. In some cases 17 codes collapsed to one
code and in another case 5 codes collapsed to one code. This is the case with other classes of Goppa
codes. So the possibility of categorizing Goppa codes through their extended versions arises. This raises
many questions; however in this paper we try to find the number of different linear codes that when
extended give the same code distinguishing between same and equivalent.
Key words: Same linear codes, equivalent linear codes, Goppa codes
The M0‐ matrix completion problem for selected 4*4 matrices, p=4 and q =4; Digraphs without completion.
Peter Waweru, Cecilia Mwathi, JKUAT, Kenya and Bernard Kivunge, Kenyatta University.
Abstract: A real m × m matrix is said to be an M0‐ matrix if all its principal minors are non negative and all of its non‐diagonal entries are non positive. In this paper, we consider the M0‐ matrix completion problem. We show that any diagraph for p = 4 and q = 4 which is either a cycle or a clique does not have M0 ‐ completion.
Key words: Matrix, M0‐ completion, Diagraph, Cycle and Clique
On the unit groups of completely primary finite rings of characteristic p. Owino Maurice Oduor, Maseno University Abstract: Consider a commutative completely primary ˉnite ring R with the unique maximal ideal J such that J3 = (0), J2 6= (0). Then the residue ˉeld R=J is isomorphic to the Galois ˉeld of order pr and the characteristic of R is pk, where 1 ∙ k ∙ 3 for some prime p and positive integer r: Suppose R0 = GR(pkr; pk) is a Galois subring of R so that R = R0 © U © V © W, where U; V and W are ˉnitely generated R0‐modules with s; t and ¸ as the numbers of elements in the generating sets for U, V and W respectively. In this paper, we determine in general the structure of the unit group R¤ of R for any positive integer s, t = 1; ¸ ¸ 1 and characteristic of R is p: Key words: unit groups, completely primary finite rings
Necessary conditions for the existence of a friend of 38. R. K. R. K. Gachimu, C. W. Mwathi, JKUAT, Kenya and I. N. Kamuti, Kenyatta University.
Abstract: This paper sets out to, systematically, use properties of the abundancy index function to prove that a friend of 38 must be an odd non‐square multiple of which is not divisible by 3, and that every
prime factor of such that has an even exponent in the prime factorization of . In addition,
if the power of 19 in is 2, then in which case the power of 127 must be even, larger than 2 and
not equal 8, and if the power of 19 is 6, both 701 and 70841 would be compulsory prime factors of ,
where the power of 701 cannot equal 1 or 3. The paper also establishes that it is not possible to have 8 as the power of 19 in the prime factorization of .
Key words: abundancy index, sum of divisors, friend, prime factor, power.
The Limitations in the Convectional ISBN‐10 Code. Peter Waweru, Cecilia Mwathi, JKUAT, Kenya and Benard Kivunge, Kenyatta University.
Abstract: The International Standard Book Number system (ISBN‐10) which was in operation until 2007 uniquely identified every book published internationally. The code had the ability to detect and correct single errors, to detect and correct some transposition errors and also detect multiple errors. This paper discusses some major limitations in the code and shows how error detection and correction capabilities affected the total dictionary on the code.
Key words: Code, Dictionary, ISBN‐10, Error detection, Error Correction.
Formulas for the computation of the Tutte polynomial of a graph with parallel classes. Eunice Mphako‐Banda, University of Witwatersrand, South Africa. Abstract: We give a relationship of the Tutte polynomials of graphs in the same parallel class. In particular we give the Tutte polynomial of a graph with some sets of parallel elements in terms of the Tutte polynomial of its simplification and the minors of its simplification.
Transforming attitudes and performance of management students statistical training Bernard Manyalla, Siaya Institute of Technology, David Stern, Maseno University.
Abstract: With the increasing use of computer‐ based statistical teaching in the educational and training domain, many instructors have recognized the significance of evaluating its effects on student’s outcomes such as learning, performance and attitude. Often, these outcomes are compared to those of conventional classroom instruction in order to determine which method of instruction is ‘better’. Computer Assisted Statistics Textbooks (CAST) that make extensive use of interactive and dynamic diagrams to teach statistics offers learners unparalleled much needed access to instructional resources far surpassing the reach of traditional classroom. This paper describes the use of CAST in teaching management statistics course at the Kenya Institute of Management Kisumu. The Kenya Institute of Management (KIM) trains about twenty thousand students a year in various locations throughout Kenya in courses which include a module on statistics. This module is taught according to a standardized syllabus with a centrally administered examination In particular there is solid quantitative evidence that the teaching with CAST transformed the student’s attitude to statistics and their performance in their examination using an analysis based on the results obtained by the experimenter before and after CAST usage over multiple exam series and different teachers using identical syllabus where potential possible causal effects are accounted for.
Key words: Statistics, Computers, Education, Management
Adjusting rainfall for climate change scenarios as inputs for crop simulation models,.
Mawora Thomas Mwakudisa and David Stern, Maseno University.
Abstract:
Crop simulation models, such as APSIM (Agricultural Productions Systems sIMulator) are used to
evaluate the impact of scenarios of climate change on the growth and yield of a wide range of crops.
These models require daily rainfall and temperature data among others. Typical scenarios are for
minimum and maximum temperatures to increase by 2 degrees and rainfall amount to increase, or
decrease, by 10%. Calculating the adjusted temperatures is straightforward, but rainfall can be changed
in a variety of different ways. These include changing the rainfall amounts, but leaving the rain days as
before, or changing the days, and leaving the amounts. This paper investigates how the pattern of
rainfall can be changed and the implications of these changes on crop yields.
A Secured Hybrid Architecture Model for Mobile Financial Services: A Case Study of Zimbabwe. Weston D. Govere and Dumisani Sibanda, Midlands State University, Zimbabwe. Abstract: Mobile banking has made it easy to carry out personal or business financial transactions without going to a bank and at any suitable time. This facility enables to send and receive money, regardless of whether one has a bank account or not. It also enable customers to perform service transactions such as bill payments, airtime top‐up and Point‐of‐Sale purchases. This transformation to a cashless society has been enabled by the advent of mobile phones which are now more affordable and available to anyone anywhere worldwide. NetOne (Zimbabwe) has taken the lead and recently launched One Wallet mobile banking facility enabling customers to send and receive cash via an SMS based transaction. One Wallet also enable customers to top up their airtime, top up another person's airtime account and pay utility bills via mobile phone. Another notable example is CellCard by Kingdom Bank. However, in order to maintain privacy and to avoid any misuse of these services, it is necessary to follow a secured architecture model which ensures the privacy and integrity of the transactions and provides confidence on mobile banking. In this research paper, a secured hybrid architecture model for mobile banking using hyperelliptic curve cryptosystem (HECC) and MD5 is described. This hybrid model is implemented with the hyperelliptic curve cryptosystem and it performs the encryption and decryption processes in an efficient way merely with an 80‐bit key size. The various screen shots given in this paper shows that the hybrid model which encompasses HECC and MD5 can be considered in the mobile banking environment to enrich the privacy and integrity of the sensitive data transmitted between the clients and the application server. Keywords: Mobile banking, hyperelliptic curve cryptosystem, MD5, Authentication, Con_dentiality, Integrity, Nonrepudiation, Privacy.
Making Markov chain modeling of rainfall data accessible.
J.O. Ong’ala, The Great Lakes University of Kisumu,Kenya. D.A. Stern, Maseno University, Kenya and
R.D. Stern, University of Readings, UK.
Abstract: Rainfall is of critical importance for many people in Kenya particularly those whose livelihoods are
dependent on rain fed agriculture. Methods of analysis of daily rainfall records based on Markov chain
models have been available for many years and their value is widely recognized. However they are
rarely used in Kenya because of the complexity of the analysis. This paper describes how these models
are being made more accessible through a series of specially written procedures and menus in GenStat,
a widely available statistics package.
Keywords: Markov Model, Climatic analysis, Rainfall data, GenStat Convergence of Ishikawas's Iteration Method for Finite Family of Pseudocontractive Mappings. Habtu Zegeye, University of Botswana, Gaborone, Botswana. Abstract:
Let C be a nonempty, closed and convex subset of a real Hilbert space H . Let
,,,2,1,: NiCCTi be a finite family of Lipschitz pseudocontractive mappings with Lipschtz
constants NiLi ,,2,1: . Let a sequence nx be generated from an arbitrary Cx 0 by
(0.1)
nnnnnn
nnnnnn
yTxx
xTxy
)1(
)1(
1
where )(mod Nnn TT and )1,0(, nn satisfying the following conditions:
(i) ;0, nnn (ii) ;inflim onn
(iii)
,0,11
1sup
21
n
Ln
n
for NiLiL ,,2,1:max: . It is proved that the sequence nx converges strongly to a common
fixed point of NiTi ,,2,1: provided that interior of common fixed points is nonempty. No
compactness assumption is imposed either on T or on C . Moreover, computation of closed convex set
nC for each 1n is not
required. The results obtained in this paper improve most of the results that have been proved for this class of nonlinear mappings.
On a more generalized class of bounded starlike functions of complex order. Biodun Oladipupo, Ladoke Ankintola University of Technology, Nigeria.
Abstract: Using the new concept of analytic functions, we denote the class of analytic functions of complex order
in the open unit disk 1: zzU by ),,(,, BAH blm defined by a new extension of Salagean
operator.Coefficient estimates and other properties are investigated for this class of functions. Consequence of the parameter involved are also discussed.
Pricing Asian currency options in a developing financial market: A modified lattice approach. Carolyne Ogutu , University of Nairobi, Kenya.
Abstract : Risk is the uncertainty of an outcome and it can bring unexpected gains but can also cause unforeseen losses, even catastrophes. Over the years, financial derivatives have been popular in financial since they are considered very good risk management tools. Currency derivatives markets are the hallmark of industrialized countries but this is seldom true for less developed countries. Even though currency
forward contracts are available in the less developed countries, they are only deemed as forward‐cover insurance schemes.
A financial derivative is a financial instrument whose value depends on other fundamental financial assets, called underlying assets, such as stocks, indexes, currencies, commodities, bonds, mortgages and other derivatives. A path‐dependent financial derivative is one whose value depends nontrivially on the price path of the underlying asset. Asian options are an example of path‐dependent financial derivatives. There are no closed form solutions for some of these path‐dependent options especially the arithmetic average Asian options. Thus numerical methods have been developed to come up with pricing solutions for them. One such method is the lattice method.
In this paper I use the moment matching technique to develop parameterization for the trinomial lattice and use an adjusted lattice to price European Asian currency options based on the arithmetic average of the underlying currency. Then, I will the backward recursion to find the option price.
Key words: Lattice models, currency options.
A Mathematical comparative model of HIV/AIDS, poverty and development.
Cecilia Musyoka, South Eastern University College, Kenya.
Abstract:
Kenya’s economic and technological development partly depends on her ability to sustain itself in
disease control and education. For her to
achieve the 2030 MDGs the challenges experienced have to be faced head on. Many interventions have
been implemented on the eradication of the HIV/AIDS pandemic. However, the impacts of the pandemic
remain to be felt world over. In Kenya, more often than not majority of female youth who bear children
at their tender age flee their homes in search of employment in major towns. Because of their low level
of education, poverty and ignorance, they end up engaging in sex so as to provide economic sustenance
for themselves and their dependants. This makes them enter into the class of Most At Risk Population of
HIV/AIDS (MARPS). This paper aims at giving a comparative study between poverty levels, disease
(mostly HIV/AIDS) prevalence and development using a mathematical comparative model. The
underlying assumptions are that rural development is highly contributed by females and that
development is dependent on education and disease control.
Key words: MARPS, HIV/AIDS, Education, poverty, mathematical comparative model.
Distributed controllability of a cochlea model.
Isaac Chepkwony, Kenyatta University, Kenya.
This article analyses the controllability structure of a cochlea. Many cochlea models found in literature
are based upon an incompressible, inviscid fluid inside the cochlea cavity, which consists of two
chambers separated by the basilar membrane. In the two –dimensional model, the basilar membrane
becomes a one‐dimensional elastic division between the two fluid filled chambers. The basilar
membrane is modeled as an infinite array of springs. Well‐posedness results are obtained for the model
in an appropriate function space. The approximate controllability, with control active on an open set on
the basila membrane is also demonstrated.
Key words: distributed control, cochlea, basilar membrane.
Solution of the third order viscous wave equation using Finite Difference Method. Oganga Duncan, Shem Away, Masinde Muliro University and Michael O. Okoya, Maseno University.
Abstract: A wave motion is the transmission of energy from one place to another through a material or a vacuum. Wave motion may occur in many forms such as water waves, sound waves, radio waves, light waves among other forms. In this paper, we consider sound waves. Sound is transmitted through gases, plasma and liquids as longitudinal (compression and rarefaction) waves. Through solids however, it can be transmitted as both longitudinal and transverse waves. Longitudinal sound waves are waves of alternating pressure deviations from the equilibrium pressure causing local regions of compression and rarefaction while transverse waves in solids are waves of alternating shear stress. In this paper, we have addressed the propagation of sound waves in viscous fluids, which is represented by the third order partial differential equation where is the distance
along the axis of propagation, denotes time and is the speed of sound in the absence of viscosity.
Dean Duffy solved the non‐dimensionalized version of this equation by Laplace Transforms and got a solution which is not easy to interpret. For this reason, we have solved the same equation solved by Dean subject to same boundary and initial conditions using the finite difference method. This method involves replacing the equation, its associated initial and boundary conditions by their finite difference analogues, analyzing the stability of the resulting finite difference schemes and then solving numerically. We came up with two schemes namely Forward Difference Scheme and Crank Nicolson Scheme and used matrix stability analysis method to analyze stability. The results that we have obtained can be interpreted easily.
Demystifying mathematics: Handling learning difficulties in mathematics among low achievers in
Kenyan schools.
Gladys Gakenia Njoroge, Kenyatta University, Kenya.
Abstract:
Mathematics is a compulsory subject in both primary and secondary schools in Kenya. However, poor
performance in Kenya national examinations year in, year out remains a serious concern for teachers of
Mathematics, parents, curriculum developers, and the general public. This is particularly worrying
because of the importance attached to the subject in national development hence the need to find out
what could be affecting Mathematics learning in Kenyan schools.
The research on which this paper is based sought to (i) examine the factors that influence performance
in Mathematics in Kenyan primary schools (ii) identify the characteristics of Mathematics learning
disabilities (iii) determine how the learners with such learning disabilities can be assessed and identified
and (iv) suggest interventions for these difficulties.
The target population was class Six pupils in selected primary schools in Nairobi County. The tools used
for the research were questionnaires, classroom observations, focal group discussions and an
Individualised Education Programme (IEP) developed by the teacher with the help of the researcher.
This paper therefore will highlight the findings from the research, discuss the implications and suggest
the way forward as far as teaching, learning and assessment of Mathematics in Kenyan schools is
concerned. Perhaps with the application of the right interventions, poor performance in Mathematics in
the national examinations will be a thing of the past.
Investigating the maximal determinant for an matrix.
I. A. Okello, C. W. Mwathi, JKUAT, Kenya and B. Kivunge, Kenyatta University.
Abstract: It is known that for any
matrix ; , Therefore, is an upper
bound of determinants of all matrices which satisfies the above condition.
In this paper, we determined the maximum determinant of an matrix where ,
, for 1, 2, 3, 4 and 5 using the determinant function formula and expansion using
minors. For an matrix, the maximum determinants for were found to be
and respectively. The numbers of distinct matrices attaining the maximum
determinant for are and respectively.
Keywords: ‐matrix, maximum determinant, supremum determinant, minor matrix.
Views on the role of Mathematics in development.
Jan Persens, University of the Western Cape, South Africa.
Abstract:
Development associated with economic and industrial growth, can best be described and judged by
investigating changes in real‐world phenomena. Given the inter‐dependence and interdisciplinary
nature of knowledge, especially in this new era, it is somewhat difficult to attribute certain contributions
to development to a particular discipline. Since the question is about the role of mathematics in
development, the concentration in this paper is addressing this topic. In addition to considering the
intellectual capacity in mathematics in Africa, the socio‐economic realities and governments’ capacity to
deliver in particular ways, are highlighted. In order to minimize or even combat the negative state of
developing economies, it is suggested that networks of centers of excellence in
mathematical sciences be established. In this way one can ensure continued and meaningful
contributions of mathematics in development. At this stage there are already several such centers in
Africa concentrating on particular mathematical sub‐disciplines. Investing in such centers will secure
critical masses of expertise and counter the outflow of scarce financial means on low student numbers
and researchers. Early starts in mathematics training and the introduction of mathematics of an applied
type may result in the enhancement of problem‐solving skills at an earlier age.
Implicative algebras. K.Venkateswarlu , Berhanu Bekele, Adis Ababa University, Ethiopia.
Abstract: The concept of lattice implication algebras is originally due to Y . Xu. In his paper , he introduced the concept of lattice implication algebra and quasi lattice implication algebra as a bounded lattice satisfying a system of axioms and studied certain properties. In this paper we have given an equivalent for lattice implication by simplifying the axioms of his definition. Further we have introduced two more binary operations on an implicative algebra and obtain that every implicative algebra is an autometrized algebra.
Encouraging Dialogic Learning in Mathematics’ Class Discourse. Kahenya, Paul Njoroge , Africa Nazarene University,Kenya.
Abstract: The level and quality of the human resources significantly determines the the levels and rate of socio – economic growth and development of any nation. Education ‘plays a dominant role as an instrument for large‐scale achievement and revolution in many spheres’ (IEJDICT, 2006). Human resource development in ‘an integrated sense encompasses education and training …’ ACT/EMP (1997). The quality of education is therefore paramount in determining the level of development. This implies then the quality of instruction at institution level will therefore determine the quality of manpower development. It has been credited that, ‘universities should be concerned with the solutions of the concrete problems of societal development,’ (Coleman (1994) cited in IEJDICT (2006)).
In this regard, the research looked at encouraging dialogic learning so as to promote deep learning of mathematics and hence improved understanding and application of mathematical concepts. This is hoped; it will enable learners to apply mathematical concepts in solving real life problems and also form a strong base in developing innovative technologies in computer science.
Traditionally, class discourse has always being a one‐way affair dominated by the teachers or what is referred to as IRE; Initiation – Reply – Evaluation (Mehan, 1978, 1979; Cazden, 2001 and Macbeth, 2003 cited in Polman, 2004). Encouraging dialogue in class discourse has been noted to promote meaningful learning (Alexander, 2005; Polman, 2004 among others). This study aimed at identifying the nature of discourse in class discussions and introducing strategies that encourages dialogic learning.
The strategies used included beginning class discussions with a challenging question; encouraging peer discussion; creating a friendly leaning environment; creating a seating arrangement that encourage dialogue and allowing sufficient time after asking questions.
The study focused at second year computer science students. It found out that without dialogue, the mathematically able students dominate class discourse. These students tended to impose their views of a concept on others irrespective of whether the concept is right or wrong. The study found that encouraging dialogue required a variety of strategies which can be broadly categorized into three namely learner – oriented, task – oriented and teacher/environment – oriented strategies.
The study also found out that dialogue in class discourse, among others, enhanced group cohesions, improved student’s confidence in self expression, promoted deep learning and it is mostly the learners’ oriented factors that determine the effectiveness of dialogue in class discourse.
Moving teaching beyond calculations towards mathematics through eLearning; a case study of e‐Statistics made simple. Kaleli James Musyoka, Thomas Mawora and David Stern, Maseno University, Kenya.
Abstract:
Recently there has been a call to revolutionize mathematics education through the use of computers.
The aim is to focus the education much more on student understanding of concepts and mathematical
problems and beyond calculations by using computers to do the calculations. Computers are the
ultimate calculating machine; they are capable of doing extremely complicated calculations almost
instantaneously and will always get accurate results provided the inputs and mechanisms are correct. In
statistics this revolution started forty years ago and has very slowly gained momentum, the ideas are
now well accepted internationally but implementation remains a problem.
Recently Maseno University, in collaboration with the Statistical Services Centre (SSC) University of
Reading, piloted an online statistics course called e‐Statistics Made Simple. This course targeted
professionals who need to use statistics but feel that, despite any previous training they might have
received, they need a course which assumes they do not understand any statistics. The approach taken
is very practical focusing on understanding and interpretation; throughout the course very few formulas
are mentioned. This paper describes the course along with the experience of giving it in Kenya and the
views of the participants.
On Banach space ideal properties. Musundi Sammy Wabomba, Chuka University College, Kenya.
Abstract: Various authors in a series of papers have given a variety of definitions about an ideal. For instance the notion of an ideal in a Banach space was introduced by Godefroy, Kalton and Shaper in 1993. A closed subspace F of a Banach space E is called an ideal in E if F?, the annihilator of F in E¤, is the kernel of a norm one projection P on E¤. In this case P is called the ideal projection. We shall discuss essential Banach space ideal properties as per the deˉnitions. For Banach spaces X and Y , we denote by L(X; Y ) the Banach space of bounded linear operators from X to Y , and by K(X; Y ) its subspaces of compact operators. A subclass Lw(X; Y ) of the space L(X; Y ) is a Banach space with respect to a suitable norm stronger than the uniform operator norm, but coincides with the uniform norm on the space K(X; Y ) of compact operators. It has been shown that K(X; Y ) is an ideal in Lw(X; Y ). We shall also investigate conditions or equivalent norms on L(X; Y ) under which stronger ideal properties operators of K(X; Y ) will hold.
Mathematical modelling for road design in kenya: the mechanistic‐empirical option. S.A. Nyakiti, D. A. Stern and O. Ongati, Maseno University, Kenya. Abstract: The design of pavements is an integral part of the road management process and contributes significantly to the economic development of a region, since well‐designed roads last longer ensuring ease of movement of labour and goods. Presently, Kenya has its roads designed relying on time‐honoured manuals whose bases are purely empirical models and procedures. While these have served
well thus far, new research‐based evidence point to their limitation in accurately forecasting the future response and performance of the roads as a function of traffic growth, environmental influences and new construction materials. This paper explores the novel design paradigm, the mechanistic‐empirical (M‐E) approach. The M‐E method takes the pavement design process into the realms of mathematics, and supplements the existing engineering procedures in accurately predicting the response and performance of roads. This is achieved through dynamic mathematical modelling of the state of the road at any time throughout its design life. We assess the readiness of the country for adoption of the M‐E design, examine the existing challenges to this, outline the economic benefits Kenya will accrue in adopting it, and highlight the mathematical formulations and modelling inherent in M‐E, in particular the rigidity in the inputs required for its climatic models. The paper concludes by proposing a mathematical solution the road agencies in Kenya can employ both in the medium and long terms to ensure a successful transition to the M‐E design method. Keywords: Mathematical Modelling; Mechanistic‐Empirical Design; Pavement Design; Pavement , Performance; Economic Development.
Laser Technology as a tool for Development. Opondo, Mary A, Kenyatta University, Kenya
Abstract: Gaseous lasers especially the carbon dioxide lasers have found extensive applications and offer new solutions to the scientific, medical and industrial problems. The carbon dioxide gas laser has lasing transitions at several wavelengths in the infra red principally around 9.6 and 10.6 . To
accommodate the increasing number of applications, a variety of laser resonator designs, power levels and model configurations are needed. All lasers produce unwanted heat and as the power level of the laser increases the problem of getting rid of the heat also increases. Since the unwanted heat is generated in the gas, the problem is one of cooling and replacement of the gas and more effective means of cooling the gas without influencing the electrical excitation are needed. In this paper various factors that affect the laser operating efficiency and ways of increasing the output power are discussed. The paper emphasises how Mathematical techniques can be used to design laser models that are suitable for particular applications.
Key words: Carbon dioxide lasers, new solutions, power levels, operating efficiency.
The role of Mathematics in development of Technical and Vocational Education and Training (TVET) in
the 21st century.
Samuel Muchiri Mwangi, Gusii Institute of Technology, Kenya.
Abstract: The 21st century we are living in is characterized by a technology‐based society which requires individuals who are able to think critically about complex issues, analyze and adapt to new situations, solve problems of various kinds, and communicate their thinking effectively. In this paper the role of mathematics in development of technical and vocational education and training (TVET) in the 21st century, is discussed. It is indicated that the study of mathematics equips learners in TVET institutions with knowledge, skills, and habits of mind that are essential for successful and rewarding participation in a technology‐based society. It is observed that Mathematics has not attracted much attention as a tool in development of TVET. Concrete steps are suggested to steps required for training mathematicians to meet challenges of development in TVET.
Key words: Mathematics, Development, Technical and Vocational Education and Training.
Completely Prime Modules. D. Ssevviiri and N. J. Groenewald, Nelson Mandela Metropolitan University, South Africa.
Abstract: We generalize a notion of completely prime ideals in rings to submodules in modules. This notion is compared with that of prime and classical prime for modules which already exist in literature. Completely prime implies classical prime but not conversely in general. When M is defined over a commutative ring, completely prime and classical prime are indistinguishable. Furthermore, when M is defined over a commutative ring, prime implies completely prime but not conversely. In general, (when R is not commutative) prime does not imply completely prime. In any module over a ring R completely semiprime semiprime classical semiprime. When R is commutative the three notions of semiprimeness coincide. Mathematics is the key to sustainable development. Ms Taliba Caroline and Cyrus S. Ssebugenyi, Makerere University, Uganda.
Abstract: Mathematics is a discipline that seeks an understanding of the patterns and structures of constructs of the human mind. Understanding has no end to its depth, and mathematics seeks the highest standards of understanding by demanding rigor in its foundations and in its development. Rigor is achieved by responsible attention to the principles of logic. Therefore, mathematics occupies a crucial and unique role in the human societies and represents a strategic key in the development of the whole mankind. The ability to compute, for example, is related to the power of technology and social organisation. Geometrical understanding of space‐time, that is the physical world and its natural patterns, show the scientific and cultural role of Mathematics in the history of civilisation and in the future development of the Information Society. Mathematics has played and will continue to play a major role for the development of an increasingly global world and civilisation.
On commutants & spectral properties of ‐commuting operators.
Mutie Kavila, Kenyatta University , Kenya.
Abstract:
Let B H denote the Banach algebra of bounded linear operators on a complex Hilbert space H and let
,A B B H satisfy the relation AB BA for . We investigate the conditions under
which e eAB BA and e eA B where ( )e A denotes the essential spectrum of A. Also
given that ( , )C A B and ( , ) : ( ) ( )R A B B H B H by ( , ) C A B X AX XB and
( , ) R A B X AXB X we show that if A is one‐one and B has dense range then 2 2( , ) 0C A B X and
3 3( , ) 0C A B X imply ( , ) 0C A B X for some ( )X B H . Similarly, if 2 2( , ) 0R A B X and
3 3( , ) 0R A B X then ( , ) 0R A B X for some ( )X B H .
Key words and phrases: Pure dominant, Compact operators, Commutant, Quasiaffinity and Normal operator
Stanley decomposition of coupled N333 system. Gachigua Grace, Kimathi University College and David Malonza, Kenyatta University, Kenya.
Abstract:
The normal form of a set of systems of differential equations at equilibrium with nilpotent linear part has the structure of a module of equivariants, and is best described by giving a Stanley Decomposition of that module. We use an algorithm based on the notion of transvectants from classical invariant theory in
determining the form of Stanley Decomposition of the ring of invariants for the coupled 333N systems
when the Stanley Decomposition of the Jordan blocks of the linear part are represented separately. The
Stanley Decomposition can then be verified by developing a table function denoted by 3nT , where 3n is
the dimension of the linear part.
On Conjectures in Prime Number Theory. Peter Acquaah, University of Ghana, Ghana. Abstract: It is generally believed that for every positive integer n, there is always at least one prime number, p,
such that 22 )1( npn . We shall provide a ’strong’ sufficient condition for the following
statements to be true. (a) For every positive integer n, there is always at least two prime numbers,
),(, qpqp such 22 )1( nqpn . (b) For all kn, N, ,1,1 nkn there exists a prime
number ),( knp such that .)2(),( nkknpkn In particular, the interval nnn )2(,2 contains
two distinct prime numbers. (c) If np is nnth (, N) prime number, then
.1,121 nppp nnn (d) If baba ,,1),gcd( Z, then the set Nnbna : contains
infinitely‐many prime numbers. Keywords: Prime numbers, Andrica's Conjecture, Legendre's conjecture, short‐intervals.
How the density of leucocytes determine the lifespan of HIV patients on anti‐retroviral drugs.
Case study: Kakamega Provincial General Hospital.
Leonard David Lukhafwa, Prof Shem Away, Masinde Muliro University
Abstract: Patients suffering from HIV/AIDS respond differently to anti‐retroviral. The causative virus (HIV) lives in
the white blood cells (Leucocytes) of humans. This generally interferes with the normal functioning of
the immune system of the patient.
Blood is a fluid that is in constant circular motion that is aided by the pumping of the heart. Pressure in
fluids generally depend on the density, height and gravitational force are fluid ie. P=phg. Naturally, the
height and gravitational force are assumed to be constant in blood in its entirely. However, introduction
of anti‐retroviral drugs in the blood system alters the mass and volume of the leucocytes. This leads to a
changes in density and consequently affect the blood pressure.
There is therefore a dire need to study the relationship between the blood group and the effect of
specific anti‐retroviral drugs in order to reduce the overall blood pressure in or to elongate the lifespan
of the patient under these drugs.