Extended Degree – Mathematics

Hourly Breakdown

• Introduce the elements of commutative algebra through a study of commutative rings and modules over such rings

• Prove factorization of homomorphisms, Correspondence theorem, the classical isomorphism theorems and their consequences
• Define certain special types of ideals
• Discuss various aspects of certain special types of ideals, by means of proofs and examples
• Prove module isomorphism theorems and other results related to submodules, quotient modules, direct sum and direct product of modules
• Prove certain results related to finitely generated modules, notably Nakayama’s lemma.
• Recall various results regarding exact sequences
• Define Unique Factorization Domains (U.F.D), Principal Ideal Domains(P.I.D) and Euclidean Domains (E.D)
•  Discuss various results regarding U.F.D, P.I.D and E.D, by  means of proofs and examples
• Explain the processes of constructing the rings and modules of fractions
•  Prove certain results related to primary decomposition, notably 1^st and 2^nd uniqueness theorems
• ·Prove certain theorems regarding Noetherian and Artinian modules
•  Discuss various results regarding Noetherian and Artin rings, in particular the Hilbert basis theorem

• Lectures,  Tutorials,  Problem solving, Use of e-resources and Handouts

• In-course Assessments 30%
• End-of-course Examination 70%

• Atiyah, M.G. and MacDonald, G., AnIntroduction to Commutative Algebra, Addison-Wesley,1969
• Dummit, D. S. and Foote, R. M., Abstract Algebra, Wiley, 2006
• Sharp, R. Y., Steps in commutative Algebra, Cambridge University Press, 2001
• Rotman, J.J., Advanced Modern Algebra, JAMS, 2010.

• Provide an introduction to the general topology through a selection of topics

• Define certain notions associated with the topological spaces
• Recall the proofs of certain facts related to bases, subbases , subspaces and quotient spaces
• Prove certain results concerned with continuous functions and homeomorphisms
• Compare the product and box topologies
• Prove various results regarding convergence in the topological spaces
• Prove the Urysohn’s lemma, Urysohn’smetrization theorem, and the Tietze extension theorem
• Reproduce certain fundamental results regarding the compact spaces
• Discuss various aspects of connected, path connected and locally connected spaces

• Topological Spaces and Continuous functions: Definition of topology, bases and subbases, closed sets and limit points, subspaces, continuous functions,  homeomorphisms, the product topology, the weak topology, quotient spaces,
• Convergence: Sequences, nets, ultranets filters, ultra filters.
• Compactness and Connectedness: 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
• The Separation Axioms and Countability Axioms: space, space,  Hausdorff space, regular space, completely regular space, Tychnoff space, normal space, perfectly normal space, completely normal space,  the countability axioms, Urysohn’s Lemma,Urysohn’smetrizationtheorem, the Tietze Extension theorem.

•  Lectures, Tutorials, Problem solving, Use of e-resources and Handouts.

• In-course Assessments 30%
• End-of-course Examination 70%

• James R. Munkres, Topology, 2^nd Edition, Prentice-Hall, 2000.
• James Dugundji, Topology, University Book stall, 1972.
• Stephen Willard, General Topology, Dove publications, 2004.
• C.Wayne Patty, Foundations of Topology, PWS-Kent publishing Company, 2009.

• Introduce the fundamental concepts of normed linear space and Banach space
• Provide a clear notion of concepts of linear functional and dual space
• Understand the concepts of inner product space and Hilbert space

• Explain the fundamental concepts of normed linear space and Banach space
• Discuss equivalence of norms and its properties
• Examine the convergence and absolute convergence of a series in Banach space
• Recall the concepts of a bounded linear functional and its properties
• Analyze the properties and applications of dual spaces
• Prove Hahn Banach theorems in real and complex normed linear spaces
• Define inner product space and Hilbert space
• Prove Riesz  representation theorem, Bessel’s inequality and Parseval identity
• Make use of separability of a Hilbert space for the existence of orthonormal basis

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

• Lectures,  Tutorial discussion,  use of e-resources and Handouts

• In-course assessment                                              30%
• End of course Examination                                    70%

• John B. Conway, A Course in Functional Analysis, Springer, 1997
• Bryan P. Rynne , Martin A. Youngson, Linear Functional Analysis, Springer, 2008
• Walter Rudin, Functional Analysis, McGraw – Hill, 1991
• Erwin Kreyszig, Introductory Functional Analysis with Applications, Wiley1989
• FrigyesRiesz and BelaSz.-Nagy, Functional Analysis, Dover Publications, 1990

Hourly Breakdown

• Understand the numerical methods for solving large systems of linear equations
• Recognize the underlying mathematical concepts of computer aided numerical algorithms
• Understand the iterative methods for computing eigenvalues of large matrices

• Outline the fundamental concepts in numerical linear algebra
• Apply the matrix factorization algorithms to solve system of linear equations
• Determine bounds for relative error in the solution of a system of linear equations
• Examine the convergence of iterative methods for solving system of linear equations
• Apply  iterative methods to solve a system of linear equations
• Examine the convergence of iterative methods for computing the eigenvalues of matrix
• Apply  iterative methods to compute eigenvalues of a matrix
• Apply the Grahm-Schmidt orthogonalization process to a matrix
• Solve the linear systems by using readily available software

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

• Lectures,  Tutorials,  Problem solving, Use of e-resources and Handouts

• In-course Assessments           30%
• End-of-course Examination 70%

• Trefethen, N. andBau, D., Numerical Linear Algebra, SIAM, 1997.
• Golub, G. and Charles, V. L., Matrix Computations, John Hopkins University Press, 1996.
• Beilina, L., Karchevskii, E. and Karchevskii, M., Numerical Linear Algebra: Theory and Applications, Springer, 2017.

Hourly Breakdown

• 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.
• Enable to investigate and apply standard mathematical programming problems.

• Formulate  mathematical models for solving given word problems
• Sketch the qualitative solution of the formulated model problems involving Differential equations
• Modify  simple models for the change of environment
• Solve single species population models
• Discuss interacting two species population models
• Solve deterministic programming problems using dynamic programming algorithm
• Apply solution methods of unconstrained nonlinear extremum problems
• Make use of Lagrangian MultipliersmethodandKarush-Kuhn-Tucker(KKT) conditions to locate local minimizers
• ApplyWolfe’s algorithm for solving quadratic programming problems
• Apply separable programming algorithm to solve nonlinear programming problems

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. Lotka-Volterra prey-predator model, formulation, solution, interpretation and limitations. Lotka-Volterra 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 Karush-Kuhn-Tucker 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.

• Lectures,  Tutorials, Problem solving, Use of e-resources and Handouts

• In-course Assessments 30%
• End-of-course Examination 70%

•  Winston, W. L., Introduction of Mathematical Programming Applications and Algorithms, Duxbury press, California, 1995.
• Hillier, F. S. and Lieberman, G.J., Introduction to Operations Research, 7^th edition, McGrawHill, New York, 2001.
• Taha, H. A., Operations Research an Introduction, 8^th edition, Pearson Prentice Hall, New Jersey, 2007.
• Braun, M. ,Coleman, C. S. and Drew,D.A., Vol. 1, Vol. 2 and Vol.3 – Differential Equation Models,Springer-Verlag, New York, 1983.
• Meyer,W.,  Concepts of Mathematical Modeling, McGraw Hill, New York, 1994.

• Provide in depth knowledge of classical number theory through a study of selection of topics.
• Develop an understanding of the core ideas and concepts of fundamental and advanced counting techniques
• Recognize the power of abstraction and generalization, and apply logical reasoning to investigate some combinatorial problems

• Recall certain theorems in the theory of congruences
• Relate various number theoretic functions through  Möbius inversion formula
• Discuss various properties of Euler  function
• Prove various results regarding primitive roots of prime numbers and composite numbers
• Recall Euler’s criterion and its consequences
• Discuss various properties of Legendre symbol, notably quadratic reciprocity law
• Solve various quadratic congruences
• Solve certain non-linear Diophantine equations
• Prove various results regarding certain numbers of special form
• Solve common counting problems using elementary counting techniques involving the multiplication rule, permutations, and combinations
• Apply the Principle of inclusion and exclusion to solve variety of counting problems
• Apply the recurrence relations to model a wide variety of counting problems and solve them
• Make use of generating functions to solve many type of counting problems subject to variety of constraints, and solve recurrence relations
• Identify the best counting technique  for given counting problem

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: Inclusion-Exclusion: The principle of inclusion-exclusion, an alternative form of inclusion-exclusion, the Sieve of Eratosthenes, derangementsRecurrence Relations: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: 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.

• Lectures,  demonstration, tutorial discussion, problem solving, use of e-resources and handouts

• In course Assessments 30%
• End of course Examination  70%

• Kenneth H. Rosen, Discrete Mathematics and its Applications, 7^th edition, McGraw Hill, 2012.
• Bernard Kolman, Robert C. Busby, S. C. Ross, Discrete Mathematical Structures, 4^th edition, Prentice Hall, 2001
• Grimaldi R.P, Discrete and combinatorial mathematics: an applied introduction, 5^th edition, Pearson education Inc, 2004
• Miklos Bona, Introduction to Enumerative Combinatorics, McGraw Hill (2007).

Hourly Breakdown

Theory

Practical

• Provide an in depth understanding of compactness in the topological spaces
• Introduce the basic concepts of algebraic topology through a selection of topics

• Discuss the processes of one point and  Stone–Cechcompactifications
• Examine the relationship between various types of compactness in the matric space
• Prove the Tychonoff theorem
• Compare the topology of pointwise convergence, topology of compact convergence and compact-open topology in a function space
• Prove various  results concerned with homotopicity of maps
• Identify covering maps and covering spaces for certain topological spaces
• Recall certain results and their proofs about fundamental groups.
• Identifythe fundamental group of
• Prove the  Fundamental theorem of Algebra and theBorsuk-Ulam theorem
• Prove certain facts of the singular homology theory
• Classify various types of knots

• Compactness: 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
• The Fundamental group and Covering Spaces: 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  retracts and homotopy type

• Homology Theory:  standard – simplex, boundary operator, – cycles, homology groups, induced homomorphism.
• Introduction to Knots: types of knots, preliminary results.

• Lectures, Tutorials, Problem solving, Use of e-resources and Handouts.

•  In-course Assessments 30%
• End-of-course Examination 70%

• James R. Munkres, Topology, 2^nd Edition, Prentice-Hall, 2000.
•  B.K.Lahiri, A First course in Algebraic Topology, Narosa, 2000.
• C.Kosniowski,  A First course in Algebraic Topology, Cambridge University press, Cambridge , 2008
• A.Hatcher, Algebraic topology, Cambridge University press, Cambridge, 2002.
• SashoKalajdzievski,  Topology and Homotopy,  CRC press, 2015.