{"id":517,"date":"2020-09-29T15:12:07","date_gmt":"2020-09-29T09:42:07","guid":{"rendered":"http:\/\/maths.jfn.ac.lk\/?page_id=517"},"modified":"2021-07-20T00:01:11","modified_gmt":"2021-07-19T18:31:11","slug":"special-degree-mathematics","status":"publish","type":"page","link":"https:\/\/maths.jfn.ac.lk\/index.php\/special-degree-mathematics\/","title":{"rendered":"Special Degree – Mathematics"},"content":{"rendered":"

Level \u2013 3M<\/h3>\n

Course units effective from academic year 2016\/2017 to date<\/h4>\n
\n
<\/span>MMT301M3: Advanced Algebra I <\/div>
\n
\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Course Code<\/strong><\/td>\nMMT301M3<\/strong><\/td>\n<\/tr>\n
Course Title<\/strong><\/td>\nAdvanced Algebra I<\/strong><\/td>\n<\/tr>\n
Credit Value<\/strong><\/td>\n03<\/strong><\/td>\n<\/tr>\n
\u00a0<\/strong><\/p>\n

Hourly Breakdown<\/strong><\/td>\n

Theory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependen Independent Learning<\/strong><\/td>\n<\/tr>\n
45<\/strong><\/td>\n—<\/strong><\/td>\n105<\/strong><\/td>\n<\/tr>\n
Objectives:<\/strong><\/td>\n<\/tr>\n
\n
    \n
  • Introduce the elements of commutative algebra through a study of commutative rings and modules over such rings<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
Intended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
\n
    \n
  • Prove factorization of homomorphisms, Correspondence theorem, the classical isomorphism theorems and their consequences<\/li>\n
  • Define certain special types of ideals<\/li>\n
  • Discuss various aspects of certain special types of ideals, by means of proofs and examples<\/li>\n
  • Prove module isomorphism theorems and other results related to submodules, quotient modules, direct sum and direct product of modules<\/li>\n
  • Prove certain results related to finitely generated modules, notably Nakayama\u2019s lemma.<\/li>\n
  • Recall various results regarding exact sequences<\/li>\n
  • Define Unique Factorization Domains (U.F.D), Principal Ideal Domains(P.I.D) and Euclidean Domains (E.D)<\/li>\n
  • \u00a0Discuss various results regarding U.F.D, P.I.D and E.D, by\u00a0 means of proofs and examples<\/li>\n
  • Explain the processes of constructing the rings and modules of fractions<\/li>\n
  • \u00a0Prove certain results related to primary decomposition, notably 1st <\/sup>and 2nd<\/sup> uniqueness theorems<\/li>\n
  • \u00b7Prove certain theorems regarding Noetherian and Artinian modules<\/li>\n
  • \u00a0Discuss various results regarding Noetherian and Artin rings, in particular the Hilbert basis theorem<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
Contents:<\/strong><\/td>\n<\/tr>\n
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.<\/td>\n<\/tr>\n
\u00a0Teaching Methods:<\/strong><\/td>\n<\/tr>\n
\n
    \n
  • Lectures,\u00a0 Tutorials, \u00a0Problem solving, Use of e-resources and Handouts<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
Assessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
\n
    \n
  • In-course Assessments 30%<\/li>\n
  • End-of-course Examination 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
Recommended Readings:<\/strong><\/td>\n<\/tr>\n
\n
    \n
  • Atiyah, M.G. and MacDonald, G., AnIntroduction to Commutative Algebra, Addison-Wesley,1969<\/li>\n
  • Dummit, D. S. and Foote, R. M., Abstract Algebra, Wiley, 2006<\/li>\n
  • Sharp, R. Y., Steps in commutative Algebra, Cambridge University Press, 2001<\/li>\n
  • Rotman, J.J., Advanced Modern Algebra, JAMS, 2010.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
    <\/span>MMT302M2: Topology I<\/div>
    \n
    \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
    Course Code<\/strong><\/td>\nMMT302M2<\/strong><\/td>\n<\/td>\n<\/tr>\n
    Course Title<\/strong><\/td>\nTopology I<\/strong><\/td>\n<\/td>\n<\/tr>\n
    Credit Value<\/strong><\/td>\n02<\/strong><\/td>\n<\/td>\n<\/tr>\n
    Hourly Breakdown<\/strong><\/td>\n\u00a0<\/strong>Theory<\/strong><\/td>\n\u00a0<\/strong>Practical<\/strong><\/td>\nIndependen\u00a0\u00a0 Independent Learning<\/strong><\/td>\n<\/td>\n<\/tr>\n
    30<\/strong><\/td>\n–<\/strong><\/td>\n70<\/strong><\/td>\n<\/td>\n<\/tr>\n
    <\/td>\nObjectives:<\/strong><\/td>\n<\/tr>\n
    <\/td>\n\n
      \n
    • Provide an introduction to the general topology through a selection of topics<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    <\/td>\nIntended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
    <\/td>\n\n
      \n
    • Define certain notions associated with the topological spaces<\/li>\n
    • Recall the proofs of certain facts related to bases, subbases , subspaces and quotient spaces<\/li>\n
    • Prove certain results concerned with continuous functions and homeomorphisms<\/li>\n
    • Compare the product and box topologies<\/li>\n
    • Prove various results regarding convergence in the topological spaces<\/li>\n
    • Prove the Urysohn\u2019s lemma, Urysohn\u2019smetrization theorem, and the Tietze extension theorem<\/li>\n
    • Reproduce certain fundamental results regarding the compact spaces<\/li>\n
    • Discuss various aspects of connected, path connected and locally connected spaces<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    <\/td>\nCourse Contents:<\/strong><\/td>\n<\/tr>\n
    <\/td>\n\n
      \n
    • Topological Spaces and Continuous functions<\/strong>: Definition of topology, bases and subbases, closed sets and limit points, subspaces, continuous functions,\u00a0 homeomorphisms, the product topology, the weak topology, quotient spaces,<\/li>\n
    • Convergence:<\/strong> Sequences, nets, ultranets filters, ultra filters.<\/li>\n
    • Compactness and Connectedness<\/strong>: Compact spaces, finite intersection property, limit point compactness, compactness in the real line, connected spaces, connectedness in the real line, path connectedness, components and local connectedness<\/li>\n
    • The Separation Axioms and Countability Axioms<\/strong>: space, space,\u00a0 Hausdorff space,<\/strong> regular space, completely regular space, Tychnoff space, normal space, perfectly normal space, completely normal space,\u00a0 the countability axioms, Urysohn\u2019s Lemma,Urysohn\u2019smetrizationtheorem, the Tietze Extension theorem.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    <\/td>\n\u00a0Teaching Methods:<\/strong><\/td>\n<\/tr>\n
    <\/td>\n\n
      \n
    • \u00a0Lectures, Tutorials, Problem solving, Use of e-resources and Handouts.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    <\/td>\nAssessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
    <\/td>\n\n
      \n
    • In-course Assessments 30%<\/li>\n
    • End-of-course Examination 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
    <\/td>\nRecommended Readings:<\/strong><\/td>\n<\/tr>\n
    <\/td>\n\n
      \n
    • James R. Munkres, Topology, 2nd <\/sup>Edition, Prentice-Hall, 2000.<\/li>\n
    • James Dugundji, Topology, University Book stall, 1972.<\/li>\n
    • Stephen Willard, General Topology, Dove publications, 2004.<\/li>\n
    • C.Wayne Patty, Foundations of Topology, PWS-Kent publishing Company, 2009.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
      <\/span>MMT303M2: Functional Analysis I <\/div>
      \n
      \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
      Course Code<\/strong><\/td>\nMMT303M2<\/strong><\/td>\n<\/tr>\n
      Course Title<\/strong><\/td>\nFunctional Analysis I <\/strong><\/td>\n<\/tr>\n
      Credit Value<\/strong><\/td>\n02<\/strong><\/td>\n<\/tr>\n
      Prerequisites <\/strong><\/td>\nPMM201G3, PMM203G3<\/strong><\/td>\n<\/tr>\n
      Hourly Breakdown<\/strong><\/td>\nTheory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependentLearning<\/strong><\/td>\n<\/tr>\n
      30 <\/strong><\/td>\n—–<\/strong><\/td>\n70 <\/strong><\/td>\n<\/tr>\n
      Objectives:<\/strong><\/td>\n<\/tr>\n
      \n
        \n
      • Introduce the fundamental concepts of normed linear space and Banach space<\/li>\n
      • Provide a clear notion of concepts of linear functional and dual space<\/li>\n
      • Understand the concepts of inner product space and Hilbert space<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
      Intended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
      \n
        \n
      • Explain the fundamental concepts of normed linear space and Banach space<\/li>\n
      • Discuss equivalence of norms and its properties<\/li>\n
      • Examine the convergence and absolute convergence of a series in Banach space<\/li>\n
      • Recall the concepts of a bounded linear functional and its properties<\/li>\n
      • Analyze the properties and applications of dual spaces<\/li>\n
      • Prove Hahn Banach theorems in real and complex normed linear spaces<\/li>\n
      • Define inner product space and Hilbert space<\/li>\n
      • Prove Riesz\u00a0 representation theorem, Bessel\u2019s inequality and Parseval identity<\/li>\n
      • Make use of separability of a Hilbert space for the existence of orthonormal basis<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
      Course Contents:<\/strong><\/td>\n<\/tr>\n
      Normed linear spaces and Banach spaces: <\/strong>Normed linear spaces, Equivalent norms, Riesz lemma, Completeness and Banach spaces, Convergence and absolutely convergence of series in Banach space , Schauder basis, Separable\u00a0 normed linear spaces.<\/p>\n

      Linear functionals and Dual spaces: <\/strong>Linear operators and linear functionals, Bounded linear functional, Isometry, Isomorphism, Dual spaces and its applications, Hahn- Banach theorems<\/p>\n

      Inner product spaces and Hilbert spaces: <\/strong>Inner products, Hilbert spaces,\u00a0 Closed subspaces and orthogonal projection, Riesz Representation theorem, Orthonormal system and othonormalization, Bessel\u2019s inequality and Parseval identity, Existence\u00a0\u00a0 of orthonormal basis in a separable Hilbert space<\/td>\n<\/tr>\n

      Teaching Methods:<\/strong><\/td>\n<\/tr>\n
      \n
        \n
      • Lectures,\u00a0 Tutorial discussion, \u00a0use of e-resources and Handouts<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
      Assessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
      \n
        \n
      • In-course assessment\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 30%<\/li>\n
      • End of course Examination\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a070%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
      Recommended Readings:<\/strong><\/td>\n<\/tr>\n
      \n
        \n
      • John B. Conway, A Course in Functional Analysis, Springer, 1997<\/li>\n
      • Bryan P. Rynne , Martin A. Youngson, Linear Functional Analysis, Springer, 2008<\/li>\n
      • Walter Rudin, Functional Analysis, McGraw \u2013 Hill, 1991<\/li>\n
      • Erwin Kreyszig, Introductory Functional Analysis with Applications, Wiley1989<\/li>\n
      • FrigyesRiesz and BelaSz.-Nagy, Functional Analysis, Dover Publications, 1990<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
        <\/span>MMT304M3: Numerical Linear Algebra <\/div>
        \n
        \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
        Course Code<\/strong><\/td>\nMMT304M3<\/strong><\/td>\n<\/tr>\n
        Course Title<\/strong><\/td>\nNumerical Linear Algebra <\/strong><\/td>\n<\/tr>\n
        Credit Value<\/strong><\/td>\n03<\/strong><\/td>\n<\/tr>\n
        \u00a0<\/strong><\/p>\n

        Hourly Breakdown<\/strong><\/td>\n

        		Theory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependentLearning<\/strong><\/td>\n<\/tr>\n
        45<\/strong><\/td>\n—<\/strong><\/td>\n105<\/strong><\/td>\n<\/tr>\n
        Objectives:<\/strong><\/td>\n<\/tr>\n
        \n
          \n
        • Understand the numerical methods for solving large systems of linear equations<\/li>\n
        • Recognize the underlying mathematical concepts of computer aided numerical algorithms<\/li>\n
        • Understand the iterative methods for computing eigenvalues of large matrices<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
        Intended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
        \n
          \n
        • Outline the fundamental concepts in numerical linear algebra<\/li>\n
        • Apply the matrix factorization algorithms to solve system of linear equations<\/li>\n
        • Determine bounds for relative error in the solution of a system of linear equations<\/li>\n
        • Examine the convergence of iterative methods for solving system of linear equations<\/li>\n
        • Apply \u00a0iterative methods to solve a system of linear equations<\/li>\n
        • Examine the convergence of iterative methods for computing the eigenvalues of matrix<\/li>\n
        • Apply \u00a0iterative methods to compute eigenvalues of a matrix<\/li>\n
        • Apply the Grahm-Schmidt orthogonalization process to a matrix<\/li>\n
        • Solve the linear systems by using readily available software<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
        Course Contents: <\/strong><\/td>\n<\/tr>\n
        Direct Methods:<\/strong> Linear algebra Review, Elementary triangular matrices and Gauss elimination, Elementary permutation matrices and pivoting, Elementary Hermitian Matrices and matrix factorization, iterative refinement<\/p>\n

        Matrix Analysis:<\/strong> Canonical forms and positive definite matrices, Vector and Matrix norms, Spectral radius, Condition of problems and scaling<\/p>\n

        Norm ReducingMethods:<\/strong> Iterative methods and error bounds, convergence results for special matrices, choice of relaxation parameter, sparce matrix technique, Conjugate gradient method<\/p>\n

        Similarity Reduction methods:<\/strong>House holders\u2019 method, Eigensystems of Hessenberg matrices and tridiagonal matrices, Jacobi method, Given\u2019s method, LR method and QR method<\/p>\n

        Power Methods:<\/strong>\u00a0 Direct power method, Raleigh quotients, Deflation process, Shift of the origin, Inverse iteration<\/td>\n<\/tr>\n

        Teaching Methods:<\/strong><\/td>\n<\/tr>\n
        \n
          \n
        • Lectures,\u00a0 Tutorials, \u00a0Problem solving, Use of e-resources and Handouts<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
        Assessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
        \n
          \n
        • In-course Assessments\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 30%<\/li>\n
        • End-of-course Examination 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
        Recommended Readings:<\/strong><\/td>\n<\/tr>\n
        \n
          \n
        • Trefethen, N. andBau, D., Numerical Linear Algebra, SIAM, 1997.<\/li>\n
        • Golub, G. and Charles, V. L., Matrix Computations, John Hopkins University Press, 1996.<\/li>\n
        • Beilina, L., Karchevskii, E. and Karchevskii, M., Numerical Linear Algebra: Theory and Applications, Springer, 2017.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
          <\/span>MMT305M3: Mathematical Modeling and Programming <\/div>
          \n
          \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
          Course Code<\/strong><\/td>\nMMT305M3<\/strong><\/td>\n<\/tr>\n
          Course Title<\/strong><\/td>\nMathematical Modeling and Programming <\/strong><\/td>\n<\/tr>\n
          Credit Value<\/strong><\/td>\n03 <\/strong><\/td>\n<\/tr>\n
          \u00a0<\/strong><\/p>\n

          Hourly Breakdown<\/strong><\/td>\n

          		Theory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependent Learning<\/strong><\/td>\n<\/tr>\n
          45<\/strong><\/td>\n—<\/strong><\/td>\n105<\/strong><\/td>\n<\/tr>\n
          Objectives:<\/strong><\/td>\n<\/tr>\n
          \n
            \n
          • Provide knowledge and skills to build mathematical models of real-world problems, analyze them and make predictions about behavior of problems taken from physics, biology, chemistry, economics and other fields.<\/li>\n
          • Enable to investigate and apply standard mathematical programming problems.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
          Intended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
          \n
            \n
          • Formulate \u00a0mathematical models for solving given word problems<\/li>\n
          • Sketch the qualitative solution of the formulated model problems involving Differential equations<\/li>\n
          • Modify \u00a0simple models for the change of environment<\/li>\n
          • Solve single species population models<\/li>\n
          • Discuss interacting two species population models<\/li>\n
          • Solve deterministic programming problems using dynamic programming algorithm<\/li>\n
          • Apply solution methods of unconstrained nonlinear extremum problems<\/li>\n
          • Make use of Lagrangian MultipliersmethodandKarush-Kuhn-Tucker(KKT) conditions to locate local minimizers<\/li>\n
          • ApplyWolfe\u2019s algorithm for solving quadratic programming problems<\/li>\n
          • Apply separable programming algorithm to solve nonlinear programming problems<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
          Course Contents:<\/strong><\/td>\n<\/tr>\n
          Fundamentals of modelling and word problems:<\/strong>Basic approach of Mathematical Modeling, its needs, types of models and limitations, setting up mathematical models for simple problems in words.<\/p>\n

          Qualitative solution sketching for first order differential equations:<\/strong> Direction field, solution sketch, convexity phase portrait, equilibrium solutions, stability.<\/p>\n

          Population models for Single and <\/strong>interacting species:<\/strong> Basic concepts, Exponential growth model, formulation, solution, interpretation and limitations, Compensation and dispensation, Logistic growth model, formulation, solution, interpretation and limitations.\u00a0 Types of interaction between two species. Lotka-Volterra prey-predator model, formulation, solution, interpretation and limitations. Lotka-Volterra model of two competing species, formulation, solution, interpretation and limitations.<\/p>\n

          Dynamic programming models:<\/strong> Stage, State, Recursive equation, Developing optimal decision policy, Bellman\u2019s principal of optimality, characteristics of deterministic dynamic programming, solving linear programming problem using dynamic programming approach.<\/p>\n

          Nonlinear programming fundamentals:<\/strong> Fundamentals of optimization, types of problems, Optimality conditions for unconstrained and constrained problems, Lagrangian Multipliers method and Karush-Kuhn-Tucker conditions.<\/p>\n

          Quadratic and Separable programmings: <\/strong>Quadratic programming, Wolfe\u2019s algorithm, Wolfe\u2019s modified simplex method, separable programming, separable function, piece wise linear approximation\u00a0of separable nonlinear programming problem, separable programming algorithm.<\/td>\n<\/tr>\n

          Teaching Methods:<\/strong><\/td>\n<\/tr>\n
          \n
            \n
          • Lectures,\u00a0 Tutorials, Problem solving, Use of e-resources and Handouts<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
          Assessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
          \n
            \n
          • In-course Assessments 30%<\/li>\n
          • End-of-course Examination 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
          Recommended Readings:<\/strong><\/td>\n<\/tr>\n
          \n
            \n
          • \u00a0Winston, W. L., Introduction of Mathematical Programming Applications and Algorithms, Duxbury press, California, 1995.<\/li>\n
          • Hillier, F. S. and Lieberman, G.J., Introduction to Operations Research, 7th<\/sup> edition, McGrawHill, New York, 2001.<\/li>\n
          • Taha, H. A., Operations Research an Introduction, 8th<\/sup> edition, Pearson Prentice Hall, New Jersey, 2007.<\/li>\n
          • Braun, M. ,Coleman, C. S. and Drew,D.A., Vol. 1, Vol. 2 and Vol.3 – Differential Equation Models,Springer-Verlag, New York, 1983.<\/li>\n
          • Meyer,W., \u00a0Concepts of Mathematical Modeling, McGraw Hill, New York, 1994.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
            <\/span>MMT306M3: Number Theory and Combinatorics <\/div>
            \n
            \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
            Course Code<\/strong><\/td>\nMMT306M3<\/strong><\/td>\n<\/tr>\n
            Course Title<\/strong><\/td>\nNumber Theory and Combinatorics<\/strong><\/td>\n<\/tr>\n
            Academic Credits<\/strong><\/td>\n03 <\/strong><\/td>\n<\/tr>\n
            Hourly Breakdown<\/strong><\/td>\nTheory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependent Learning<\/strong><\/td>\n<\/tr>\n
            45<\/strong><\/td>\n—<\/strong><\/td>\n105<\/strong><\/td>\n<\/tr>\n
            Objectives :<\/strong><\/td>\n<\/tr>\n
            \n
              \n
            • Provide in depth knowledge of classical number theory through a study of selection of topics.<\/li>\n
            • Develop an understanding of the core ideas and concepts of fundamental and advanced counting techniques<\/li>\n
            • Recognize the power of abstraction and generalization, and apply logical reasoning to investigate some combinatorial problems<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
            Intended Learning Outcomes :<\/strong><\/td>\n<\/tr>\n
            \n
              \n
            • Recall certain theorems in the theory of congruences<\/li>\n
            • Relate various number theoretic functions through\u00a0 M\u00f6bius inversion formula<\/li>\n
            • Discuss various properties of Euler \u00a0function<\/li>\n
            • Prove various results regarding primitive roots of prime numbers and composite numbers<\/li>\n
            • Recall Euler\u2019s criterion and its consequences<\/li>\n
            • Discuss various properties of Legendre symbol, notably quadratic reciprocity law<\/li>\n
            • Solve various quadratic congruences<\/li>\n
            • Solve certain non-linear Diophantine equations<\/li>\n
            • Prove various results regarding certain numbers of special form<\/li>\n
            • Solve common counting problems using elementary counting techniques involving the multiplication rule, permutations, and combinations<\/li>\n
            • Apply the Principle of inclusion and exclusion to solve variety of counting problems<\/li>\n
            • Apply the recurrence relations to model a wide variety of counting problems and solve them<\/li>\n
            • Make use of generating functions to solve many type of counting problems subject to variety of constraints, and solve recurrence relations<\/li>\n
            • Identify the best counting technique \u00a0for given counting problem<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
            Contents:<\/strong><\/td>\n<\/tr>\n
            Number Theory<\/strong>: Congruences, The Chinese remainder theorem, Wilson\u2019s theorem, Fermat\u2019s little theorem, Euler \u03c6 function, Euler\u2019s theorem, Number theoretic functions, M\u00f6bius inversion formula, Primitive roots, Quadratic residues, Euler\u2019s Criterion, Quadratic reciprocity, Quadratic Congruences, Legendre symbol, Perfect numbers, Mersenne primes and Amicable numbers, Fermat\u2019s numbers, Certain nonlinear Diophantine equations<\/p>\n

            Basic Counting Techniques: <\/strong>Fundamental principles of counting:Sum, product, subtraction, and division rules. Permutations and\u00a0 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.<\/p>\n

            Advanced Counting Techniques: <\/strong>Inclusion-Exclusion: The principle of inclusion-exclusion, an alternative form of inclusion-exclusion, the Sieve of Eratosthenes, derangementsRecurrence Relations:<\/strong>Modeling with recurrence relations,solving linear homogeneous recurrence relations with constant coefficients, linear non-homogeneous recurrence relations with constant coefficients, divide-and-conquer algorithms and recurrence relations.Generating Functions:<\/strong> 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.<\/td>\n<\/tr>\n

            Teaching Methods :<\/strong><\/td>\n<\/tr>\n
            \n
              \n
            • Lectures, \u00a0demonstration, tutorial discussion, problem solving, use of e-resources and handouts<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
            Assessment \/ Evaluation Details :<\/strong><\/td>\n<\/tr>\n
            \n
              \n
            • In course Assessments 30%<\/li>\n
            • End of course Examination\u00a0 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
            Recommended Readings :<\/strong><\/td>\n<\/tr>\n
            \n
              \n
            • Kenneth H. Rosen, Discrete Mathematics and its Applications, 7th<\/sup> edition, McGraw Hill, 2012.<\/li>\n
            • Bernard Kolman, Robert C. Busby, S. C. Ross, Discrete Mathematical Structures, 4th<\/sup> edition, Prentice Hall, 2001<\/li>\n
            • Grimaldi R.P, Discrete and combinatorial mathematics: an applied introduction, 5th<\/sup> edition, Pearson education Inc, 2004<\/li>\n
            • Miklos Bona, Introduction to Enumerative Combinatorics, McGraw Hill (2007).<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
              <\/span>MMT307M2: Topology II <\/div>
              \n
              \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
              Course Code<\/strong><\/td>\nMMT307M2<\/strong><\/td>\n<\/td>\n<\/tr>\n
              Course Title<\/strong><\/td>\nTopology II<\/strong><\/td>\n<\/td>\n<\/tr>\n
              Credit Value<\/strong><\/td>\n02<\/strong><\/td>\n<\/td>\n<\/tr>\n
              \u00a0<\/strong><\/p>\n

              Hourly Breakdown<\/strong><\/td>\n

              		\u00a0<\/strong><\/p>\n

              Theory<\/strong><\/td>\n

              		\u00a0<\/strong><\/p>\n

              Practical<\/strong><\/td>\n

              		Independen\u00a0\u00a0 Independent Learning<\/strong><\/td>\n<\/td>\n<\/tr>\n
              30<\/strong><\/td>\n—<\/strong><\/td>\n70<\/strong><\/td>\n<\/td>\n<\/tr>\n
              <\/td>\nObjectives:<\/strong><\/td>\n<\/tr>\n
              <\/td>\n\n
                \n
              • Provide an in depth understanding of compactness in the topological spaces<\/li>\n
              • Introduce the basic concepts of algebraic topology through a selection of topics<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
              <\/td>\nIntended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
              <\/td>\n\n
                \n
              • Discuss the processes of one point and\u00a0 Stone\u2013Cechcompactifications<\/li>\n
              • Examine the relationship between various types of compactness in the matric space<\/li>\n
              • Prove the Tychonoff theorem<\/li>\n
              • Compare the topology of pointwise convergence, topology of compact convergence and compact-open topology in a function space<\/li>\n
              • Prove various\u00a0 results concerned with homotopicity of maps<\/li>\n
              • Identify covering maps and covering spaces for certain topological spaces<\/li>\n
              • Recall certain results and their proofs about fundamental groups.<\/li>\n
              • Identifythe fundamental group of<\/li>\n
              • Prove the\u00a0 Fundamental theorem of Algebra and theBorsuk-Ulam theorem<\/li>\n
              • Prove certain facts of the singular homology theory<\/li>\n
              • Classify various types of knots<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
              <\/td>\nCourse Contents:<\/strong><\/td>\n<\/tr>\n
              <\/td>\n\n
                \n
              • Compactness<\/strong>: compactness in metric spaces, sequential compactness, Heine-Borel Property, total boundedness, complete metric spaces, the TychonoffTheorem,local compactness, one point and Stone-Cechcompactification, pointwise and compact convergence, compact-open topology<\/li>\n
              • The Fundamental group and Covering Spaces: <\/strong>homotopy of paths, the fundamental group, covering spaces, fundamental group of the circle, the fundamental theorem of algebra, antipodes, the Borsuk-Ulam theorem, retractions and fixed points, deformation\u00a0 retracts and homotopy type<\/li>\n<\/ul>\n
                  \n
                • Homology Theory:\u00a0 <\/strong>standard – simplex, boundary operator, – cycles, homology groups, induced homomorphism.<\/li>\n
                • Introduction to Knots<\/strong>: types of knots, preliminary results.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
              <\/td>\n\u00a0Teaching Methods:<\/strong><\/td>\n<\/tr>\n
              <\/td>\n\n
                \n
              • Lectures, Tutorials, Problem solving, Use of e-resources and Handouts.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
              <\/td>\nAssessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
              <\/td>\n\n
                \n
              • \u00a0In-course Assessments 30%<\/li>\n
              • End-of-course Examination 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
              <\/td>\nRecommended Readings:<\/strong><\/td>\n<\/tr>\n
              <\/td>\n\n
                \n
              • James R. Munkres, Topology, 2nd <\/sup>Edition, Prentice-Hall, 2000.<\/li>\n
              • \u00a0B.K.Lahiri, A First course in Algebraic Topology, Narosa, 2000.<\/li>\n
              • C.Kosniowski,\u00a0 A First course in Algebraic Topology, Cambridge University press, Cambridge , 2008<\/li>\n
              • A.Hatcher, Algebraic topology, Cambridge University press, Cambridge, 2002.<\/li>\n
              • SashoKalajdzievski,\u00a0 Topology and Homotopy,\u00a0 CRC press, 2015.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n<\/div>\n

                Level \u2013 4M<\/h3>\n

                Course units effective from academic year 2016\/2017 to date<\/h4>\n
                \n
                <\/span>MMT401M4: Measure Theory <\/div>
                \n
                \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                Course Code<\/strong><\/td>\nMMT401M4<\/strong><\/td>\n<\/tr>\n
                Course Title<\/strong><\/td>\nMeasure Theory<\/strong><\/td>\n<\/tr>\n
                Credit Value<\/strong><\/td>\n04 <\/strong><\/td>\n<\/tr>\n
                \u00a0<\/strong><\/p>\n

                Hourly Breakdown<\/strong><\/td>\n

                		Theory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependent Learning<\/strong><\/td>\n<\/tr>\n
                60<\/strong><\/td>\n—<\/strong><\/td>\n140<\/strong><\/td>\n<\/tr>\n
                Objectives:<\/strong><\/td>\n<\/tr>\n
                \n
                  \n
                • Introduce the fundamental concepts of Lebesgue measure spaces and abstract measure spaces<\/li>\n
                • Develop clear ideas on the concept of Lebesgue measurable functions, integrals, and their convergence properties<\/li>\n
                • Discussthe fundamental connection between differentiation, and integration<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                Intended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
                \n
                  \n
                • Construct Lebesgue measures on the real line<\/li>\n
                • Define abstract measure space<\/li>\n
                • Illustrate the properties of abstract measure space<\/li>\n
                • Discuss the properties of measurable functions and the convergence of sequence of measurable functions<\/li>\n
                • Explain the simple function approximation of measurable functions<\/li>\n
                • Formulate integrals in a measure space<\/li>\n
                • Discuss the convergence of integrals<\/li>\n
                • Extend the measures from algebras\/semialgebra to \u03c3-algebras<\/li>\n
                • Formulateproduct measures<\/li>\n
                • Prove Fubini\u2019s theorem, and Tonelli\u2019s theorem<\/li>\n
                • Discuss the fundamental connection between differentiation, and integration<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                Course Contents:<\/strong><\/td>\n<\/tr>\n
                Measure Spaces:<\/strong>Preliminaries:Algebra and \u03c3-algebras of sets,Borel sets; Lebesgue measure: Outer measure, Measurable sets, and Lebesgue measure,Properties,Example of a non-measurable set, Borel measures; General measure: Definition of measure, Measure space, Complete measure space, Examples, Properties.<\/p>\n

                Measurable Functions:<\/strong>\u00a0 Basic properties of measurable functions, Examples, Borel measurable functions, Approximation Theorem; Littlewoods\u2019s three principles: Egoroff\u2019s theorem.<\/p>\n

                Integration:<\/strong>Integral of nonnegative functions, Integrability of a nonnegative function, Fatou\u2019s Lemma, Monotone convergence theorem, Lebesgue Convergence Theorem, Generalized Convergence Theorem.<\/p>\n

                Extension of Measure:<\/strong>Measure on an algebra, Extension of measures from algebras to \u03c3-algebras, Carath\u00e9odory\u2019s theorem, and Lebesgue-Stieltjes integral.<\/p>\n

                Product Measure:<\/strong>Measurable rectangle, Semialgebra, Construction of product measures, Fubini\u2019s theorem, and Tonelli\u2019s theorem<\/p>\n

                Differentiation and Integration:<\/strong>Differentiation of monotone functions: Vitali\u2019s lemma, Functions of bounded variations; Differentiation of an integral: Indefinite integral, and Absolutely continuous functions.<\/td>\n<\/tr>\n

                Teaching Methods:<\/strong><\/td>\n<\/tr>\n
                \n
                  \n
                • Lectures, Tutorials, Handouts, Problem solving, Use of e-resources<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                Assessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
                \n
                  \n
                • In-course Assessments\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 30%<\/li>\n
                • End-of-course Examination 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                Recommended Readings:<\/strong><\/td>\n<\/tr>\n
                \n
                  \n
                • Halsey Royden, Patrick Fitzpatrick, Real Analysis,4th<\/sup> Edition, 2010<\/li>\n
                • Walter Rudin, Real and Complex Analysis, 3rd<\/sup> Edition, 1986<\/li>\n
                • G De Barra, Measure theory and Integration, 2nd<\/sup> Edition, 2003<\/span><\/li>\n
                • Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd Edition, 2007<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
                  <\/span>MMT402M3: Advanced Algebra II<\/div>
                  \n
                  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                  Course Code<\/strong><\/td>\nMMT402M3<\/strong><\/td>\n<\/tr>\n
                  Course Title<\/strong><\/td>\nAdvanced Algebra II<\/strong><\/td>\n<\/tr>\n
                  Credit Value<\/strong><\/td>\n03 <\/strong><\/td>\n<\/tr>\n
                  \u00a0<\/strong><\/p>\n

                  Hourly Breakdown<\/strong><\/td>\n

                  		Theory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependent Learning<\/strong><\/td>\n<\/tr>\n
                  45<\/strong><\/td>\n—<\/strong><\/td>\n105<\/strong><\/td>\n<\/tr>\n
                  Objectives:<\/strong><\/td>\n<\/tr>\n
                  Provide an in-depth knowledge in the theory of groups through a selection of advanced topics.<\/td>\n<\/tr>\n
                  Intended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
                  \n
                    \n
                  • Prove the orbit-stabilizer theorem, Cauchy\u2019s theorem, the class equation and other related results<\/li>\n
                  • Define Sylow-p-subgroup<\/li>\n
                  • Prove the Sylow-theorems<\/li>\n
                  • Examine the simplicity of groups by means of Sylow theorems<\/li>\n
                  • Prove the fundamental theorem of finite abelian groups<\/li>\n
                  • Discuss the simplicity of Alternative Group An<\/sub><\/li>\n
                  • Construct a group that is free on a given non-empty set<\/li>\n
                  • Analyse the presentations of certain groups, in particular the dihedral groups and the quaternion groups<\/li>\n
                  • Prove the Zassenhaus lemma, theorems of Schreier and Jordan-Holder and certain other results related to the chain conditions<\/li>\n
                  • Discuss various properties of the Frattini subgroup of a given group<\/li>\n
                  • Define nilpotent group, soluble group, supersoluble group and residually nilpotent group<\/li>\n
                  • Examine various properties of nilpotent groups, soluble groups, supersoluble groups and residually nilpotent groups<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                  Course Contents:<\/strong><\/p>\n

                  Direct products, group acting on sets, p-groups, the Sylow theorems, finite abelian groups, symmetric groups, simplicity of An<\/sub>,\u00a0 free groups, presentations of groups, series of subgroups, composition series, chain conditions, Frattini subgroup, nilpotent groups, soluble groups, super soluble groups, residually nilpotent groups.<\/td>\n<\/tr>\n

                  Teaching Methods:<\/strong><\/td>\n<\/tr>\n
                  \n
                    \n
                  • Lectures,\u00a0 Tutorials, Handouts, Problem solving, Use of e-resources<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                  Assessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
                  \n
                    \n
                  • n-course Assessments 30%<\/li>\n
                  • End-of-course Examination 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                  Recommended Readings:<\/strong><\/td>\n<\/tr>\n
                  \n
                    \n
                  • Abstract algebra, Joha A Beachy William D Blair, 4th<\/sup> edition, Waveland, 2019.<\/li>\n
                  • Contemporary Abstract Algebra Joseph.A.Gallian, 9th<\/sup> edition, Cengage, 2016.<\/span><\/li>\n
                  • \u00a0Abstract Algebra, David S Dummit,Richard M. Foote,4th<\/sup> edition, Wiley, 2018.<\/span><\/li>\n
                  • \u00a0A course in the theory of groups \u2013 Derek.J.S.Robinson 2nd<\/sup> edition, Springer, 1995.<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
                    <\/span>MMT403M4: Functional Analysis II<\/div>
                    \n
                    \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                    Course Code<\/strong><\/td>\nMMT403M4<\/strong><\/td>\n<\/tr>\n
                    Course Title<\/strong><\/td>\nFunctional Analysis II<\/strong><\/td>\n<\/tr>\n
                    Credit Value<\/strong><\/td>\n04 <\/strong><\/td>\n<\/tr>\n
                    Hourly Breakdown<\/strong><\/td>\nTheory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependent Learning<\/strong><\/td>\n<\/tr>\n
                    60 hours<\/strong><\/td>\n—–<\/strong><\/td>\n140 hours<\/strong><\/td>\n<\/tr>\n
                    Objectives:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><\/td>\n<\/tr>\n
                    \n
                      \n
                    • Introduce the fundamental theorems for normed and Banach spaces<\/li>\n<\/ul>\n
                        \n
                      • Provide a clear notion of concepts of\u00a0 compact linear operators and adjoint operators on Normed spaces<\/li>\n<\/ul>\n
                          \n
                        • Acquaint with the concepts of spectral theory of bounded self adjoint linear operators in Normed spaces<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                    Intended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
                    \n
                      \n
                    • Demonstrate uniform boundedness principle<\/li>\n
                    • Prove open mapping theorem, bounded inverse theorem and closed graph theorem<\/li>\n
                    • Define reflexive normed linear spaces<\/li>\n
                    • Discuss the strong and weak convergence of sequence in a normed space<\/li>\n
                    • Prove Banach fixed point theorem<\/li>\n
                    • Define Self adjoint, Unitary, Normal and Compact linear operators<\/li>\n
                    • Develop\u00a0 Resolvent and spectrum<\/li>\n
                    • Discuss the spectral properties of bounded linear operators<\/li>\n
                    • Construct the spectral family of a bounded self-adjoint linear operator<\/li>\n
                    • Extend the spectral theorem to continuous functions<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                    Course Contents:<\/strong><\/td>\n<\/tr>\n
                    Fundamental theorems for Normed and Banach spaces: <\/strong>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<\/p>\n

                    Compact linear operators and Adjoint operators on Normed spaces: <\/strong>Hilbert-Adjoint Operator , Self-adjoint, Unitary and Normal operators,\u00a0 Compact linear operators on normed Spaces and its properties<\/p>\n

                    Spectral theory of bounded self adjoint linear operators in Normed Spaces<\/strong> :<\/strong> Resolvent and spectrum of linear operator, Spectral theory in finite dimensional normed spaces,\u00a0 Spectral properties of bounded operators and self-adjoint\u00a0 operators ,<\/strong> Spectral properties of Compact linear operators ,Positive operators, Spectral family of a bounded self-adjoint linear operator, Extension of the spectral theorem to continuous functions.<\/td>\n<\/tr>\n

                    Teaching Methods:<\/strong><\/td>\n<\/tr>\n
                    \n
                      \n
                    • Lectures,\u00a0 Tutorial discussion, Handouts and use of e-resources<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                    Assessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
                    \n
                      \n
                    • In-course assessment\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a030%<\/li>\n
                    • End of course Examination\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                    Recommended Readings:<\/strong><\/td>\n<\/tr>\n
                    \n
                      \n
                    • John B. Conway, A Course in Functional Analysis, Springer, 1994<\/li>\n
                    • Bryan P. Rynne , Linear Functional Analysis, Springer, 2007<\/li>\n
                    • Walter Rudin, Functional Analysis, Mac Graw \u2013 Hill, 1977<\/li>\n
                    • Erwin Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1989<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
                      <\/span>MMT404M3: Advanced Complex Analysis<\/div>
                      \n
                      \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                      Course Code<\/strong><\/td>\nMMT404M3<\/strong><\/td>\n<\/tr>\n
                      Course Title<\/strong><\/td>\nAdvanced Complex Analysis<\/strong><\/td>\n<\/tr>\n
                      Credit Value<\/strong><\/td>\n03<\/strong><\/td>\n<\/tr>\n
                      Hourly Breakdown<\/strong><\/td>\nTheory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependent Learning<\/strong><\/td>\n<\/tr>\n
                      45hours<\/strong><\/td>\n—–<\/strong><\/td>\n105 hours<\/strong><\/td>\n<\/tr>\n
                      Objectives:<\/strong><\/td>\n<\/tr>\n
                      \n
                        \n
                      • Provide a rigorous treatment of holomorphic and meomorphic\u00a0 functions<\/li>\n
                      • Acquaint with the concepts of conformal mapping and analytic continuation<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                      Intended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
                      \n
                        \n
                      • Recall the definition and results related to functions representable by power series<\/li>\n
                      • Prove Cauchy\u2019s\u00a0 theorem, Cauchy\u2019s formula, Morera\u2019 s theorem, residue theorem, open mapping theorem and global\u00a0 Cauchy\u2019s\u00a0 theorem<\/li>\n
                      • Discuss various properties of Poisson\u2019s kernel<\/li>\n
                      • Establish Harnack\u2019s theorem and mean value property<\/li>\n
                      • Examine the maximum modulus theorem for unbounded regions<\/li>\n
                      • Define the notion of a conformal mapping and conformal equivalence<\/li>\n
                      • Construct explicit conformal equivalences between specified complex regions<\/li>\n
                      • Prove Schwarz’s lemma and Riemann mapping theorem.<\/li>\n
                      • Define an infinite product and give product formulas for standard mathematical functions.<\/li>\n
                      • Construct entire functions with specified zero sets<\/li>\n
                      • Develop automorphisms with specified behaviour.<\/li>\n
                      • Discuss the Schwarz reflection principle as a form of analytic continuation<\/li>\n
                      • Find analytic continuation of given function along a given curve<\/li>\n
                      • Prove Monodromy theorem<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                      Course Contents:<\/strong><\/td>\n<\/tr>\n
                      Holomorphic functions: <\/strong>Complex differentiation, Integration over paths, Local and global Cauchy theorem, Calculus of residues.<\/p>\n

                      Harmonic functions: <\/strong>Cauchy-Riemann equations, Poisson integral, Mean value property, Boundary behavior of Poisson integrals ,Representation theorems<\/p>\n

                      Maximum modulus principle: <\/strong>Schwarz lemma, Phragmen-Lindelof method, An interpolation theorem, Converse of\u00a0 the maximum modulus theorem<\/p>\n

                      \u00a0<\/strong>Conformal mapping<\/strong> :<\/strong>Preservation of angles, Linear fractional transformation, Normal families, Riemann mapping theorem, Continuity at the boundary, Conformal mapping of an annulus<\/p>\n

                      Zeros of Holomorphic functions: <\/strong>Infinite products, Weierstrass factorization theorem, An interpolation theorem, Jensen\u2019s formula, Blaschke products, Muntz-Szasz theorem<\/p>\n

                      Analytic continuation: <\/strong>Regular points and singular points, Schwarz reflection principle, Continuation along curves, Monodromy theorem, Construction of modular function, Picard theorem<\/td>\n<\/tr>\n

                      Teaching Methods:<\/strong><\/td>\n<\/tr>\n
                      \n
                        \n
                      • Lectures,\u00a0 Tutorial discussion, Handouts and use of e-resources<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                      Assessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
                      \n
                        \n
                      • In-course assessment \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a030%<\/li>\n
                      • End of course Examination\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                      Recommended Readings:<\/strong><\/td>\n<\/tr>\n
                      \n
                        \n
                      • Walter Rudin , Real and Complex Analysis, McGraw-Hill, 1987<\/li>\n
                      • John B. Conway,<\/strong>Functions of one complex variable, Springer,1994<\/li>\n
                      • Bruce P. Palka, An Introduction to Complex function theory, Springer-Verlag, 1991<\/li>\n
                      • Robert E. Greene, Steven G. Krantz,\u00a0 Function Theory of One Complex Variable,American\u00a0 Mathematical\u00a0 Society, 2006<\/li>\n
                      • Wilhelm Schlag, A Course in Complex Analysis and Riemann Surfaces, American\u00a0 Mathematical\u00a0 Society,2014<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
                        <\/span>MMT405M3: Mathematical Physics<\/div>
                        \n
                        \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                        Course Code<\/strong><\/td>\nMMT405M4<\/strong><\/td>\n<\/tr>\n
                        Course Title<\/strong><\/td>\nMathematical Physics<\/strong><\/td>\n<\/tr>\n
                        Credit Value<\/strong><\/td>\n04 <\/strong><\/td>\n<\/tr>\n
                        \u00a0<\/strong><\/p>\n

                        Hourly Breakdown<\/strong><\/td>\n

                        		Theory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependent Learning<\/strong><\/td>\n<\/tr>\n
                        60<\/strong><\/td>\n—<\/strong><\/td>\n140<\/strong><\/td>\n<\/tr>\n
                        Objectives:<\/strong><\/td>\n<\/tr>\n
                        \n
                          \n
                        • Introduce the central concepts and principles in quantum mechanics, such as the Schr\u00f6dinger equation, the wave function and its statistical interpretations.<\/li>\n
                        • Develop a theory of continuum mechanics and understand fundamental concepts of stress and strain.<\/li>\n
                        • Familiarize the fundamental principles of the general theory of relativity.<\/li>\n
                        • Describe the properties and dynamics of charged particles and electromagnetic fields.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                        Intended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
                        \n
                          \n
                        • Explain the basic principles of quantum mechanics.<\/li>\n
                        • Solve the Schrodinger equation to obtain wave functions for some basic, physically important types of potential in one dimension.<\/li>\n
                        • Explain the operator formulation of quantum mechanics.<\/li>\n
                        • Obtain the Heisenberg’s uncertainty principle in quantum mechanics.<\/li>\n
                        • Describe motion, velocity and acceleration using spatial and material descriptions.<\/li>\n
                        • Define the concepts of strain and stress measures for finite deformations.<\/li>\n
                        • Solve specific technical problems of displacement, strain and stress in continuous materials.<\/li>\n
                        • Obtain the equations of motion and conservation laws for continuum materials.<\/li>\n
                        • Define the properties of isotropic linear elastic materials.<\/li>\n
                        • Explain the fundamental concepts of special relativity.<\/li>\n
                        • Derive the Lorentz transformations.<\/li>\n
                        • Explain the concepts of length contraction and time dilation.<\/li>\n
                        • Define the relativistic notation for 4-vectors.<\/li>\n
                        • Recall the important laws and principles of electric, magnetic, and electromagnetic fields.<\/li>\n
                        • Solve problems in electrostatic, magnetostatic, and electromagnetic fields<\/li>\n
                        • Construct the necessary equations related to Faraday\u2018s law, induced electromotive force and Maxwell\u2018s equations.<\/li>\n
                        • Apply Maxwell\u2018s equations to solutions of problems relating to transmission lines and uniform plane wave propagation<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                        Course Contents:<\/strong><\/td>\n<\/tr>\n
                        Quantum mechanics: <\/strong>Wave mechanics, Schr\u00f6dinger\u2019s equation, probability interpretation, Simple quantum-mechanical systems, Poisson brackets, State vectors, statistical aspects, Heisenberg\u2019s uncertainty principle.<\/p>\n

                        Continuum mechanics:\u00a0 <\/strong>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.<\/p>\n

                        Relativity: <\/strong>Galilean relativity, inertial frames, Galilean transforms, Newtonian relativity, Constancy of speed of light, simultaneity, space-time diagrams, proper time, Lorentz transforms, length-contraction and time-dilation, transformation of velocities. 4- vectors: displacement, velocity, acceleration, frequency, Conservation of 4- momentum, mass-energy equivalence.<\/p>\n

                        Electromagnetism:<\/strong> Charge, Coulomb\u2019s law, electric field and potential, Electric flux and density, Electro dynamics, Magnetostatics fields, Faradays law, Displacement current, Time-varying potentials, Derivation of Maxwell\u2019s equations, Electromagnetic wave propagation.<\/td>\n<\/tr>\n

                        Teaching Methods:<\/strong><\/td>\n<\/tr>\n
                        \n
                          \n
                        • Lectures,\u00a0 Tutorials, Handouts, Problem solving, Use of e-resources<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                        Assessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
                        \n
                          \n
                        • In-course Assessments\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 30%<\/li>\n
                        • End-of-course Examination 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                        Recommended Readings:<\/strong><\/td>\n<\/tr>\n
                        \n
                          \n
                        • F. Schwabl, Quantum Mechanics<\/em>. Springer-Lehrbuch, Heidelberg, 2002.<\/li>\n
                        • J. J. Sakurai, Modern Quantum Mechanics<\/em>. Addison Wesley Longman, 1994.<\/span><\/li>\n
                        • R. Chatterjee, Mathematical Theory of Continuum Mechanics, <\/em>Narosa Pub, 2019<\/span><\/li>\n
                        • C.S. Jog, Continuum Mechanics, (3rd edition)<\/em>, Cambridge-IISc Press, 2015.<\/span><\/li>\n
                        • J. B. Hartle, Gravity: An Introduction to Einstein’s General Relativity,<\/em> Addison-Wesley, 2003.<\/span><\/li>\n
                        • \u00a0B. Schutz: A First Course in General Relativity<\/em>, Cambridge University Press, 2009.<\/span><\/li>\n
                        • D.J.R. Griffiths,.,Introduction to Electrodynamics (4th<\/sup>\u00a0 ed.),<\/em>Cambridge University Press, 2017.<\/span><\/li>\n
                        • \u00a0D. Fleisch, A Student’s Guide to Maxwell’s Equations, Cambridge University Press, 2008.<\/li>\n
                        • R.K.Wangsness, Electromagnetic Fields (2nd<\/sup> ed.), Wiley, 2007.<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
                          <\/span>MMT406M3: Differential Equations<\/div>
                          \n
                          \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                          Course Code<\/strong><\/td>\nMMT406M3<\/strong><\/td>\n<\/tr>\n
                          Course Title<\/strong><\/td>\nDifferential Equations<\/strong><\/td>\n<\/tr>\n
                          Credit Value<\/strong><\/td>\n03<\/strong><\/td>\n<\/tr>\n
                          \u00a0<\/strong><\/p>\n

                          Hourly Breakdown<\/strong><\/td>\n

                          		Theory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependent Learning<\/strong><\/td>\n<\/tr>\n
                          45<\/strong><\/td>\n—<\/strong><\/td>\n105<\/strong><\/td>\n<\/tr>\n
                          Objectives:<\/strong><\/td>\n<\/tr>\n
                          \n
                            \n
                          • \u00a0Familiarize theunderlying theoretical concepts in differential equations<\/li>\n
                          • \u00a0Understand the Analytical Aspects of the solutions of differential equations<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                          Intended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
                          \n
                            \n
                          • Outline the analytical concepts in differential equations<\/li>\n
                          • Prove Gronwall\u2019s inequality<\/li>\n
                          • Prove the existence and uniqueness of the solution of the first order differential equation<\/li>\n
                          • Determine the\u00a0\u00a0 solution of a first order ordinary differential equation<\/li>\n
                          • Construct the Wronksian for a higher order ordinary differential equation<\/li>\n
                          • Elaborate the linear independenceness of the solutions<\/li>\n
                          • Construct linear independent solution using reduction of order<\/li>\n
                          • Define various types of\u00a0 equilibriums<\/li>\n
                          • \u00a0Classify the equlibria of a ordinary differential equation<\/li>\n
                          • Make use of Routh-Hurwitz condition to determine the equilibria of a system of ordinary differential equation<\/li>\n
                          • Formulate the nullclines of a system of ordinary differential equations<\/li>\n
                          • Build a phase plane diagram\u00a0\u00a0 for a system of ordinary differential equations<\/li>\n
                          • Construct the solution for a first order partial differential equation<\/li>\n
                          • Analyze the continuity of the solution of a first order partial differential equation<\/li>\n
                          • Classify a second order partial differential equation into hyperbolic, parabolic and elliptic<\/li>\n
                          • Formulate the canonical form of the second order partial differential equation<\/li>\n
                          • Establish the De Alembert\u2019s solution of a second order partial differential equation<\/li>\n
                          • Prove the uniqueness of the solution of a second order partial differential equation<\/li>\n
                          • Formulate the solution of a second order partial differential equation<\/li>\n
                          • Apply Green\u2019s theorem<\/li>\n
                          • Show the symmetry of Green\u2019s function<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                          Course Contents: <\/strong><\/td>\n<\/tr>\n
                          Ordinary Differential Equations:<\/strong> Initial Value Problems, Boundary Value Problems, Equivalence and matrix notation, Grownwall\u2019s inequality, Existence and Uniqueness of Solutions,\u00a0 Picard\u2019s Theorem,\u00a0 Non-uniqueness of solutions, boundedness of solutions, Solutions to the second order systems<\/p>\n

                          Linear System of Ordinary Differential Equations: <\/strong>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\u2019s function<\/p>\n

                          Stability Analysis of Ordinary Differential equations:\u00a0 <\/strong>Autonomous\u00a0 systems, linearization, Linear systems, Routh Hurwitz criterion, classification of equilibria, Phase plane analysis, Poincare-Bendixon theorem, Lyapunov stability<\/p>\n

                          First order Partial Differential Equations: <\/strong>First order quasi linear scalar equations, method of characteristics, systems of equations, concepts of order and dimension<\/p>\n

                          Second order Linear Partial Differential Equations: <\/strong>Classification of equations, canonical forms of hyperbolic, parabolic and elliptic equations, Boundary conditions, the Laplacian, wave equation and heat equation, Poisson\u2019s equation, uniqueness and the maximum principal, integral transform solutions, application of Laplace transform, and Fourier transforms, Hankel Transforms.<\/td>\n<\/tr>\n

                          Teaching Methods:<\/strong><\/td>\n<\/tr>\n
                          \n
                            \n
                          • Lectures,\u00a0 Tutorials, Handouts, Problem solving, Use of e-resources<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                          Assessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
                          \n
                            \n
                          • In-course Assessments\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 30%<\/li>\n
                          • End-of-course Examination 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                          Recommended Readings:<\/strong><\/td>\n<\/tr>\n
                          \n
                            \n
                          • D. P. Andrei, F. Z. Valentin, Handbook of Ordinary Differential Equations: Exact Solutions, Methods and Problems, CRC Press, 2018<\/li>\n
                          • E. A. Coddington, An introduction to Ordinary Differential Equations, Dover Publications, 1961<\/li>\n
                          • S. Ahmad, A. Ambrosetti, A textbook on Ordinary Differential Equations, Springer, 2014<\/li>\n
                          • P. Hartman, Ordinary Differential Equations, SAIM, 2002<\/li>\n
                          • L. C. Evans, Partial Differential Equations, AMS, 2010<\/li>\n
                          • M. Renardy, R. C. Rogers, An introduction to partial differential equations, Springer, 2004<\/li>\n
                          • Y. pinchover, J. Rubinstein, An introduction to partial differential equations, Cambridge University Press, 2005<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
                            <\/span>MMT407M3: Numerical Differential Equations<\/div>
                            \n
                            \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                            Course Code<\/strong><\/td>\nMMT407M3<\/strong><\/td>\n<\/tr>\n
                            Course Title<\/strong><\/td>\nNumerical Differential Equations <\/strong><\/td>\n<\/tr>\n
                            Credit Value<\/strong><\/td>\n03<\/strong><\/td>\n<\/tr>\n
                            \u00a0<\/strong><\/p>\n

                            Hourly Breakdown<\/strong><\/td>\n

                            		Theory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependent Learning<\/strong><\/td>\n<\/tr>\n
                            45<\/strong><\/td>\n—<\/strong><\/td>\n105<\/strong><\/td>\n<\/tr>\n
                            Objectives:<\/strong><\/td>\n<\/tr>\n
                            \u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Understand the theoretical concepts of various numerical methods applied to solvedifferential equations<\/p>\n

                            \u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Familiarize the underlying mathematical concepts of computer aided numerical algorithms<\/p>\n

                            \u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Acquaint with the stability theory concepts of the methods<\/td>\n<\/tr>\n

                            Intended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
                            \n
                              \n
                            • Illustrate the fundamental concepts involving norms and matrix inequalities<\/li>\n
                            • Apply implicit and explicit Euler methods<\/li>\n
                            • Estimate the error bounds for the solutions obtained using Euler methods<\/li>\n
                            • Recall the order, consistency, stability, convergence and the characteristic polynomials associated with a linear multistep method<\/li>\n
                            • \u00a0Prove the relationships among the order, consistency, stability,\u00a0 convergence and the characteristic polynomials of a linear multistep method<\/li>\n
                            • Examine the consistency, stability and convergence of a linear multistep method<\/li>\n
                            • Outline various types of predictor-corrector methods<\/li>\n
                            • Apply a given predictor-corrector method to solve a differential equation<\/li>\n
                            • Demonstrate the stability of a predictor-corrector method<\/li>\n
                            • Illustrate Runge-Kutta method<\/li>\n
                            • Recall the order, consistency, stability and convergence of Runge-Kutta method<\/li>\n
                            • Construct Runge-Kutta method using order conditions<\/li>\n
                            • Define various types of stability associated with the Runge-Kutta method<\/li>\n
                            • Interpret the relationships among various types of stabilities<\/li>\n
                            • Determine a difference formula to solve a boundary value problem<\/li>\n
                            • Construct\u00a0 Finite difference methods applied to boundary value problems<\/li>\n
                            • Estimate the truncation error for a finite difference method<\/li>\n
                            • Apply finite difference methods to solve a boundary value problem<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                            Course Contents: <\/strong><\/td>\n<\/tr>\n
                            Euler methods: <\/strong>Explicit and Implicit Euler methods,\u00a0\u00a0 Convergence, local and global truncation errors, order of the method, error bounds<\/p>\n

                            Linear multistep methods:<\/strong> 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, Predictor-Corrector methods<\/p>\n

                            Runge-Kutta methods: <\/strong>General Runge-kutta methods, implicit and explicit methods, local and global truncation errors, order of the method, order conditions, derivation of the method<\/p>\n

                            Stability: <\/strong>A- stability, B \u2013 stability, algebraic stability<\/p>\n

                            Boundary Value Problems: <\/strong>FiniteDifference methods, Shooting methods, truncation error estimation, Convergence of the method, Stability of the method<\/td>\n<\/tr>\n

                            Teaching Methods:<\/strong><\/td>\n<\/tr>\n
                            \n
                              \n
                            • Lectures,\u00a0 Tutorials, Handouts, Problem solving, Use of e-resources<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                            Assessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
                            \n
                              \n
                            • In-course Assessments\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 30%<\/li>\n
                            • End-of-course Examination 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                            Recommended Readings:<\/strong><\/td>\n<\/tr>\n
                            \n
                              \n
                            • M. H. Holmes, Introduction to Numerical Methods in Differential Equations, Springer-Verlag NewYork, 2007<\/li>\n
                            • D. Griffiths and D. J. Higham,\u00a0 Numerical Methods for Ordinary Differential Equations, Springer-Verlag London, 2010<\/li>\n
                            • K. Atkinson, W. Han and D. E. Stewar, Numerical Solution of Ordinary Differential Equations, Wiley, 2009<\/li>\n
                            • J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer-Verlag NewYork, 1995<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
                              <\/span>MMT408M3: Category Theory<\/div>
                              \n
                              \n\n\n\n\n\n\n\n\n\n\n\n
                              Course Code<\/strong><\/td>\nMMT408M3<\/strong><\/td>\n<\/tr>\n
                              Course Title<\/strong><\/td>\nCategory Theory<\/strong><\/td>\n<\/tr>\n
                              Credit Value<\/strong><\/td>\n03 <\/strong><\/td>\n<\/tr>\n
                              \u00a0<\/strong><\/p>\n

                              Hourly Breakdown<\/strong><\/td>\n

                              		Theory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependent Learning<\/strong><\/td>\n<\/tr>\n
                              45<\/strong><\/td>\n—<\/strong><\/td>\n105<\/strong><\/td>\n<\/tr>\n
                              Objectives:<\/strong><\/td>\n<\/tr>\n
                              \n
                                \n
                              • Provide an introduction to category theory, as a language of unification with an emphasis on applications.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                              Intended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
                              \n
                                \n
                              • Define the Category, Functor, Natural transformations, Limits and Colimits, adjoints and monads<\/li>\n
                              • Construct\u00a0 various examples of category\u00a0 and other concepts in category theory such as\u00a0\u00a0 Functor, Natural transformations, Limits and Colimits, adjoints and monads<\/li>\n
                              • Analyze the properties of special types of objects and morphisms<\/li>\n
                              • Prove Yoneda Lemma,<\/li>\n
                              • Construct initial algebra and final colgebra by means of the Adamek\u2019s theorem and Bar\u2019s theorem respectively.<\/li>\n
                              • Explain the notions of sources, sinks and generalized limits and colimits.<\/li>\n
                              • Solve problems using category theoretic concepts.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n\n\n\n
                                Course Contents:<\/strong><\/td>\n<\/tr>\n
                                Foundations:-<\/strong> Basic definitions, sets, classes, conglomerates<\/p>\n

                                Categories and Functors:-<\/strong> Categories,\u00a0 examples, commutative diagrams, Definitions , duality Principle, isomorphisms,\u00a0 Fuctors, examples of Functors, Composition of functors, Hom-sets, Subcategory, Equivalences of categories.<\/p>\n

                                Natural transformations:-<\/strong> Natural transfromation between two functors, Naturality condition, Examples of Natural transformations,\u00a0\u00a0 Functor categories, Representable functor, Composition of natural transformations, The category of small categories,\u00a0 Yoneda Lemma<\/p>\n

                                Objects and Morphisms:-<\/strong> Initial Object and Final Object, separator and coseparator, sections, retraction, monomorphismsm, epimorphisms, regular and extremalepimorphism, equalizer, coequalizer,<\/p>\n

                                Limits and Colimits :-<\/strong> Products and Coproducts, Sinks and Sources, pullback and pushout, Limit and Colimit of functors, preservation of Limits, Lifting of Limits<\/p>\n

                                Applications :-<\/strong> Algebras and\u00a0 Coalgebras:-Category of algebras and coalgebras, Initial algebras and final coalgebras, applications to computer science.<\/p>\n

                                Optional Topics:- Adjoints and Monads (according to the avaliblity of time)<\/td>\n<\/tr>\n

                                Teaching Methods:<\/strong><\/td>\n<\/tr>\n
                                \n
                                  \n
                                • Lectures, tutorial discussions and e-learning<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                                Assessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
                                \n
                                  \n
                                • In-course assessment\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 30%<\/li>\n
                                • End of course Examination\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                                Recommended Readings:<\/strong><\/td>\n<\/tr>\n
                                \n
                                  \n
                                • Ad\u00e1mek, Ji\u0159\u00ed, Horst Herrlich, and George E. Strecker. “Abstract and concrete categories. The joy of cats.”,2004.<\/li>\n
                                • Awodey, Steve. Category Theory. Oxford University Press, 2010.<\/li>\n
                                • An Invitation to Applied Category Theory, Online version:-https:\/\/arxiv.org\/abs\/1803.05316<\/li>\n
                                • Crole, Roy L. Categories for types. Cambridge University Press, 1993.<\/li>\n
                                • Spivak, David I. “Category theory for scientists.” arXiv preprint arXiv:1302.6946,2013.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n
                                  <\/span>MMT409M3: Research Project<\/div>
                                  \n
                                  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                  Course Code<\/strong><\/td>\nMMT409M6<\/strong><\/td>\n<\/tr>\n
                                  Course Title<\/strong><\/td>\nResearch Project<\/strong><\/td>\n<\/tr>\n
                                  Credit Value<\/strong><\/td>\n06<\/strong><\/td>\n<\/tr>\n
                                  \u00a0<\/strong><\/p>\n

                                  Hourly Breakdown<\/strong><\/td>\n

                                  		Theory<\/strong><\/td>\nPractical<\/strong><\/td>\nIndependent Learning<\/strong><\/td>\n<\/tr>\n
                                  <\/td>\n<\/td>\n300 Hours<\/td>\n<\/tr>\n
                                  Objectives:<\/strong><\/td>\n<\/tr>\n
                                  Provide an opportunity for the students to participate ina research project in Mathematics to develop research skills.<\/td>\n<\/tr>\n
                                  Intended Learning Outcomes:<\/strong><\/td>\n<\/tr>\n
                                  \n
                                    \n
                                  • Identify a research question, problem, or design through a literature survey.<\/li>\n
                                  • Apply basic principles and knowledge found in the literature related to the research problem<\/li>\n
                                  • Demonstrate appropriate research methodologies<\/li>\n
                                  • Identify relevant theory and concepts, relate these to appropriate methodologies<\/li>\n
                                  • Develop a research proposal to address or resolve a specific research question or problem<\/li>\n
                                  • Demonstrate the research findings in written and verbal forms<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                                  Course Description:<\/strong><\/td>\n<\/tr>\n
                                  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.<\/td>\n<\/tr>\n
                                  Teaching Methods:<\/strong><\/td>\n<\/tr>\n
                                  \n
                                    \n
                                  • Guided independent study, Discussion with the supervisor, Use of e-resources.<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                                  Assessment\/ Evaluation Details:<\/strong><\/td>\n<\/tr>\n
                                  \n
                                    \n
                                  • Presentation\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 30%<\/li>\n
                                  • Project Report\u00a0\u00a0\u00a0\u00a0 70%<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n
                                  Recommended Readings:<\/strong><\/td>\n<\/tr>\n
                                  \n
                                    \n
                                  • Lianghuo Fan,Luc Trouche,Chunxia Qi,Sebastian Rezat,Jana Visnovska, Research on Mathematics Textbooks and Teachers\u2019 Resources: Advances and Issues, \u00a0Springer International Publishing, \u00a02018.<\/li>\n
                                  • Robert Gerver, Writing Math Research Papers: A Guide for Students and Instructors, 2th<\/sup> Edition, Information Age Publishing, Inc, 2014.<\/span><\/li>\n
                                  • George Gr\u00e4tzer, More Math Into LaTeX 5th Edition,Springer International Publishing, 2016<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

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