Pure and Applied Mathematics
https://repository.maseno.ac.ke/handle/123456789/103
2024-03-28T16:58:25ZMathematical modeling of Dual protection and art Adherence for a high risk HIV Population
https://repository.maseno.ac.ke/handle/123456789/5986
Mathematical modeling of Dual protection and art Adherence for a high risk HIV Population
ORIEDO, Indayi Samson
The spread of HIV/AIDS remains a major concern to public health enthusiasts
world over. In spite of interventions such as medical male circumcision,
condom use, treatment using Antiretroviral Therapy (ART), as
well as use of Pre-Exposure Prophylaxis, the number of new HIV/AIDS
infections in Sub-Sahara Africa remains high. This may be attributed
to factors such as PrEP failure and inconsistency in condom use especially
among the high risk group. The e ectiveness of condoms depends
on quality and proper use, while the success of ART largely depends on
adherence. Mathematical models for these interventions exist in literature.
However the challenges associated with the use of a single approach
consequently necessitate the use of dual protection for better outcome
against infection especially for the high risk population. In this study, a
mathematical model for dual protection, incorporating PrEP and Condom
use, and ART adherence is formulated, based on a system of ordinary
di erential equations and analyzed. The results obtained from stability
analysis indicate that provided the basic reproductive number (R0) is less
than unity, the disease free equilibrium point is both locally and globally
asymptotically stable, while provided that R0 is greater than unity,
the endemic equilibrium point is locally asymptotically stable. Sensitivity
analysis showed that the most sensitive parameter is 1, the mean contact
rate with undiagnosed infectives. Numerical simulation results revealed
that dual protection and ART adherence are key in the ght against the
spread of HIV among the high risk population. These ndings will help in
reducing the number of new HIV infections as well as lower the infectivity
of those who are already infected.
Master's Thesis
2023-01-01T00:00:00ZOperation of mutation on Polar quivers
https://repository.maseno.ac.ke/handle/123456789/5261
Operation of mutation on Polar quivers
OYENGO, Michael Obiero Michael Obiero
In recent times, there has been a lot of interest in the study of quivers,
both by mathematicians and theoretical physicists. We introduce a new
concept of polar quivers and their mutation. The idea of polar quivers
arises from the concept of anomaly free R-charges in theoretical physics.
Mutation of polar quivers is build on mutation quivers with potential,
which was defined by Derksen, Weyman and Zelevinsky. An R-charge assigns
angles to the arrows of a quiver. In a polar quiver we assign angles
and positive non-zero integers to vertices and impose conditions equivalent
to the anomaly conditions for R-charges. We then establish that
mutation of a polar quiver will give a polar quiver if and only if a simple
additional condition is satisfied. We use families of quivers linked by mutation,
from the work of Stern, as our source of examples. The results of
this study have applications in geometry and theoretical physics.
2009-01-01T00:00:00ZNorms of tensor products and elementary Operators
https://repository.maseno.ac.ke/handle/123456789/5260
Norms of tensor products and elementary Operators
ODERO, Beatrice Adhiamb
In this thesis, we determine the norm of a two-sided symmetric operator
in an algebra. More precisely, .we investigate the lower bound of the operator
using the injective tensor norm. Further, we determine the norm of the inner
derivation on irreducible C*-algebra and confirm Stampfli's result for these
algebras.
2009-01-01T00:00:00ZOn norms of elementary operators
https://repository.maseno.ac.ke/handle/123456789/5257
On norms of elementary operators
NYAARE , Benard Okelo
The study of elementary operators has been of great interest to many
mathematicians for the past two decades. Of special interest has been to
determine the norms of these operators. The norm problem for elementary
operators involves finding a formula which describes the norm of an elementary
operator in terms of its coefficients. The upper estimates of these
norms are easy to find but approximating these norms from below has
proved to be difficult in generaL Several mathematicians have produced
known results for special cases on the lower estimates, for example, Mathieu
found that for prime C*-algebras, the coefficient is ~, Stacho and Zalar
obtained 2( v'2-1) for standard operator algebras on Hilbert spaces, Cabrera
and Rodriguez obtained 20!I2 for JB* -algebras while Timoney came up
with a formula involving the tracial geometric mean to calculate the norm
of elementary operators. An operator T: A ~ A is called an elementary
operator if T can be expressed in the formZ'[z) = L~=Iaixbi, \j x E A
where A is an algebra and tu, b; fixed in A. The norm of an operator T
is defined by IITII= sup{IITxll : x E H, Ilxll = I} where H is a Hilbert
space. The purpose of this study therefore, has been t,o determine the
lower estimate of the norm of the basic elementary operator on a' C*-
algebra through tensor products. To do this we needed to have a good
background knowledge on functional analysis, general topology, operator
theory and C*-algebras by understanding the existing theorems and relevant
examples especially on tensor product norms. We used the approach
of tensor products in solving our particular problem. We hope that the
results obtained shall be useful to applied mathematicians and physicists
especially in quantum mechariics.
2009-01-01T00:00:00ZMathematics of Pesticide Adsorption in a Porous Medium: Convective-Dispersive Transport with steady state water flow In two Dimension
https://repository.maseno.ac.ke/handle/123456789/5251
Mathematics of Pesticide Adsorption in a Porous Medium: Convective-Dispersive Transport with steady state water flow In two Dimension
WETOYl, A.Seth Harrisson
The transport of solutes through porous media where chemicals undergo adsorption or
change process on the surface of the porous materials has been a subject of research over
years. Usage of pesticides has resulted in production of diverse quantity and quality for
the market and disposal of excess. material has also become an acute problem. The
concept of adsorption is essential in determining the movement pattern of pesticides in
soil in order to asses the effect of migrating chemical, from their disposal sites, on the
quality of ground water. In the study of movement of pesticides in the soil, the
mathematical models so far developed only consider axial movement. The contribution of
radial movement to the overall location of solutes in the porous media seems to have
been disregarded by researchers in this field. The objective of this study is to close this
gap by developing a mathematical model to determine the combine radial and axial
movement of pesticides due to Convective - Dispersive transport of pesticides with
steady - state water flow in a porous media.
The methodology will involve determining the comprehensive dispersion equation
accounting for both axial and radial movement of solutes in the porous media and finding
the solution of the governing equation using finite difference methods. The solution of
this equation will be applied to the data from experiments carried out on adsorption and
movement of selected pesticides at hi~h concentration by soil department, University of
Florida, Gainesville U.S.A. We will confme our study to single - Region Flow and
Transport.
2007-01-01T00:00:00ZNumerical solution of Korteweg-de vries equation
https://repository.maseno.ac.ke/handle/123456789/5250
Numerical solution of Korteweg-de vries equation
ONAM, Joel Otieno
The Kotteweg-de Vrr-es(KdV)is a mathematical model of waves on shallow
water surfaces. The mathematical theory behind the KdV equation
is rich and interesting, and, in the broad sense, is a topic of active mathematical
research. The equation is named after Diederik Korteweg and
Gustav de Vries,
It has long been known that conservative discretization schemes for
the KdV and other nonlinear equations tend to become numericrtlly unstable.
Although finite difference approximations have been used, there
are always instabilities of the solutions obtained,
In this work we solved the Korteweg-ds Vries (KdV) equation using an
explicit finite difference method, subject. to various boundery conditions
which are travelling wave solutions to the KdV equation. The methodology
involved carefully designing conservative finite difference discretization
that can remain stable and deliver sharp solution profiles fora long
time. We then determined the accuracy of the finite diffurence scheme by
comparing the graphical outputs of the numerical results.
2008-01-01T00:00:00ZOn a generalized q-numerical Range
https://repository.maseno.ac.ke/handle/123456789/5249
On a generalized q-numerical Range
Musundi, Sammy Wabomba
We 'consider numerical ranges of a bounded linear operator on complex
Hilbert spaces. Many properties of the classical numerical range are
known. We investigate the properties of the q-numerical range in relation
to those of the classical numerical range. We also establish the
relationship between the q-numerical range and the algebra q-numerical
range. Furthermore, we extend the results of the classical numerical range
and q-numerical range to the C-numerical range and investigate how the
C-numerical range is an explicit generalization of both the classical numerical
range and q-numerical range.
2008-01-01T00:00:00ZMathematical modelling of flood Wave: a case study of Budalang'i Flood plain basin in Busia county, Kenya
https://repository.maseno.ac.ke/handle/123456789/5230
Mathematical modelling of flood Wave: a case study of Budalang'i Flood plain basin in Busia county, Kenya
MUSINDAYI, Stephen Miheso
Flooding is a worldwide problem with more adverse e ects in developing countries.
In Kenya, severe
ooding is experienced on the lower tributaries of Lake Victoria,
mainly Budalang'i area. This is indicated in the historical
oods of 2003, 2007,
2017 and 2019, leading to mass displacement of people and property destruction.
This has attracted attention of researchers worldwide and application of di erent
measures to curb
ood in the study regions. Mathematical modeling of
ood wave
has however not been adopted in Budalang'i
ood plain. Therefore this study
formulated, analyzed and simulated the 2D
ood wave model with incorporation
of a sink to the Budalangi
ood plain. Formulation was applied on existing Navier
Stokes equations with the addition of a sink term on continuity equation. Analy-
sis of the shallow water model entailed transforming the equations using Jacobian
transformation and assessing the nature of
ow using Froude number. For simula-
tions of the 2D shallow water model, the study adopted a nite di erence scheme
to make approximations which solved the system of equations and displayed in the
gures . It is realized that in the formulation of the 2D shallow equations, appro-
priate model for Budalang'i
ood plain is easily derived from the 3D Navier Stokes
equations under
ood plain assumptions and addition of a sink term is necessary
for modelling in the
ood plain. Assessment of the properties reveals that super-
critical
ows are dominant. Addition of a sink term ensures steady state velocity
thus reducing higher frequency and turbulence as well as over bank
ows while
incorporating coriolis term has signi cant e ect on the turbulence. The study
concludes that addition of a sink term to the 2D shallow water model will enable
control of the
oods in the area of study. The ndings will aide disaster manage-
ment stakeholder to come up with a more reliable
ood prevention technique and
new knowledge on how source terms can help reduce
ood risk.
2022-01-01T00:00:00ZNorms of elementary operators.
https://repository.maseno.ac.ke/handle/123456789/5205
Norms of elementary operators.
RUTO, Peter Kiptoo
The norm of an elementary operator has been investigated over long period
by several mathematician under various special circumstances. Timoney
working on algebra of bounded linear operators on Hilbert spaces,
established the lower bound of norms of eleementary operators on Calkin
algebra.
Similary, mathieu studied norm properties of elementary operators on
Calkin algebra and established a result whose key basis is the Haagerup
tensor norm. We joined results from these eminent mathematicians to
establish norms of elementary operators, particularly determine the lower
bounds of elementary operators.
2010-01-01T00:00:00ZOn completely bounded Operators
https://repository.maseno.ac.ke/handle/123456789/5203
On completely bounded Operators
AMBOGO, David Otieno
Calculating norms of matrices when the entries are not constants is the
first problem tackled in this thesis. We have considered the space of matrices
with entries from the algebra of bounded linear operators and have
managed to approximate norm in this space. The basic idea has been to
identify this space with the space of bounded operators from H" (where
'H" is the orthogonal sum of n-copies of 7-{)to 'H'" and calculating the
norm of an operator on H", This forms the content of chapter two. The
notion of completely bounded operators is a fairly new and developing
area in Mathematics. It started its life in the early 1980's following Stinespring's
and Arveson's work on completely positive operators. It later
gave rise to operator spaces, a new branch in operator algebra. Progress
in this new area of Mathematics has been rapid and it is difficult to say
which results motivated others. We have investigated the norm of completely
bounded operators and have shown that they form an increasing
sequence. The idea was to apply Hilbert-Schmidt norm to the definition
of these operators. We have also given examples of these operators for
illustration, something which is missing in the available literature. We
have also investigated operator spaces, especially their algebraic tensor
product. Specific interest has been in the matricial tensor product.
2010-01-01T00:00:00Z