Level – 3M
Course units effective from academic year 2016/2017 to date
Course Code  MMT301M3  
Course Title  Advanced Algebra I  
Credit Value  03  
Hourly Breakdown 
Theory  Practical  Independen Independent Learning 
45  —  105  
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Contents:  
Ring homomorphisms, factorization theorems, isomorphism theorems , nil and nilpotent ideals, prime and maximal ideals, primary ideals, nilradical and Jacobson radicals, operations on ideals, integral domains, principal ideal domain(P.I.D), unique factorization domain(U.F.D), Euclidean domain (E.D), modules, module homomorphisms, submodules and quotient modules, module isomorphism theorems, direct sum and direct product of modules, finitely generated modules, exact sequences, rings and modules of fractions, localization, primary decomposition , chain conditions, Noetherian rings, Artin rings.  
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Course Code  MMT302M2  
Course Title  Topology I  
Credit Value  02  
Hourly Breakdown  Theory  Practical  Independen Independent Learning  
30  –  70  
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Course Code  MMT303M2  
Course Title  Functional Analysis I  
Credit Value  02  
Prerequisites  PMM201G3, PMM203G3  
Hourly Breakdown  Theory  Practical  IndependentLearning 
30  —–  70  
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Course Contents:  
Normed linear spaces and Banach spaces: Normed linear spaces, Equivalent norms, Riesz lemma, Completeness and Banach spaces, Convergence and absolutely convergence of series in Banach space , Schauder basis, Separable normed linear spaces.
Linear functionals and Dual spaces: Linear operators and linear functionals, Bounded linear functional, Isometry, Isomorphism, Dual spaces and its applications, Hahn Banach theorems Inner product spaces and Hilbert spaces: Inner products, Hilbert spaces, Closed subspaces and orthogonal projection, Riesz Representation theorem, Orthonormal system and othonormalization, Bessel’s inequality and Parseval identity, Existence of orthonormal basis in a separable Hilbert space 

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Course Code  MMT304M3  
Course Title  Numerical Linear Algebra  
Credit Value  03  
Hourly Breakdown 
Theory  Practical  IndependentLearning 
45  —  105  
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Course Contents:  
Direct Methods: Linear algebra Review, Elementary triangular matrices and Gauss elimination, Elementary permutation matrices and pivoting, Elementary Hermitian Matrices and matrix factorization, iterative refinement
Matrix Analysis: Canonical forms and positive definite matrices, Vector and Matrix norms, Spectral radius, Condition of problems and scaling Norm ReducingMethods: Iterative methods and error bounds, convergence results for special matrices, choice of relaxation parameter, sparce matrix technique, Conjugate gradient method Similarity Reduction methods:House holders’ method, Eigensystems of Hessenberg matrices and tridiagonal matrices, Jacobi method, Given’s method, LR method and QR method Power Methods: Direct power method, Raleigh quotients, Deflation process, Shift of the origin, Inverse iteration 

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Course Code  MMT305M3  
Course Title  Mathematical Modeling and Programming  
Credit Value  03  
Hourly Breakdown 
Theory  Practical  Independent Learning 
45  —  105  
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Course Contents:  
Fundamentals of modelling and word problems:Basic approach of Mathematical Modeling, its needs, types of models and limitations, setting up mathematical models for simple problems in words.
Qualitative solution sketching for first order differential equations: Direction field, solution sketch, convexity phase portrait, equilibrium solutions, stability. Population models for Single and interacting species: Basic concepts, Exponential growth model, formulation, solution, interpretation and limitations, Compensation and dispensation, Logistic growth model, formulation, solution, interpretation and limitations. Types of interaction between two species. LotkaVolterra preypredator model, formulation, solution, interpretation and limitations. LotkaVolterra model of two competing species, formulation, solution, interpretation and limitations. Dynamic programming models: Stage, State, Recursive equation, Developing optimal decision policy, Bellman’s principal of optimality, characteristics of deterministic dynamic programming, solving linear programming problem using dynamic programming approach. Nonlinear programming fundamentals: Fundamentals of optimization, types of problems, Optimality conditions for unconstrained and constrained problems, Lagrangian Multipliers method and KarushKuhnTucker conditions. Quadratic and Separable programmings: Quadratic programming, Wolfe’s algorithm, Wolfe’s modified simplex method, separable programming, separable function, piece wise linear approximation of separable nonlinear programming problem, separable programming algorithm. 

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Course Code  MMT306M3  
Course Title  Number Theory and Combinatorics  
Academic Credits  03  
Hourly Breakdown  Theory  Practical  Independent Learning 
45  —  105  
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Contents:  
Number Theory: Congruences, The Chinese remainder theorem, Wilson’s theorem, Fermat’s little theorem, Euler φ function, Euler’s theorem, Number theoretic functions, Möbius inversion formula, Primitive roots, Quadratic residues, Euler’s Criterion, Quadratic reciprocity, Quadratic Congruences, Legendre symbol, Perfect numbers, Mersenne primes and Amicable numbers, Fermat’s numbers, Certain nonlinear Diophantine equations
Basic Counting Techniques: Fundamental principles of counting:Sum, product, subtraction, and division rules. Permutations and combinations, the Binomial Theorem, Pascal’s identity and Triangle, Vandermonde’s Identity, permutations and combinations with limited and unlimited repetition, multinomial theorem and its applications, generating permutations and combinations,The Pigeonhole Principle and some of its interesting applications. Advanced Counting Techniques: InclusionExclusion: The principle of inclusionexclusion, an alternative form of inclusionexclusion, the Sieve of Eratosthenes, derangementsRecurrence Relations:Modeling with recurrence relations,solving linear homogeneous recurrence relations with constant coefficients, linear nonhomogeneous recurrence relations with constant coefficients, divideandconquer algorithms and recurrence relations.Generating Functions: Extended binomial theorem, counting problems and generating Functions, using generating functions to solve recurrence relations. Exponential generating functions and its applications on solving some permutation problems. 

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Course Code  MMT307M2  
Course Title  Topology II  
Credit Value  02  
Hourly Breakdown 
Theory 
Practical 
Independen Independent Learning  
30  —  70  
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Level – 4M
Course units effective from academic year 2016/2017 to date
Course Code  MMT401M4  
Course Title  Measure Theory  
Credit Value  04  
Hourly Breakdown 
Theory  Practical  Independent Learning 
60  —  140  
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Course Contents:  
Measure Spaces:Preliminaries:Algebra and σalgebras of sets,Borel sets; Lebesgue measure: Outer measure, Measurable sets, and Lebesgue measure,Properties,Example of a nonmeasurable set, Borel measures; General measure: Definition of measure, Measure space, Complete measure space, Examples, Properties.
Measurable Functions: Basic properties of measurable functions, Examples, Borel measurable functions, Approximation Theorem; Littlewoods’s three principles: Egoroff’s theorem. Integration:Integral of nonnegative functions, Integrability of a nonnegative function, Fatou’s Lemma, Monotone convergence theorem, Lebesgue Convergence Theorem, Generalized Convergence Theorem. Extension of Measure:Measure on an algebra, Extension of measures from algebras to σalgebras, Carathéodory’s theorem, and LebesgueStieltjes integral. Product Measure:Measurable rectangle, Semialgebra, Construction of product measures, Fubini’s theorem, and Tonelli’s theorem Differentiation and Integration:Differentiation of monotone functions: Vitali’s lemma, Functions of bounded variations; Differentiation of an integral: Indefinite integral, and Absolutely continuous functions. 

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Course Code  MMT402M3  
Course Title  Advanced Algebra II  
Credit Value  03  
Hourly Breakdown 
Theory  Practical  Independent Learning 
45  —  105  
Objectives:  
Provide an indepth knowledge in the theory of groups through a selection of advanced topics.  
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Course Contents:
Direct products, group acting on sets, pgroups, the Sylow theorems, finite abelian groups, symmetric groups, simplicity of A_{n}, free groups, presentations of groups, series of subgroups, composition series, chain conditions, Frattini subgroup, nilpotent groups, soluble groups, super soluble groups, residually nilpotent groups. 

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Course Code  MMT403M4  
Course Title  Functional Analysis II  
Credit Value  04  
Hourly Breakdown  Theory  Practical  Independent Learning 
60 hours  —–  140 hours  
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Course Contents:  
Fundamental theorems for Normed and Banach spaces: Uniform boundedness principle, Open mapping theorem, Bounded inverse theorem, Closed linear operators and Closed graph theorem, Reflexive normed linear spaces, Strong and weak convergence, Banach fixed point theorem
Compact linear operators and Adjoint operators on Normed spaces: HilbertAdjoint Operator , Selfadjoint, Unitary and Normal operators, Compact linear operators on normed Spaces and its properties Spectral theory of bounded self adjoint linear operators in Normed Spaces : Resolvent and spectrum of linear operator, Spectral theory in finite dimensional normed spaces, Spectral properties of bounded operators and selfadjoint operators , Spectral properties of Compact linear operators ,Positive operators, Spectral family of a bounded selfadjoint linear operator, Extension of the spectral theorem to continuous functions. 

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Course Code  MMT404M3  
Course Title  Advanced Complex Analysis  
Credit Value  03  
Hourly Breakdown  Theory  Practical  Independent Learning 
45hours  —–  105 hours  
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Course Contents:  
Holomorphic functions: Complex differentiation, Integration over paths, Local and global Cauchy theorem, Calculus of residues.
Harmonic functions: CauchyRiemann equations, Poisson integral, Mean value property, Boundary behavior of Poisson integrals ,Representation theorems Maximum modulus principle: Schwarz lemma, PhragmenLindelof method, An interpolation theorem, Converse of the maximum modulus theorem Conformal mapping :Preservation of angles, Linear fractional transformation, Normal families, Riemann mapping theorem, Continuity at the boundary, Conformal mapping of an annulus Zeros of Holomorphic functions: Infinite products, Weierstrass factorization theorem, An interpolation theorem, Jensen’s formula, Blaschke products, MuntzSzasz theorem Analytic continuation: Regular points and singular points, Schwarz reflection principle, Continuation along curves, Monodromy theorem, Construction of modular function, Picard theorem 

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Course Code  MMT405M4  
Course Title  Mathematical Physics  
Credit Value  04  
Hourly Breakdown 
Theory  Practical  Independent Learning 
60  —  140  
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Course Contents:  
Quantum mechanics: Wave mechanics, Schrödinger’s equation, probability interpretation, Simple quantummechanical systems, Poisson brackets, State vectors, statistical aspects, Heisenberg’s uncertainty principle.
Continuum mechanics: Strain: Lagrangian tensor, Eulerian tensor, small strain, shearing, volume deformation, strain vector. Stress: force, stress tensor, stress vector, conservation of mass, balance of linear momentum, balance of angular momentum, conservation of energy. Relativity: Galilean relativity, inertial frames, Galilean transforms, Newtonian relativity, Constancy of speed of light, simultaneity, spacetime diagrams, proper time, Lorentz transforms, lengthcontraction and timedilation, transformation of velocities. 4 vectors: displacement, velocity, acceleration, frequency, Conservation of 4 momentum, massenergy equivalence. Electromagnetism: Charge, Coulomb’s law, electric field and potential, Electric flux and density, Electro dynamics, Magnetostatics fields, Faradays law, Displacement current, Timevarying potentials, Derivation of Maxwell’s equations, Electromagnetic wave propagation. 

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Course Code  MMT406M3  
Course Title  Differential Equations  
Credit Value  03  
Hourly Breakdown 
Theory  Practical  Independent Learning 
45  —  105  
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Course Contents:  
Ordinary Differential Equations: Initial Value Problems, Boundary Value Problems, Equivalence and matrix notation, Grownwall’s inequality, Existence and Uniqueness of Solutions, Picard’s Theorem, Nonuniqueness of solutions, boundedness of solutions, Solutions to the second order systems
Linear System of Ordinary Differential Equations: First order Scalar equations, Linear ordinary differential equations with constant coefficients, Linear independence and the Wronskian, Inhomogeneous equations, Variation of parameters, Matrix notations, Linear ordinary differential equations with variable coefficient. Series solutions, Legendre equations, equations with regular singular points, Bessel equations, Boundary Value problems and Green’s function Stability Analysis of Ordinary Differential equations: Autonomous systems, linearization, Linear systems, Routh Hurwitz criterion, classification of equilibria, Phase plane analysis, PoincareBendixon theorem, Lyapunov stability First order Partial Differential Equations: First order quasi linear scalar equations, method of characteristics, systems of equations, concepts of order and dimension Second order Linear Partial Differential Equations: Classification of equations, canonical forms of hyperbolic, parabolic and elliptic equations, Boundary conditions, the Laplacian, wave equation and heat equation, Poisson’s equation, uniqueness and the maximum principal, integral transform solutions, application of Laplace transform, and Fourier transforms, Hankel Transforms. 

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Course Code  MMT407M3  
Course Title  Numerical Differential Equations  
Credit Value  03  
Hourly Breakdown 
Theory  Practical  Independent Learning 
45  —  105  
Objectives:  
· Understand the theoretical concepts of various numerical methods applied to solvedifferential equations
· Familiarize the underlying mathematical concepts of computer aided numerical algorithms · Acquaint with the stability theory concepts of the methods 

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Course Contents:  
Euler methods: Explicit and Implicit Euler methods, Convergence, local and global truncation errors, order of the method, error bounds
Linear multistep methods: General methods, order of the methods, error constant, convergence of the method, consistency of the method, stability of the method, Construction of a linear multistep method, PredictorCorrector methods RungeKutta methods: General Rungekutta methods, implicit and explicit methods, local and global truncation errors, order of the method, order conditions, derivation of the method Stability: A stability, B – stability, algebraic stability Boundary Value Problems: FiniteDifference methods, Shooting methods, truncation error estimation, Convergence of the method, Stability of the method 

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Course Code  MMT408M3  
Course Title  Category Theory  
Credit Value  03  
Hourly Breakdown 
Theory  Practical  Independent Learning 
45  —  105  
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Course Contents: 
Foundations: Basic definitions, sets, classes, conglomerates
Categories and Functors: Categories, examples, commutative diagrams, Definitions , duality Principle, isomorphisms, Fuctors, examples of Functors, Composition of functors, Homsets, Subcategory, Equivalences of categories. Natural transformations: Natural transfromation between two functors, Naturality condition, Examples of Natural transformations, Functor categories, Representable functor, Composition of natural transformations, The category of small categories, Yoneda Lemma Objects and Morphisms: Initial Object and Final Object, separator and coseparator, sections, retraction, monomorphismsm, epimorphisms, regular and extremalepimorphism, equalizer, coequalizer, Limits and Colimits : Products and Coproducts, Sinks and Sources, pullback and pushout, Limit and Colimit of functors, preservation of Limits, Lifting of Limits Applications : Algebras and Coalgebras:Category of algebras and coalgebras, Initial algebras and final coalgebras, applications to computer science. Optional Topics: Adjoints and Monads (according to the avaliblity of time) 
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Course Code  MMT409M6  
Course Title  Research Project  
Credit Value  06  
Hourly Breakdown 
Theory  Practical  Independent Learning 
300 Hours  
Objectives:  
Provide an opportunity for the students to participate ina research project in Mathematics to develop research skills.  
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Course Description:  
Students are expected to carry out an independent research project in the field of Mathematics under the supervision of a senior staff member in the department. Students need to give presentations in the beginning, middle, and the end of their research. At the completion of the research project students are expected to write a comprehensive report. During the research, students are expected to maintain a research diary.  
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