1
Manjak Nibron
Haggai
A study of entire
Transcendental
Function and
Picard’s Sets
2
Yusuf Haruna
On Criteria For
The Oscillation
And
Nonoscillation
Of Second Order
Linear
Differential
Equations
Oct,94 The theory of Picard can be traced back to the year 1876, when K. Weiestrass
showed that in the vicinity of an Isolated essential singularity (I.e.s) a
meromorphic function f(z) approaches every given value a arbitrary closely.
In 1879 E. Picard even proved that surprising fact that a meromorphic
function takes in the vicinity of an I. e.s every finite or infinite value G. with
two exceptions at most. Hence the search for Picards' sets lead to the
emergence of this project. The project is in three parts. Chapter one deals with
analytic functions, singularities and maximum modulus principle and goes on
to define the concept of order for entire transcendental function. The
fundamental theorem of Algebra is introduced as a lead to the picard theorem
since no elementary proof of the theorem exists, special cases of the theorem
were considered in order to preserve elementary proofs of these special cases.
Three of the many different proofs of the Picard theorems are presented in
chapter two.In the process the notion families of functions is dealt with.
Profund results like the theorems of Schotlky is presented and the
Nevanlinna theory is reviewed and displayed in our proofs. The chapter is
concluded with a review of a result of zakman who attempted to present an
elementary proof of the Picard theorems. The last chapter attempts to look at
Picard's sets for entire transcendental function. The Picard's set is introduced
as a generalisation of the Picard's theorem. A critical review of the literature
is made in an attempt to improve existing results which are not sharp and
new proof of existing results are established.
Oct,94 The project deals with criteria for the oscillation and non oscillation
of linear second-order differential equations, particularly those with
integrable coefficients. The entire work has been divided into four
chapters. In the first chapter we give background information on
how this research work started. Also considered in this chapter are
examples of oscillatory and nonoscillatory second-order linear
differential equations and the classical results of Sturm
(Comparison and Separation theorems). The chapter ends with
transformation of second order linear differential equations. In
chapter two we have a review of the literature.
Selected portions of the general theory on oscillation and
nonoscillation criteria established by authors like Kwong [1982],
Hille [1948], Wong [1983] and others are presented. Chapter three
starts with definitions of some important terms used in this report.
This is followed by a discussion of results of Kwong (1982J on
second order linear oscillation and certain Riccati integral
equations. Chapter four is devoted to a study of Wong's results.
Oscillation and nonoscillation criteria are singled out for a detailed
study of their application to his example of second- order linear
differential equations with integrable coefficients. This serves as a
motivation and a basis for proposing examples and testing the
applicability or nonapplicability of his results to these examples. By
replacing either sine by cosine or cosine by sine in his examples the
results on criteria for oscillation and nonoscillation remain valid. In
the case of
q( t)
Wong's criteria are not applicable.
Who says that pure mathematics is strictly academic? In theory of
numbers, an old theorem due to Mobius has unexpectedly proved
to be a way of solving physical problems of inversion that may
have important applications. In this project, we apply a Modified
Mobius Inversion formula to the calculations of phonon density
of states of Aluminium(AI), Copper (Cu) and Lead (Pb), The
results obtained agreed reasonably well with experiment. This
shows the possibility of the application of Mobius Inversion
formula to physical science problems. Other applications are also
highlighted in the project, especially in condensed matter physics.
3
Uba Ahmed
Ali
Mobius
Transforma And
Some Inverse
Problems In
Condensed
Matter Physics
Mar.,
94
4
Yakubu D.G
Shingu
Studies On
Fourth And Fifth
Order RungeKutta Methods
For Initial Value
Problems
Oct,94 The Studies review some well known fourth and fifth
order Runge-Kutta methods. An entirely new fifth order
Runge-Kutta formula is derived which gives a result
better than the well known Runge-KuttaFehlberg
methods (1964). In addition the newly derived fifth
order Runge-Kutta method does not require the use of
error control strategy. Modifications are also made to
the well known fourth order Runge-Kutta methods
resulting in some new formula which proves to be quite
accurate and provides estimate of truncation error
5
Mshelia I.
Bello
Conjugacy
Problems For
Hyperbolic
Toral
Automorphism
without any extra function evaluation. The idea follows
from the fact that two numerical solutions of the same
order can be obtained by using the Arithmetic mean
(A.M.) and the Geometric mean (G. M.) averaging of
the functional values. The numerical results are
compared with those obtained using the well known
fourth and fifth order Runge-Kutta methods. The results
obtained confirmed that the modified fourth order
methods are suitable for use as error control strategies.
The outline of the thesis are as follows:- Chapter one
contains the introduction. In chapter two the review of
the relevant related literature is presented. Chapter three
contains discussion of algorithms of the various
methods and their applications to a simple problem.
Analyses of the results are presented in chapter four and
in chapter five we present the summary.
Oct,94 In this Project we investigate some of the rich Mathematical
structures associated with hyperbolic toral automorphism. In
particular, explicit results are obtained concerning structural and
topological stability. Chapter one is an introductory survey, where we
describe many interesting facts about hyperbolic toral automorphism
with examples.
In Chapter two, a literature review on hyperbolic toral automorphism
is given and its relevance. Chapter three reviews the proofs that
hyperbolic toral automorphism is structurally stable and gives an
explicit size for the stability neighborhood of the automorphism. The
detailed account we give is for the two torus which leads to
perturbation result. The final chapter is concerned with the integral
similarity of hyperbolic linear maps. We show, using quadratic forms
that all hyperbolic toral automorphisms of trace 4 and determinant 1
are conjugate We pick a matrix [ 2 3 ]
1 2
Which is different from the common [ 2 1 ]
1
1
which occurs so
frequently in literature.
6
Ozovehe Mary
Oziohu
7
Djibo
HadizatouMag
agi
8
Ajie
Ikechukwu
The Stability Of
Dynamical
Systems On 2Manifolds
00324
On Convention
Heat Transfer
Oct,95 In this project, we study stability concepts of ordinary differential
equations also called vector fields or dynamical systems defined on
manifolds. The main result is the Poincare - bandixson theorem and
its analogue on 2- manifolds. A careful study of this theorem and its
analogue is followed by establishing analogues of two theorems on
the stability of periodic solutions
July
The subject of this project is based on the study of natural
,95
convection of free convection heat transfer in laminar flow.
Application Of
The
Traditionally, engineering curricula have included courses in
fluid mechanics. Thus a fundamental description of fluid
mechanics had been employed in the first chapter to increase
the reader's understanding on the flow to be analyzed. The
knowledge of differential equations is revealed in chapter
two for the development of some important equations, such
as the equation of continuity, momentum, energy and mass
transfer, together with some definitions of dimensionless
parameters. The third chapter carried out the basic
mechanism of energy transfer in each of the heat transfer
mode (Conduction, Radiation and Convection). Convection
mechanism is considered in depth with the study revealed in
different types of flows. Considerable attention had been
given to the study of heat and mass transfer by natural or free
convection. The next chapter pointed out the investigation
made by different authors on diverse area. In chapter five.
We have studied the oscillatory flow through porous medium
by .ne presence of free convection flow and oscillatory
temperature. The analytical solutions for the velocity and
temperature fields have been obtained. The velocity profiles
for different parameters have been shown on graphs followed
by fruitful discussion.
Oct,95 For many Practical applications, the resulting equations are
partial differential equations. Unfortunately, some of these
9
James
Decomposition
Methods For
The Solution Of
Deterministic
Partial
Differential
Equations
Muhammad
Manga Adamu
A Finite
Difference
Method For
Solving
Incompressible
Viscous Fluid
Flow Problems
Dec,
96
types of equations are difficult to analyse or to solve. Although
various approaches have been proposed only numerical
answers are possible and analytical solutions are usually
impossible to obtain. Since there is need to analyse the
behavior of the system, an analytical or even an approximate
representation of the analytical solution would be very
useful.In this project, we reviewed the decomposition method
due to Adomain which appears to offer such an analytical
approximate approach. The decomposition method supplies an
analytical approximate solution to the original partial
differential equation. We used it to sole linear, non linear,
initial or boundary- value problems.We decomposed the
original problem into an invertible part and the reminder. For
equations involving non linear terms, a special type of
polynomial- the An polynomials were used to decompose the
part.
We also reviewed the concept of asymptotic decomposition
and used it to solve initial/boundary-value problems.
The project consists of introduction to Fluid Mechanics as a
subject, the equations involved in Fluid Mechanics
(Integral/Differential Form). But, the differential form is
considered here in this work; which arises and gives NavierStokes equation. The work examines: The tracking of motions of
fluids past objects or through objects, in oceans or in molecules,
here on earth or in distant galaxies and the behaviour of liquids,
gases, and plasma - of everything that is not solid. The theory of
Fluid mechanics gives the formulation of literally dozens of fields
within science and engineering: for meteorology, oceanography,
astronomy, aerodynamics etc. A numerical method for solving
incompressible viscous fluid flow problem is introduced. This
method uses the Leaf-Frog and DuFort-Frankel's scheme, and its
principles of the method lies in the introduction of a steady
incompressible two dimensional boundary layer flow in presence
of transverse magnetic field.
10
Abdulhamid
BAla Ma’aji
The Stability Of
Periodic
Solutions Of
Non-Linear
Autonomous
Differential
Equations
11
Atureta
Mohammed
Salawu
The Steppind
Stone Algorithm
For Solving
Transportation
Problems A
Case Study Of
The Royal Brick
Industry, Jos
Nigeria
Sept
,97
The stability of periodic solutions of non-linear differential
equations was studied. The emphasis was on two
dimensional non-linear autonomous systems of
differential equation. Three examples were selected to
illustrate the stability of periodic solutions. The results applied
to the linear approximation or linear variational equations
were based on the Linearization Theorem and Floquet's
Theory. The first of these examples was constructed from one
found in a text. The constructed example turned out to possess
a limit cycle which was asymptotically orbitally stable. The
second example had more than one limit cycle, each
asymptotically orbitally stable from the exterior or positively
stable from the outside. The third example had a periodic
solution which was stable, but not asymptotically orbitally
stable.
Oct,97 This study is on the transportation problem of the Royal Brick
Industry, Jos, Nigeria. The main objective of this research
work is to provide solution that gives the minimum cost of
transportation, and the production schedule of the industry.
The stepping stone algorithm is used to provide solution to the
real life transportation problem. The problem considered in
this study is that of distributing finished product from three
factories to five warehouses at a minimum cost; and at
appropriate production schedule. The transportation problem is
first expressed as a linear programming problem. The vogel
approximation method is used to generate initial solution to the
problem. The stepping stone algorithm is then used to improve
the initial solution. After a number of iterations, an optimal
(final) solution was obtained. The optimal solution represents
the solution of the transportation problem. The numerical
solution is useful to the management of the industry, especially
for carrying out decisions regarding transportation or
distribution policies. Furthermore, the significance of the
12
Adamu
Muhammad
Sanda
13
Haladu
Yakubu Ado
14
Muktari
MuhammedBa
maina
Some Aspects
Of Fluid Flow
Problems
0001/40
4071
Survey of
Numerical
Errors in Finite
Element
Methods
Conditions For
Oscillation And
Nonoscillation
Of First Order
Linear Delay
Differential
Equations
solution is that it will enable management to facilitate plan for
product distribution during a specified time interval.
Oct,97 There are relatively few viscous flow problems for which
analytical solutions can be obtained in closed form; hence the
method of solution is important. A few classic examples of
incompressible laminar flows were considered. The interest
here is to obtain detailed information about the velocity field.
Knowledge of the velocity field permits calculations of many
other parameters. The complete differential equations of
motion were used in this work. The approach used to solve
such problems is to take the geometry in such a manner that
the non-linear convective term disappears and thus explicit
solution is possible. Flows between parallel plates were
considered where the velocity field was calculated and thus the
calculation of load was possible.
04
The project report consists of a survey of types of numerical
errors, arising from the applications of finite element methods
for solving differential equations. Some of the errors reviewed
include interpolation error, optimal and quasi- optimal errors
and discretization error. Analyses are also made of their nature
and magnitudes. To undertake these, a review of various finite
element methods is made in the text.
June,
This project studies oscillation and non oscillation conditions
04
for first order linear delay differential equations of the form.
n
y’(t) + ∑ p1 y(t-T1)= 0, where p1>0 and Ti > 0
t=1
One method often used in establishing tile conditions for
oscillation and non oscillation for this type or equations
includes the use of characteristic associated with them. In this
project we present as the main result a necessary and
sufficient condition for the oscillation and non oscillation of
all solutions of the above equation. for n = 2 ,t2 = 3t1 as
15
Magaji
YunbungaAda
mu
PGS/
0001/40
4048
Some A-Stable
Block Hybrid
Methods For
Initial Value
Problem In
Ordinary
Differential
Equations
16
Yusuf Ibrahim
Gwanda
0001/40
4045
On the Stability
of Linear and
Almost Linear
Differential
Equations
>0
Examples of oscillatory and non oscillatory equations were
constructed and the condition obtained applied on them. Other
existing conditions were also applied on the same examples
and they seem to confirm the validity of the condition that has
been obtained.
Dec,
The hybrid methods as they tend to possess certain
05
characteristics of both Runge-Kutta methods and linear
multistep methods are used to develop block hybrid methods
for solving stiff and non-stiff differential systems, The
resulting methods with continuous coefficients are evaluated at
some off grid points to obtain A-stable discrete schemes which
are well known for their adequate accuracies and good stability
properties. Numerical experiments were compared with
analytic solution to determine the degree of accuracy of the
newly derived methods.
Jun,06 The criteria for the stability of certain class of linear and
almost linear differential equations were Investigated.
Conditions under which the perturbed systems behave more or
less like the linearized ones are investigated. In achieving this,
the method of linear approximation was employed. A
procedure by which the liapunov method could be used to
linearly approximate the stabilities of a given differential
17
Tahir Alhaji
0102/40
4093
Criterion For
Oscillation And
Non Oscillation
Of First
Orderations
equation even at the critical point was devised.
In this research, the criteria for the stability of an almost linear
two dimensional system of the form X'1= g1 (X1 ,X2 ), X'2 = g2
(X1 ,X2 ) Where g1 (0,0)= g2 (0,0) and g1 and g2 are nonlinear
in terms of X1 and X2 and are continuously differentiable in
the vicinity of the origin, were studied. We proved, using
relevant theorems that, if the eigenvalues of A in the linear part
of X’1 = a11 X1 + a12 X2 + R1 (X1 ,X2 ) X'2 = a22 X1 + a22 X2 +
R2 (X1 ,X2 ) are different from zero and have non zero real
part, then the stability properties of the above system are the
same as those of the linearized counter part
X’1=AX.
Examples illustrating the relevance of various theorems
employed are discussed.
Oct,06 This work is entitled a Criterion for Oscillation and Non
oscillation of first order linear delay differential equations.
The main objective of this work is to establish the necessary
and sufficient conditions for the oscillation of all the solutions
of the equation
t
r
x’ (t ) + ∑ P1 ( t ) x ( t – Ti ( t ) ) - ∑qj ( t ) x ( t - ○j ( t)) = 0
1=1
j=1
For i=1, 2,…,l, j=1,2,…,r
The methods used in establishing those conditions are quite
justifiable and being applied in so many physical problems.
Examples are later constructed to test the applicability or
otherwise of the Conditions being established.
Finally, suggestion is made for further research.
18
Badmus
Ademola
Mudashiru
0102/40
40127
A Reformation
Of Some PStable Schemes
In Continous
Form For
Sept,
07
In this thesis, we re-formulate the Ritchmyer and Morton P-Stable
scheme in its continuous form. Also the derivation of a self-starting
hybrid block scheme for solution of yn = f(x, y) in both continuous
and discrete form. The order, error constants and the region of
absolute stability of the schemes are discussed. The implementation
19
Markus
Samaila
0405/40
40168
20
Anthony Peter
0405/40
40163
21
Agom Euman
Ugbeshe
0102/40
Second Order
Initial Value
Problems In
Ordinary
Differential
Equation
On Some
Symmetric
Uniformly
Accurate GaussRungekutta
Methods For
First Order
Initial Value
Problems
strategies of the schemes with numerical results are included to
illustrate the efficiency and accuracy of the proposed methods
Aug
,07
Symmetric methods are particularly attractive for solving stiff
ordinary differential equations. In project report, by the
selection of Gauss-points for both interpolation and
collocation, we high order symmetric single-step GaussRunge-Kutta collocation methods for accurate solution of
ordinary differential equations. The resulting symmetric
methods with continuous coefficient are evaluated for the
proposed block methods for accurate solution of ordinary
differential equations. More interestingly, the block methods
are self-starting with adequate absolute stability intervals
which are capable of producing simultaneously dense
approximation to the solution of ordinary differential equations
at a block of points. The use of these methods leads to a
maximal gain in efficiency as well as in minimal function
evaluations per step.
On The
Construction Of
Bol loops Of
Order 15
Sept,
07
In this project, a method combining both manual
multiplication of permutations and a computer programming
language (GAP) was used to construct Bol loop of order 3p,p
being an odd prime. A specific case for order 15 was
constructed by right regular representations. Two Bol loops of
order 15 were found tip to isomorphism; one satisfying the
automorphic inverse property, the other docs not. From the
Cayley tables constructed for order 15, the centre, centrum,
conjugacy classes, nucleus among other properties were given
alongside our results.
Mathematical
Modelling Of
Aug,
07
In this study, a mathematical model to predict the total amount of a
Therapeutic Agent (TA) injected intravenously into human tissues
4094
Tehe Diffusion
Of Therapeutic
Agents In
Human Tissues
22
Abdulhameed
Mohammed
Gazali
0506/40
40367
A class of
General Linear
Methods with
Almost Rungekutta Stability
for stiff and non
Stiff initial
Value problems
July,
08
23
Amina Hamza
0405,40
40167
Some Uniform
Order Five
Rungekutta
Collocation
Methods For
Initial Value
July,
08
was formulated. The total concentration of it in two compartments of
the Human biological system was modeled. Data from texts were
used to justify the model and the results were found to be valid as it
provides potential evidence in estimating the total amount of a TA In
human tissues qualitatively as a function of time. The model may be
useful to Health workers for estimating the concentration of a TA in
human in accordance with physiological and pharmacological
realities. The model IS, however, a theoretical prediction which is
only approximate
General Linear methods were originally introduced as a platform for
the unified study of consistency, stability and convergence for a
wide variety of class of numerical methods for initial value ordinary
differential equation problems. We have found that a multistep
collocation formula exists that is based on the general matrix inverse
from earlier works. This provides continuous inter polant for dense
output and has a unifying scope for many families of traditional
methods in continuous or discrete forms. This extremely broad class
of methods, besides containing Runge-Kutta and Linear multistep
methods as special cases, also contains hybrid methods and pseudo
Runge-Kutta methods, two step Runge-Kutta methods, Almost
Runge-Kutta methods and so. We here derive a special class of these
very useful and interesting methods that are generating interest in
the scientific world. We first obtained the direct block collocation
methods associated with the continuous scheme, then converted the
block collocation methods which are not A-stable to A-stable
uniformly high order accurate general linear methods that are
acceptable for solving stiff and non-stiff initial value ordinary
differential equation problems. For illustrative purposes, some
standard test examples of stiff and non-stiff initial value problems
were solved.
By the selection of equal number of interpolation and
collocation points inside the step (Xn, Xn+1)we derive RungeKutta collocation methods for a variety of problems in
ordinary differential equations. The main interest is to
upgrade the internal off-step points to have uniformly
accurate orders throughout the interval of integration. The
For future research,
derivation
and
analysis of one-step
Gauss-Radau-RungeKutta
collocation
methods of orders
Problems
24
Watifa Daniel
Mshelia
0102/40
40108
25
Achi Ismailu
0506/40
Collocation
Methods For
Third Order
Ordinary
Differential
Equations
Stability in Prey
Predator systems
resulting methods with continuous coefficients are evaluated
for the proposed block hybrid methods. The hybrid methods if
desired can be applied simultaneously as block methods for
moving the integration process forward at a time. The block
methods based on the hybrid formulation can also be
converted to Runge-Kutta methods which are superior in
terms of accuracy and stability interval than the well known
conventional method of the same order in current use. Some
test problems are used to illustrate the new integrators using a
fixed step size h.
July
,08
Linear multi-step methods for the direct solution of special third
order ordinary differential equations are considered. The approach
for the development of these methods is based on collocation of the
differential system generated from the basis functions. The order,
error constant and the region of absolute stability of the scheme with
the numerical experiments are fully discussed
Aug,
08
The stability of positive equilibrium of solutions in preypredator systems of nonlinear autonomous equations was
2m, m = 3,4,5, ... are
suitable area') to look
at.
Showing that the
methods derived in
(a) above are all
linked
with
continuous
Gauss
Radau-Runge-Kutta
methods
including
super-convergence
for
multi-step
methods, is worth
investigating.
Finally, apply the
derived methods to
real life situations
like Heat transfer
problems,
wave
equations and a host
of others, is also
suggested (in general
second order initial
value
ordinary
differential equation
problems).
40365
of Non linear
Autonomous
Equations
26
Abimbola
Nurudeen
Adeshina
0405/40
40176
A Comparative
Study Of The
Branch Bound
And Gomory
Cutting Plane
Methods Of
Solving Integer
Programming
Problems
Nov,
09
27
Muhammed
Idris Daya
9798/40
4010
Stability Of
Delay
Differential
Equations By
Lyapunov
Method
Sept,
11
studied. The emphasis was on two-dimensional nonlinear
autonomous systems of ordinary differential equations of preypredator system with Ivlev's functional response. Examples
were presented to illustrate the asymptotic stability using the
theorem of stability criteria for plane autonomous systems and
a necessary and sufficient condition for a unique stable limit
cycle. The examples presented turned out to be asymptotically
stable at the positive equilibrium point. The conditions for both
local and global stability of a prey-predator system with time
delays were investigated and examples were also presented.
Also studied is Lotka- Volterra equations in which the
example considered turned out to be asymptotically stable by
using method of linearization
In this work, a comparative study of Branch - Bound method
and Gomory Cutting plane method or solving integer
programming problems is considered. The research is aimed at
Finding the comparative advantages of one method over the
other. This was measured in terms of number of iterations of
each method to reach an optimum solution and the efficiency
of each method for various forms or integer problems. A
program produced by Tara corporation, India to solve branch bound and a program was also written in Matlab language to
solve gomory cutting plane were used to solve the problems.
Results were compared using the two methods. in this
research, it was discovered that Gomory cutting plane method
is more efficient when solving problems that involves slack
variables while Branch - Bound method is more efficient when
solving problems that involves surplus variables.
This thesis studied the stability of one-dimensional delay
differential equation of the form x(t) = a(t)f(x(t-r(t))) , where
a: [0,(0) --+ [0, ∞)→[0, ∞] f: R → R, and r: [0, ∞ ]→ [0, q)
for some q>0, xf(x) >0 for x = 0 and a(t), f(x) and r(t) are
continuous.
For future research,
construction
of
Lyapunov function
different from that
of Y oneyama (1986)
By utilizing the Lyapunov method, examples were given to to prove theorem 4.1
illustrate the stability conditions obtained by Yoneyama is a suitable area to
(1986).
look upon. Also
choosing values for
a(t) and r(t) different
from the ones in this
research work to
establish
the
conditions
for
stability given by
Yoneyama(1986) is
also an area to look
upon.
Finally, by applying
the Lyapunov second
method to real life
situation such as
dynamical
models
and host of others is
an area for further
investigation.
28
Mohammed
Ardo
Mohammed
0102/40
4090
On Construction
Of P- Groups
And Their
Representations
By Character
Tables
Sept,
11
In this work, we applied some group concepts to construct
some p-groups as well as to display their nature, which
represents them by character tables. It was observed that if g
and g-1 € G belong to different conjugacy classes say Ci and Cj
then the all the entries in the character table for Cj are complex
conjugates of the corresponding entries for Cj. Also, if g and
g-1 € G belongs to the same conjugacy class, say C, then the
entries in the character table for Cj are real valued. Similarly,
if g and g-1 € G belong to distinct conjugacy class say Ci and
Cj then the group has at least one pair' complex conjugate of
the irreducible representation. Moreover, if there is an element
For further research,
the character tables of
p-group of order pn for
all P, where p>3 are
suitable
area
for
research.
Character table of
Sylow p-subgroups is
worth investigation.
g € G for which g-1 does not belong to any conjugacy class,
then the entries in the character table are real valued. Also,
groups of order 2n for 3<n<6 are non abelian and simple while
group of orders 2n for n<2 are abelian and simple. The Groups
Algorithms - Programming (GAP) version 4.4.12 was applied
to assist us toward the validation of our results.
29
Aliyu Danladi
Hina
0607/40
40381
The Effect Of
Field Extension
On The Group
Structure Of
Elliptic Curves
Sept,,
12
An elliptic curve E defined over a finite field K, E(K) is the set
of solutions to the general Weierstrass polynomial E: y2 +
a1xy+ a3y = x3+ a2x2+ a4x: + a6 where the coefficients a1, a2,
a3, a4, a € K. there exist a well defined addition ofpoints on
each curve such that the points form an abelian group under
the addition operation. This group is either cyclic or
isomorphic to the product of two cyclic groups. These set of
solutions that form the group lie in the closure of the field K
over which the curve is defined. If we allow the set to lie only
in a particular extension of K, the addition operation is well
defined there too. Therefore we can associate a group to every
extension K' of the field K denoted by E(K'). Will the
structure of the group defined over the base field K, be
affected if the same group is made to lie in the extension K' of
K?
30
Tsok Samuel
Hwere
0607/40
40380
Application of
Group theory to
Games
Mar,
12
Some popular mathematical games have a group - theoretic
foundation which is unknown to many and in some cases,
group theory helps provide strategies to win these games. In
fact Group theory can be applied to many games, very
effectively. In this work we will discuss bow the different
facets of the definition relate with the different ways in which
groups turn in games and we finally present the physical
application of real life situation of a particular game discussed.
More research can be
conducted
by
students of group
theory to discover
more
fields
that
group
theory
is
applicable, this will
confirm the fact that
group theory is an
applied course, rather
than
viewing
it
otherwise.
31
Mohammed
Kabir Adamu
Rakiya
0607/40
40388
Analytic
Mar,
Soplutions
Of 13
Canonical
Equations Using
Adomian
Decomposition
Method
Mathematical modelling of many frontier physical systems
leads to systems of ordinary or partial differential equations
linear or non linear. Unfortunately some of these types of
equations require a tedious analysis of solution. Several
numerical methods have been proposed to solve these
equations but only numerical solutions which are approximate
solutions have been obtained. Analytic solutions are usually
very difficult to obtain. Since there is a need to analyze the
behaviour of the systems, numerical solutions with high
accuracy and analytic solutions to such models are of
fundamental importance. In this project work, we reviewed the
Adomian decomposition method which appears to offer such
analytic approach. We considered the use of the method for the
solutions of Canonical equations linear or nonlinear partial
differential equations. The solutions are calculated in the form
of series with easily computable components. The validity of
the approach is verified by comparing the results obtained with
the exact solutions. In most cases, the series solutions
converge rapidly to the exact solutions with only few terms.