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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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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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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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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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