Scheme of Studies M .Sc Mathematics

Total Semesters = 4

Duration of Each Semester = 18 Weeks

Cumulative Credits of M Sc (4 Semesters) = 68

COMPULSORY COURSES

SEMESTER-I

No Course Code Course Title Cr. Hours

1 MATH-501 Real Analysis-I 3

2 MATH-503 Abstract Algebra 3

3 MATH-505 Topology 3

4 MATH-507 Vector & Tensor Analysis 3

5 MATH-509 Set Theory and ODEs 3

6 MATH-511 Introduction to Computers and its Applications 2+1

Total 18

SEMESTER-II

No Course Code Course Title Cr. Hours

1 MATH-502 Real Analysis-II 3

2 MATH-504 Linear Algebra 3

3 MATH-506 Functional Analysis 3

4 MATH-508 Mechanics 3

5 MATH-510 Complex Analysis 3

6 MATH-512 Computer Programming with C++ 2+1

Total 18

SEMESTER-III

No Course Code Course Title Cr. Hours

1 MATH-601 Differential Geometry 3

2 MATH-603 Partial Differential Equations 3

3 MATH-605 Numerical Analysis-I 3

Elective Courses

Pure Group Applied Group

No Course Code Course Title Cr. Hours No Course Code

Course Title

Cr. Hours

1 MATH- XXX Elective-I* 3 1 MATH- XXX

Elective-I**

3

2 MATH- XXX Elective-II* 3 2 MATH- XXX

Elective-II**

3

Computational Group

No Course Code Course Title Cr. Hours

1 MATH- XXX Elective-I*** 3

2 MATH- XXX Elective-II*** 3

Total 15

SEMESTER-IV

No Course Code Course Title Cr. Hours

1 MATH-602 Probability Theory 3

2 MATH-604 Integral Equations 3

3 MATH-606 Numerical Analysis-II 3

4 MATH-608 History of Mathematics 2

Elective Courses

Pure Group Applied Group

No Course Code Course Title Cr. Hours No Course Code

Course Title

Cr. Hours

1 MATH-XXX Elective-I* 3 1 MATH- XXX

Elective-I** 3

2 MATH-XXX Elective-II* 3 2 MATH- XXX

Elective-II**

3

Computational Group

No Course Code Course Title Cr. Hours

1 MATH-XXX Elective-I*** 3

2 MATH-XXX Elective-II*** 3

Total 17

Common Elective Courses

No Course Code Course Title Cr. Hours

1 MATH-610 Special Functions 3

2 MATH-611 Computer Programming with Fortran 2+1

3 MATH-612 Advanced Programming for Scientific Computing 2+1

List A Courses of Pure Mathematics

No Course Code Course Title Cr. Hours

1 MATH-613 Analytic Number Theory 3

2 MATH-614 Algebraic Number Theory 3

3 MATH-615 Advanced Group Theory –I 3

4 MATH-616 Advanced Group Theory –II 3

5 MATH-617 Algebraic Topology-I 3

6 MATH-618 Algebraic Topology-II 3

7 MATH-619 Category Theory –I 3

8 MATH-620 Category Theory –II 3

9 MATH-621 Rings and Fields 3

10 MATH-622 Theory of Modules 3

11 MATH-623 Lie Algebra 3

12 MATH-624 Advanced Functional Analysis 3

13 MATH-625 Galois Theory 3

14 MATH-627 Measure Theory 3

List B Courses of Applied Mathematics

No Course Code Course Title Cr. Hours

1 MATH-629 Fluid Mechanics-I 3

2 MATH-630 Fluid Mechanics-II 3

3 MATH-631 Quantum Mechanics-I 3

4 MATH-632 Quantum Mechanics-II 3

5 MATH-633 Electromagnetic Theory –I 3

6 MATH-634 Electromagnetic Theory –II

3

7 MATH-635 Special Relativity 3

8 MATH-636 Elasticity Theory 3

9 MATH-637 Analytical Dynamics 3

10 MATH-639 Astronomy-I 3

11 MATH-640 Astronomy-II 3

List C Courses of Computational Mathematics

No Course Code Course Title Cr. Hours

1 MATH-641 Operations Research-I 3

2 MATH-642 Operations Research-II 3

3 MATH-643 Methods of Optimization-I 3

4 MATH-644 Methods of Optimization-II 3

5 MATH-645 Theory of Splines-I 3

6 MATH-646 Theory of Splines-II 3

7 MATH-647 Graph Theory 3

8 MATH-648 Theory of Automata 3

9 MATH-649 Control Theory 3

10 MATH-650 Applied Matrix Theory 3

11 MATH-651 Finite Element Analysis 3

* These courses are optional and can be selected from list A or common electives. ** These courses are optional and can be selected from list B or common electives. *** These courses are optional and can be selected from list C or common electives.

Course Contents for M.Sc Mathematics Course contents for M. Sc mathematics are given below semester wise.

SEMESTER-I

Course Title: Real Analysis-I Course Code: MATH-501 Credit Hours: 3 Objectives of the course: This is the first course in analysis. It develops the fundamental ideas of analysis and is aimed at developing the students’ ability in reading and writing mathematical proofs. Another objective is to provide sound understanding of the axiomatic foundations of the real number system, in particular the notions of completeness and compactness. Course Contents: Number Systems: Ordered fields, rational, real and complex numbers, Archimedean property, supremum, infimum and completeness. Topology of real numbers: Convergence, completeness, completion of real numbers, open sets, closed sets, compact sets, Heine Borel theorem, connected sets. Sequences and Series of Real Numbers: Limits of sequences, algebra of limits. Bolzano Weierstrass theorem, Cauchy sequences, liminf, limsup, limits of series, convergences tests, absolute and conditional convergence, power series. Continuity: Functions, continuity and compactness, existence of minimizers and maximizers, uniform continuity, continuity and connectedness, intermediate mean value theorem, monotone functions and discontinuities. Differentiation: Mean value theorem, L’Hopital’s Rule, Taylor’s theorem. Recommended Books:

1. Walter Rudin, Principles of Mathematical Analysis, 3rd Ed, 1976. 2.T.M. Apostal, Mathematical Analysis, Addison Wesley, 1957. 3.W.Kaplan, Advanced calculus, Addison Wesley, 2002. 4. R.L. Rabenstein, Elements of Ordinary differential equations, Academic Press,

1984. 5. Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis, Wiley,

3rd Ed, 1999. 6. Halsey Royden, Real Analysis, 3rd Ed. Prentice Hall, 1988. 7. H.L. Royden, Real Analysis, 3rd Ed, 1989. 8. S. Lang, Analysis I, Addison-Wesley Publ. Co., Reading, Massachusetts, 1968. 9. G. M. Habibullah, Real Analysis, Ilmi Kitab Khana, Lahore, Pakistan, 2002. 10. A. Kumar, S. Kumaresan, A Basic Courese in Real Analysis, CRC Press, 2014.

Course Title: Abstract Algebra Course Code: MATH-503 Credit Hours: 3 Course Contents:

Cyclic groups, Cosets decomposition of a group, Lagrange’s theorem and its

consequences, Conjugacy classes, Centralizers and Normalizers, Normal Subgroups,

Homomorphism of groups, Cayley’s theorem, Quotient groups, Fundamental theorem of

homomorphism, isomorphism theorems, Endomorphism and automorphisms of groups,

Commutator subgroups, Permutation groups, p-Subgroups, Sylow Theorems,Definition

and examples of Rings, Special classes of rings, Fields, Ideals, Ring homomorphism.

Recommended Books:

1. J.J.Rottman, The Theory of Groups: An Introduction, Allyn & Bacon, Boston,

1965.

2. J.Rose, A Course on Group Theory, C.U.P. 1978.

3. I.N. Herstein, Topics in Algebra, 2nd Ed, Wiley, 1975.

4. I.D. MacDonald, The Theory of Groups, Oxford University Press, 1968. 5. Paul M. Cohn, Basic Algebra, Springer, 2002.

6. D.Burton, Abstract and Linear Algebra, Addison-Wesley publishing Co. 1972.

7. P.B. Battacharya, S.K.Jain and S.R.Nagpaul, Basic abstract Algebra, C.U.P.

1995.

8. N. Jacobson, Basic Algebra, Vol. l & II, WH Freeman NY, 1989. 9. Joseph A. Gallian, Contemporary Abstract Algebra, Houghton Mifflin College

Div, 6 Student Ed, 2004.

Course Title: Topology Course Code: MATH-505 Credit Hours: 3 Course Contents:

Topological spaces, bases and sub-bases, first and second axiom of countability,

separability, continuous functions and homeomorphism, finite product space.

Separation axioms (T0, T1, T2), Techonoff space, Regular spaces, completely regular

spaces, normal spaces, compact spaces, connected spaces. Recommended Books:

1. Sheldon W. D., 2005. Topology. 1st ed. NY: McGraw Hill.

2. Lipschutz S. 1968. General Topology, Schaum’s outline series. NY:McGraw Hill.

3. Munkers J.R., 2006.Topology. 2nd ed. NJ: Pearson Prentice Hall.

4.Simon G.F. 1963.Introduction to Topology and Modern Analysis. 1st ed. NY:

McGraw Hill.

5. J. Willard,1970. General Topology. 1st ed. NY:Addison-Wesley.

6. Armstrong M.A., 1979. Basic Topology. 1st ed. NY: McGraw Hill.

Course Title: Vector and Tensor Analysis

Course Code: MATH-507 Credit Hours: 3 Objectives of the course: This course shall assume background in calculus. It covers basic principles of vector analysis, which are used in mechanics Course Contents: Vector Analysis: Gradient, divergence and curl of point functions, expansion formulae, curvilinear coordinates, line, surface and volume integrals, Gauss’s, Green’s and Stoke’s theorems. Cartesian Tensors: Summation convention, proper and improper transformation, transformation equations, orthogonally conditions, Kronecker tensor and Levi-civita tensor, tensors of different ranks, inner and outer products, contraction, quotient theorems, symmetric and anti symmetric tensors, Application to Vector Analysis.

Recommended Books: 1. M.R. Spiegel, Vector and an Introduction to Tensor Analysis, Mcgraw Hill Book

company, 2009. 2. F. Chorlton, vector and Tensor Methods, Eills Horwood Publisher, Chichester,

U.K. 1977. 3. Dr. Nawazish Ali Shah, Vector and Tenser Analysis, 4th Ed, 2005. 4. E.C. Young, Vector and Tensor Analysis, Mareel Dekker, Inc, 1993.

Course Title: Set Theory and ODEs Course Code: MATH-509 Credit Hours: 3

Set Theory: Equivalent sets, countable and uncountable sets, the concept of cardinal

numbers, addition and multiplication of cardinals, Cartesian product as sets of functions,

addition and multiplication of ordinals, partially ordered sets, axiom of choice,

Special Functions: The Gamma function , The Beta Function Hyper geometric, Solution

in series of Bessel differential equation, recurrence formulas for Jn (x), series solution of

legendre differential equation, rodrigues formula for polynomial Pn (x), Generating

function for Pn (x), recurrence relations and the orthogonality of Pn (x) functions.

Recommended Books:

1. Kezysztof C, Set Theory for Working Mathematician, 1st Ed. Cambridge University

Press. 1997.

2. Felix Hausdorff, Set Theory, 1st Ed. AMS Chelsea Publishing. 2005.

3. Patrick Suppes, Axiomatic Set Theory, Dover Publications, Inc, New York. 1972.

4. P.R.Halmos, Naïve Set Theory, Springer. 1998.

5. Nico M. Temme, Special Functions: An Introduction to the Classical Functions of

Mathematical Physics. 1st Ed. Wiley-Interscience. 1995.

6. Z.X. Wang, D.R. Guo, Special Functions. 1st Ed. World Scientific Pub. Co. inc.

1989.

Course Title: Introduction to Computers and its Applications Course Code: MATH-511 Credit Hours: 3 Fundamental Concepts of Computer Systems: Basic Computer Organization, Number Systems and codes, Processor and Memory, Secondary Storages, Input(output) Units, Computer Softwares, Internet Basic, History and Classification of Computers, Practically using Windows, practically using Software Packages (MS Word and MS Excel), Practically using some Computer Algebra System CAS (Matlab/Maple etc). Recommended Books:

1. P.K Sinha, Computer Fundamentals, 3rd Ed. SAMS Publications for MS Windows and MS Office, 2004.

2. Gilate, A., MATLAB: An Introduction with Applications, 2nd Ed. John Wiley & Sons Inc. 2004.

3. Amos Gilat, MATLAB: An Introduction with Applications - Paperback (Jan 2, 2008).

4. Robert Lafore, The Waite Group's C Programming Using Turbo C++/Book and Disk (The Waite Group), (Paperback - Oct 1993)

5. Shan Sun Kuo, Computer Applications of Numerical Methods, (Hardcover - Jun 1972)

SEMESTER-II Course Title: Real Analysis II Course Code: MATH-502 Credit Hours: 3 Objectives of the course: A continuation of Real Analysis I, this course will continue to cover the fundamentals of real analysis, concentrating on the Riemann-Stieltjes integrals, Functions of Bounded Variation, Improper Integrals, and convergence of series. Emphasis would be on proofs of main results. Course Contents: The Riemann-Stieltjes Integrals: Definition and existence of integrals, properties of integrals, Real Valued Functions of Several Variables: Continuous real valued functions, Partial derivatives and differentials, Geometric interpretation of differentiability, Chain rule, Taylor’s theorem. Maxima and Minima, Vector Valued Functions of Several Variables Linear transformations and matrices, Continuous and differentiable transformations, Chain rule for transformations, Inverse function theorem, Implicit function theorem, Jacobians, Method of Lagrange multipliers. Functions of Bounded Variation: Definition and examples, properties of functions of bounded variation. Improper Integrals:Types of improper integrals, tests for convergence of improper integrals, absolute and conditional convergence of improper integrals. Sequences and Series of Functions: Power series, definition of point-wise and uniform convergence, uniform convergence and continuity, uniform convergence and differentiation, examples of uniform convergence. Recommended Books:

1. Walter Rudin, Principles of Mathematical Analysis, 3rd Ed. 1976. 2. T.M. Apostal, Mathematical Analysis, 2nd Ed. Addison Wesley, 1974. 3. W.Kaplan, Advanced calculus, 5th Ed. Addison Wesley, 2002. 4. R.L. Rabenstein, Elements of Ordinary differential equations, Academic Press,

1984. 5. Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis, 3rd

Ed. 1999. 6. Walter Rudin, Principles of Mathematical Analysis, 3rd Ed. McGraw-Hill Inc.

1976.

Course Title: Linear Algebra Course Code: MATH-504 Credit Hours: 3

Subspaces, Bases, Dimension of a vector space, Quotient space, Change of bases,

Linear Transformation and matrices, Inner Product Spaces and Orthognality, Orthogonal

subspaces, Rank and Nullity of linear transformation, Eigen values and Eigen vectors,

Characteristic equation, Similar matrices, Diagonalization of matrices, Orthogonal and

Orthonormal sets, Gramm Schmidt process of orthognalizations, Characteristic equation,

Dual spaces.

Recommended Books:

1. I.N. Herstein, Topics in Algebra, Xerox Publishing Company, 1975.

2. Dr. Malik, J.N. Mordeson, M.R. Sen, Fundamental of Abstract Algebra,

McGraw Hill companies, Inc. 1987.

3. P.M. Cohn, Algebra, Vol.I, London: John Wiley, 1989.

4. D. Burton, Abstract and Linear Algebra, Addison-Wesley publishing Co. 1986.

5. N. Jacobson, Basic Algebra, Vol.II Freeman, 1989.

6. Dr. Karamat Hussain, Linear Algebra, 1st edition, 2007.

Course Title: Functional Analysis Course Code: MATH-506 Credit Hours: 3 Objectives of the course: This course extends methods of linear algebra and analysis to spaces of functions, in which the interaction between algebra and analysis allows powerful methods to be developed. The course will be mathematically sophisticated and will use ideas both from linear algebra and analysis. Course Contents: Metric Spaces: Convergence, Cauchy’s sequences and examples, Completeness of metric space, completeness proofs. Banach Spaces: Normed linear Spaces, Banach Spaces, Quotient Spaces, Continuous and bounded linear operators, Linear functional, Linear operator and functional on finite dimensional Spaces. Hilbert Spaces: Inner product Spaces, Hilbert Spaces (definitions and examples), conjugate spaces, representation of linear functional on Hilbert space, reflexive spaces. Recommended Books:

1. E. Kreyszig, Introduction to Functional Analysis with Applications, John Wiley

and sons, 1989.

2. N. Dunford and J.T. Schwartz, Linear Operators (part-1 General theory),

Interscience publishers, New York, 1958.

3. Seymour Lipschutz, Outline of General Topology, 2011.

Course Title: Mechanics Course Code: MATH-508 Credit Hours: 3 Objectives of course: To provide solid understanding ofclassical mechanics and enable the students to use thisunderstanding while studying courses on quantum mechanics,statistical mechanics, electromagnetism, fluid dynamics, and astrodynamics. Course Outline: General introduction. Projectile motion with air resistance. Applications of the principle of conservation of energy. Centre of mass of a system of particles, linear momentum, angular momentum and K.E. with respect to the centre of mass. Motion of a rigidbody, translation and rotation, linear and angular velocity of a rigid body about a fixed axis,moments and products of inertia. Parallel and perpendicular axistheorems.General

motion ofrigid bodies in space. Angular momentum and moment of inertia.Principal axes and principal moments of inertia. Determination ofprincipal axes by diagonalizing the inertia matrix.Equimomental systems.Rotating axes theorem. Euler’s dynamical equations. Free rotation of arigid body with three different principal moments of inertia,torque free motion of a symmetrical top.Eulerian angles and rigid body motion. Orbital motion. Recommended Books:

1.DiBenedettoE, (2011) Classical Mechanics. Theory and Mathematical Modeling,Birkhauser Boston.

2. John R. Taylor,(2005)Classical Mechanics, University of Colorado. 3. Goldstein H, (1980) Classical Mechanics, Addison-Wesley PublishingCo. 4. SpiegelM. R,(2004)Theoretical Mechanics, 3rd Edition, Addison-Wesley

Publishing Company. 5. FowlesG. R. and CassidayG. L,(2005)Analytical Mechanics, 7thedition,

Thomson Brooks/COLE, USA, 6. Richard Fitzpatrick,(2006) Classical Mechanics.The University of Texas at

Austin. 7. K. Sankara Rao, (2005) Classical Mechanics.New Delhi-11001 8. Mir K.L. (2007) Theoretical Mechanics:IlmiKetabKhana. Lahore.

Complex Analysis Course Code: MATH-510 Prerequisite(s): Calculus-II Credit Hours: 3+0 Objectives of the course: This is an introductory course in complex analysis, giving the basics of the theory along with applications, with an emphasis on applications of complex analysis and especially conformal mappings. Students should have a background in real analysis (as in the course Real Analysis I), including the ability to write a simple proof in an analysis context. Course Contents: Introduction: The algebra of complex numbers, Geometric representation of complex numbers, Powers and roots of complex numbers. Functions of Complex Variables: Definition, limit and continuity, Branches of functions, Differentiable and analytic functions. The Cauchy-Riemann equations, Entire functions, Harmonic functions, Elementary functions: The exponential, Trigonometric, Hyperbolic,Logarithmic and Inverse elementary functions, Open mapping theorem. Maximum modulus theorem. Complex Integrals: Contours and contour integrals, Cauchy-Goursat theorem, Cauchy integral formula, Lioville’s theorem, Morerea’s theorem. Series: Power series, Radius of convergence and analyticity, Taylor’s and Laurent’s series, Integration and differentiation of power series. Singularities, Poles and residues: Zero, singularities, Poles and Residues, Types of singular points, Calculus of residues, contour integration, Cauchy’s residue theorem with applications. Mobius transforms, Conformal mappings and transformations. Recommended Books:

1. R. V. Churchill, J. W. Brown, Complex Variables and Applications ,5th edition, McGraw Hill, New York, 1989.

2. J. H. Mathews and R. W. Howell, Complex Analysis for Mathematics and Engineering, 2006.

3. S. Lang, Complex Analysis, Springer-Verlag, 1999. 44. R. Remmert, Theory of Complex

Functions, Springer-Verlag, 1991. 5. W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.

Course Title: Computer Programming with C++ Course Code: MATH-512 Credit Hours: 3 Objectives of the course: The purpose of this course is to introduce students to operating systems and environments Course Contents:

Introduction to operating systems, C language, building blocks, variables, input/output, loops (FOR, WHILE, DO), decisions (IF, IF ELSE, ELSE IF) construct switch statement, conditional statement, function hat returns a value using argument to pass data to another function, external variable, arrays and strings, pointers, structure, files and introduction to C++ Recommended Books:

1. Aho A.V. , Ulman J.D., 1995. Foundation of Computer Science. 1st ed. NY: Computer Science Press, WH Freeman.

2. Hein J.L., 1996. Theory of Computation: An Introduction. 1st ed. Boston: Jones and Bartlett.

Semester III

Course Title: Differential Geometry Course Code: MATH-601 Credit Hours: 3 Objectives of the course: The course provides a foundation to solve partial differential equations with special emphasis on wave, heat and laplace equations. Formulation and some theory of these equations are also intended. Course Contents:

Space Curves: Arc length, Tangent, Normal and Binormal, Curvature and Torsion of a

Curve, Tangent Surface, Spherical Indicatrix, Involutes and Evolutes, Envelopes,

Existence Theorem for a Space Curve, Helices, Curves on Surfaces, Surfaces of

Revolution, Helicoids, Families of Curves, Developable associated with Space Curves,

Developable associated with Curves on Surfaces, The First and Second Fundamental

form, Principle Curvatures, Lines of Curvature, Geodesics. Recommended Books:

1. R.S. Millman and G.D. Parker, Elements of Differential Geometry, Prentice

Hall, 1977.

2. T.J. Wilmore, An Introduction to Differential Geometry, Oxford Calarendon

Press, 1959.

3. C.E. Weatherburn, Differential Geometry, Cambridge University Press, 1955.

4. A. Pressley, Elementary Differential Geometry, Springer Verlag. 2001.

5. D. Somasundaran, Differential Geometry, Narosa Publishing House New Delhi,

2005.

Course Title: Partial Differential Equations Course Code: MATH-603 Credit Hours: 3+0 Objectives of the course: Partial Differential Equations (PDEs) are at the heart of applied mathematics and many other scientific disciplines. The course aims at developing understanding about fundamental concepts of PDEs theory, identification and classification of their different types, how they arise in applications, and analytical methods for solving them. Special emphasis would be on wave, heat and Laplace equations. Course Contents: First order PDEs: Introduction, formation of PDEs, solutions of PDEs of first order, The Cauchy’s problem for quasilinear first order PDEs, First order nonlinear equations, Special types of first order equations Second order PDEs: Basic concepts and definitions, Mathematical problems, Linear operators, Superposition, Mathematical models: The classical equations, the vibrating string, the vibrating membrane, conduction of heat solids, canonical forms and variable, PDEs of second order in two independent variables with constant and variable coefficients, Cauchy’s problem for second order PDEs in two independent variables .Methods of separation of variables: Solutions of elliptic, parabolic and hyperbolic PDEs in Cartesian and cylindrical coordinates .Laplace transform: Introduction and properties of Laplace transform, transforms of elementary functions, periodic functions, error function and Dirac delta function, inverse Laplace transform, convolution theorem, solution of PDEs by Laplace transform, Diffusion and wave equations. Fourier transforms: Fourier integral representation, Fourier sine and cosine representation, Fourier transform pair, transform of elementary functions and Dirac delta function, finite Fourier transforms, solutions of heat, wave and Laplace equations by Fourier transforms.

Recommended Books: 1. Humi M, Miller W.B; Boundary Value Problems and Partial Differential

Equations. PWS- KENT Publishing Company, 1991. 2. Myint UT, Partial Differential Equations for Scientists and Engineers, 3rdedition,

North Holland, Amsterdam, 1987. 3. Dennis G. Zill, Michael R. Cullen, Differential equations with boundary value

problems, Brooks Cole, 2008. 4. John Polking, Al Boggess, Differential Equations with Boundary Value

Problems, 2nd Edition, Pearson,July 28, 2005.

5. J. Wloka, Partial Differential Equations, Cambridge University press, 1987.

Course Title: Numerical Analysis-I Course Code: MATH-605 Credit Hours: 3 Objectives of the course: This course is designed to teach the students about numerical methods and their theoretical bases. The course aims at inculcating in the students the skill to apply various techniques in numerical analysis, understand and do calculations about errors that can occur in numerical methods and understand and be able to use the basics of matrix analysis. It is optimal to verifying numerical methods by using computer programming (MatLab, Maple, C++, etc)

Course Contents: Error analysis: Floating point arithmetic, approximations and errors. Methods for the solution of nonlinear equations: Bisection method, regula-falsi method, fixed point iteration method, Newton-Raphson method, secant method, error analysis for iterative methods. Interpolation and polynomial approximation: Forward, backward and centered difference formulae, Lagrange interpolation, Newton’s divided difference formula, Interpolation with a cubic spline, Hermite interpolation, least squares approximation. Numerical differentiation and Integration: Forward, backward and central difference formulae, Richardson’s extrapolation, Newton-Cotes formulae, Numerical integration: Rectangular rule, trapezoidal rule, Simpson’s 1/3 and 3/8 rules, Boole’s and Weddle’s rules, Gaussian quadrature. Numerical solution of a system of linear equations: Direct methods: Gaussian elimination method, Gauss-Jordan method; matrix inversion; LU-factorization; Doolittle’s, Crout’s and Cholesky’s methods, Iterative methods: Jacobi, Gauss-Seidel and SOR. Eigenvalues problems: Introduction, Power Method, Jaccobi's Method. The use of software packages/ programming languages for above mentioned topics is recommended. Recommended Books:

1. C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis, Pearson Education,

Singapore, 2005. 2. R. L. Burden and J. D. Faires: Numerical Analysis, latest edition, PWS Pub. Co.

3. J.H. Mathews, Numerical Methods for Mathematics, latest Edition, Prentice Hall International. 4. S. C. Chapra and R. P. Canale: Numerical Methods for Engineers, 6th edition,

McGraw Hill. 5. Sankara K. 2005. Numerical Methods for Scientists and Engineers. 2nd ed. New

Delhi: Prentice Hall.

SEMESTER-IV

Probability Theory Course Code: MATH-602 Prerequisite(s): None Credit Hours: 3+0 Objectives of the course: A prime objective of the course is to introduce the students to the fundamentals of probability theory and present techniques and basic results of the theory and illustrate these concepts with applications. This course will also present the basic principles of random variables and random processes needed in 24 applications. Course Contents: Finite probability spaces: Basic concept, probability and related frequency, combination of events, examples, independence, random variables, expected value, standard deviation and Chebyshev's inequality, independence of random variables, multiplicatively of the expected value, additivity of the variance, discrete probability distribution.Probability as a continuous set function: Sigma-algebras, examples, continuous random variables, expectation and variance, normal random variables and continuous probability distribution. Applications: De Moivre-Laplace limit theorem, weak and strong law of large numbers, the central limit theorem, Markov chains and continuous Markov process. Recommended Books:

1. M. Capinski, E. Kopp, Measure, Integral and Probability, Springer-Verlag, 1998. 2. R. M. Dudley, Real Analysis and Probability, Cambridge University Press, 2004. 3. S. I. Resnick, A Probability Path, Birkhauser, 1999.

4. S. Ross, A first Course in Probability Theory, 5th ed., Prentice Hall, 1998. 5. Robert B. Ash, Basic Probability Theory, Dover. B, 2008. 6. Chaudhry, S.M. and Kamal, S. (2008), Introduction to Statistical Theory, Part I,

II, 8th ed, Ilmi Kitab Khana, Lahore, Pakistan.

Course Title: Integral Equations Course Code: MATH-604 Credit Hours: 3+0 Objectives of the course: Many physical problems that are usually solved by differential equation methods can be solved more effectively by integral equation methods. This course will help students gain insight into the application of advanced mathematics and guide them through derivation of appropriate integral equations governing thebehavior of several standard physical problems.

Course Contents: Linear integral equations of the first kind, Linear integral equations of the second kind. Relationship between differential equation and Volterra integral equation. Neumann series. Fredholm Integral equation of the second kind with separable Kernels. Eigenvalues and eigenvectors. Iterated functions. Quadrature methods. Least square methods. Homogeneous integral equations of the second kind. Fredholm integral equations of the first kind. Fredholm integral equations of the second kind. Abel’s integral equations. Hilbert Schmidt theory of integral equations with symmetric Kernels. Regularization and filtering techniques. Recommended Books:

1.A. J. Jerri; Introduction to Integral Equations with Applications second edition. Sampling Publishing, 2007. 2 .W. V. Lovitt, Linear Integral Equations, Dover Publications, 2005 3. C. T. H. Baker, Integral Equations, Clarendon Press, 1977. 4. F. Smithies, Integral Equations, Cambridge University Press, 1989. 5. A. M. Wazwaz, A first Course in Integral Equations, World Scientific Pub., 1989.

Course Title: Numerical Analysis II Course Code: MATH-606 Credit Hours: 3 Objectives of the course: This course is designed to teach the students about numerical methods and their theoretical bases. The students are expected to know computer programming to be able to write program for each numerical method. Knowledge of calculus and linear algebra would help in learning these methods Course Contents: Difference and Differential Equation: Formulation of difference equations, solution of linear (homogeneous and inhomogeneous) difference equations with constant coefficients. The Euler and modified Euler method, Runge-Kutta methods and predictor-corrector type methods for solving initial value problems along with convergence and instability criteria. Finite difference, collocation and variational method for boundary value problems.

Books Recommended:

1. C.Gerald, Applied Numerical Analysis, Addison-Wesley publishing company,

1984.

2. A.Balfour & W.T.Beveridge, Basic Numerical Analysis with FORTARAN,

Heinmann Educational Books Ltd. 1977.

3. Shan and Kuo, Computer Applications of Numerical Methods, Addison-Wesley,

National Book Foundation, Islamabad.

Course Title: History of Mathematics Course Code: MATH-608 Credit Hours: 2 History of Numerations: Egyptian, Babylonian, Hindu and Arabic contributions. Algebra: Including the contributions of Al-Khwarzmi and Ibn Kura. Geometry: the areas, the work of Al-Toussi on Euclud’s axioms, Analysis. The Calculus: Newton, Leibniz and Gauss, The concept of limit, Laplace. Recommended Books:

1. C.B. BOYER AND U.V. MERSBACH, The history of mathematics, 2nd Ed. John Wiley.

2. David M. Burton, The History of Mathematics: An Introduction, 7th Ed. McGraw-Hill, 2010.

Common Elective Courses Course Title: Special Functions Course Code: MATH-610 Credit Hours: 3 Course Contents: The Gamma function: The Weierstrass gamma function, Euler integral representation of gamma function, relations satisfied by gamma function, Euler’s constant, the order symbols o and O, properties of gamma function. Beta function: Definition, integral representation of beta function, relation between gamma and beta functions, properties of beta function, Legendre’s duplication formula, Gauss’ multiplication theorem. Hypergeometric function : Hypergeometric series, the functions F(a,b;c;z) and F(a,b;c;I), integral representation of hypergeometric function, the hypergeometric differential equation, the contiguous relations, simple transformations, a theorem due to Kummer. Confluent Hypergeometric Function: Confluent hypergeometric series, integral representation of confluent hypergeometric function, the confluent hypergeometric differential equation, Kummer’s first formula. Orthogonal polynomials: Simple sets of polynomials, orthogonality, the three term recurrence relation, the Christofell-Darboux formula, normalization, Bessel’s inequality, generating functions, differential equations, recurrence relations . Recommended Books:

1. Rainville, E.D., 1971. Special Functions. 2nd ed. Chelsea Publishing Co. 2. Lebedev, N.N., 1972. Special Functions and their Applications. 2nd ed. Prentice

Hall, 3. Whittaker & Watson, 1978. A Course in Modern analysis. 2nd ed. Cambridge,

University Press.

Course Title: Computer Programming with Fortran Course Code: MATH-611 Credit Hours: 3

Bisection method, Regula Falsi Method, Newton-Raphson method for solving non-linear equations. Gaussian elimination with different pivoting strategies, Jacobi and Gauss-Seidal Iterative methods for systems of simultaneous linear equations. Trapezoidal rule, Simpson’s rule and Gaussian method of numerical integration. Modified and improved Euler’s methods, predictor corrector methods for finding the numerical solution of IVP’s involving ODE’S. Note: Practical examination will be of two hours duration in which one or more computational projects will be examined. Recommended Books:

1. M.L Abell and J.P. Braselton, Mathematica hand book, New York, 1992 T.J. Akai, Applied Numerical Methods, j. Willey, 1994

2. J. Mathews, Numerical Methods for Computer Science, Engineering and

Mathematics, Printice Hall, 1987.

Course Title: Advanced Programming for Scientific Computing Course Code: MATH-612 Credit Hours: 3 Introduction to Object Oriented Programming (OOP), Using OOP Features: Overloading Overriding Inheritance Polymorphism, Writing Programs for Numerical Methods Mixed Programming with FORTRAN/C++/Matlab. Recommended Books:

1. Deitel, H.M. and Deitel, P.J., C++ How to Program, 5th Ed. Deitel Associates, 2007.

2. Chapenman, S.J., FORTRAN 90/95 for Scienctists and Engineers, 2nd Ed. McGraw Hill. Co.

3. Press, W.H. Numerical Recips in FORTRAN 2nd Edition, Cambridge University Press. 1989.

4. Gilate, A. Matlab and Introduction with Applications, John Wiley & Sons Inc. 2008.

COURSES OF PURE MATHEMATICS

Course Title: Analytic Number Theory Course Code: MATH-613 Credit Hours: 3

Divisibility, Euclid’s theorem, Congruences, Elementary properties, Residue classes and

Euler’s function. Linear congruence and congruence of higher degree, Congruences with

prime moduli, The theorems of Fermat, Euler and Wilson., Primitive roots and indices,

Integers belonging to a given exponent, composite moduli Indices, Quadratic Residues,

Composite moduli, Legendre symbol, Law of quadratic reciprocity, the Jacobi symbol,

Number-Theoretic Functions, Mobius function, the function [x], Diophantine Equations,

Equations and Fermat’s conjecture for n = 2, n = 4.

Recommended Books:

1. W.J. Leveque, Topics in Number Theory, Vols.I Addison-Wesley publishing

company. 1956.

2. H. Griffin, Elementary Theory of Numbers, Mc Graw Hill Companies,

Inc, New York. 1970.

3. William J. LeVeque, Topics in Number Theory, Volumes I and II, (Paperback - Nov 7, 2002)

Course Title: Algebraic Number Theory Course Code: MATH-614 Credit Hours: 3

Review of polynomials, irreducible polynomials, Algebraic numbers and integers, Units

and Primes in R [v] ideals. Arithmetic of ideals congruencies, the norm of an ideal. Prime

ideals, Units of algebraic number field.

Application to Rational Number Theory: Equivalence and class number, Cyclotomic field

Kp, Fermat’s equation, Kummer’s theorem, The equation x2 + 2 =y3, pure cubic

fields, Distribution of primes and Riemann’s zeta function. Books Recommended:

1. W.J. Leveque, Topics in Number Theory, Vols. II, Addison-Wesley publishing

company, 1956.

2. I.N. Stewart and D.O. Tall Algebraic Number Theory, 2nd Ed. Chapman and

Hall/CRC Press, 1987. 3. William J. LeVeque, Topics in Number Theory, Volumes I and II, 2002.

Course Title: Advanced Group Theory-I Course Code: MATH-615 Credit Hours: 3

Group of automorphisms, direct products and normal products of groups, holomorph of a

group, characteristic and fully invariant subgroups, cyclic permutations and orbits, the

alternating groups, generators of symmetric and alternating groups, permutation groups,

Simple groups, simplicity of nA , 5n , series in groups, the stabilizer subgroups,

Zassenhau’s Lemma, normal series and their refinements, composition series, principal

or chief series, finitely generated abelian groups, double Cosets, Sylow’s theorems,

applications of Sylow Theorem. Recommended Books:

1. Rottman J.J., 1965.The Theory of Groups: An Introduction. 1st ed. Boston :Allyn

&

Bacon,

2. MacDonald I., 1968. The Theory of Groups. 1st ed. Oxford University Press.

3. Cohn P.M., 1974.Algebra, Vol.I, London: John Wiley.

4. Burton D., 1972.Abstract and Linear Algebra. 1st ed. Addison-Wesley.

5. Battacharya P.B., Jain S.K. and Nagpaul S.R., 1994. Basic abstract Algebra. 2nd

ed. C.U.P.

Course Title: Advanced Group Theory-II Course Code: MATH-616 Credit Hours: 3 Course Contents:

Solvable groups definition and examples, theorems on solvable groups, nilpotent groups,

characterization of finite nilpotent groups, upper and lower central series, the Frattini

subgroups, free groups, basic theorems, definition and examples of free products of

groups, linear groups, types of linear groups, representation of linear groups, group

algebras and representation modules. Recommended Books:

1. MacDonald I., 1968. The Theory of Groups. 1st ed. Oxford University Press.

2. Cohn P.M., 1974.Algebra, Vol.I, London: John Wiley.

3. Burton D., 1972.Abstract and Linear Algebra. 1st ed. Addison-Wesley.

4. Battacharya P.B., Jain S.K. and Nagpaul S.R., 1994. Basic abstract Algebra. 2nd

ed. C.U.P.

5. Jacobson .N.,1989. Basic Algebra, Vol.II. 2nd ed. Freeman.

Course Title: Algebraic Topology-I Course Code: MATH-617 Credit Hours: 3 Homotopy theory, Homotopy theory of path and maps, Fundamental group of circle, Covering spaces, Lifting criterion, Loop spaces and higher homotopy group. Homotopy Theory: Affin spaces, Singular theory, Chain complexes, Homotopy invariance of homology, Relation between n, and H,relative homology The exact homology sequence. Books Recommended:

1. Kosniowski C.A First course in algebraic Topology, C.U.P. 1980. 2. Greenberg M.J & Harper,J.R,Algebraic Topology, A First Course, The Bonjan

Cunning Pub. Co. 3. Croom F.H., Basic Concept of algebraic theory, Spinger-Verlag, New York,

1978. Course Title: Algebraic Topology-II Course Code: MATH-618 Credit Hours: 3 Relative homology, The exact homology sequences, Excion theorem and application to spheres, Mayer Victoris sequences, Jordan-Brouwer separation theorem, Spherical complexes, Betti number and Euler characteristic, Cell Complexes and adjunction spaces. Recommended Books:

1. Kosniowski C., A First course in algebraic Topology, C.U.P. 1980. 2. Greenberg,M.J & Harper,J.R,Algebraic Topology, A First Course, The Bonjan

Cunning Pub, Co. 3. Croom, F.H., Basic Concept of algebraic theory, Spinger-Verlag, New York,

1978. Course Title: Category Theory-I Course Code: MATH-619 Credit Hours: 3 Basic concepts of category, Definition of category, examples, Epimorphism, Monomorphism, Retractions, Initial, Terminal, and null objects, Category of graphs, Limits in categories, Equalizers, Pull backs, Inverse images and intersections, Constructions with kernel pairs, Functions and adjoint Functions, Functions, Bifunctions, Natural transformations, Diagrams, Limits, Colimits, Universal problems and adjoint functions. Recommended Books:

1. Jaap van Oosten, Basic Category Theory, University of Otrecht, 2007. 2. D.E. Rydeheard, R. M. Burstall, Computational Theory, 2001. 3. Michael Barr, Charles Wells, Category Theory Lecture Notes, 1990.

4. Peter Freyd, Abelian Categories: An Introduction to the Theory of Funtors, Harper and Row, 1964.

5. Arbib MA. & Manes, E.G., arrows, structure and functions, Academic press New York, 1977.

6. Ehrig H. and P fender, Kategorien and Automation Walter de Gruyter, berlin, New York.

7. Herrlich H & Strecker G.E Category Theory, Allyn and Becon Inc., Boston, 1973.

Course Title: Category Theory-II Course Code: MATH-620 Credit Hours: 3 Subjects, Quotient objects and factorization, (E,M) Categories, (Epi external mono) and (external epi mono) Categories, (Generating external mono) factorization. Pointed categories: Normal and exact categories, Additive categories, Abelian categories, Definition of automation and examples, Category of automata, Epimorphism, Monomorphism, initial, terminal and null objects in Aut. Congruences and factor automata, Automata with constant input and output. Recommended Books:

1. Jaap van Oosten, Basic Category Theory, University of Otrecht, 2007. 2. D. E. Rydeheard, R. M. Burstall, Computational Theory, 2001. 3. Michael Barr, Charles Wells, Category Theory Lecture Notes, 1990. 4. Peter Freyd, Abelian Categories: An Introduction to the Theory of Funtors,

Harper and Row, 1964. 5. Arbib MA. & Manes, E.G., arrows, structure and functions, Academic press

New York 1973. 6. Ehrig H. and P fender, Kategorien and Automation Walter de Gruyter, berlin,

New York. 7. Herrlich H & Strecker G.E Category Theory, Allyn and Becon Inc., Boston,

1973. Course Title: Rings and Fields Course Code: MATH-621 Credit Hours: 3 Definitions and basic concepts, homomorphism theorems, polynomial rings, Quotient rings, unique factorization domain, factorization theory, Noetherian and Artinian rings, Euclidean domain, arithmetic in Euclidean domain, extension fields, algebraic and transcendental elements, simple extension. Books Recommended:

1. Louis Halle Rowen, Graduate Algebra: Noncommutative view, AMS, 2008. 2. Louis Halle Rowen, Rings Theory, Voume I and II, Academic Press, 1988. 3. P. M. Cohn, Free Ideal Rings and Localization in General, Cambridge

University Press, 2006. 4. Fraleigh, J.A., A first course in Abstract Algebra, Addison Wesley Publishing

Company, 1982.

5. Herstein, I.N., Topics in Algebra, John Wiley & sons. 6. Lang, S. Algebra, Addison Wesley, 2005. 7. Hartley, B., and Hawkes, T.O., Ring, Modules and Linear Algebra, Chapman

and Hall, 1970. 8. Donald S. Paseman, A course in Ring Theory, AMS, 1991.

Course Title: Theory of Modules Course Code: MATH-622 Credit Hours: 3 Elementary notions and examples, Modules, submodules, quotient modules, finitely generated and cyclic modules, exact sequences and elementary notions of homological algebra, Noetherian and Artinian rings and modules, radicals, semisimple rings and modules, tensor product of modules, bimodules, algebra and coalgebra, torsion module, primary components, invariance theorem. Recommended Books: 1. Adamson, J.,1976. Rings and modules 1st ed. NY: Chelsea.

2. Blyth, T.S., 1977. Module Theory. 1st ed. Oxford University Press. 3. Hartley, B. and Hawkes, T.O., 1980. Rings, Modules and Linear algebra. 1st ed. Chapman and Hall. 4. Herstein I.N, 1995. Topics in Algebra with Application. 3rd ed. Books/Cole.

5. Jacobson .N.,1989. Basic Algebra, Vol.II. 2nd ed. Freeman.

Course Title: Lie Algebra Course Code: MATH-623 Credit Hours: 3 General Theory, Definitions and First Examples, Ideals and homomorphisms, Isomorphism

Theorems, Lie algebra of derivations, Nilpotent Lie Algebras. Engel’s Theorem, Solvable Lie

Algebras, Lie’s Theorem, Radical, Semi-simplicity, Killing form, Cartan’s Criterion, Jordan,

Chevalley Decomposition, Representations, Inner derivations, Course Units, Maximal Toral

Subalgebras and Roots, Orthogonality Properties, Integrality Properties, Classification, Simple

lie algebras and irreducible root systems, The Lie Algebra of Type G2 And Octonions,

Representations Conjugacy theorems.

Recommended Books: 1. Brain C. Hall, Lie Algebra, 2003 2. J.E. Huomphry, Introduction to Lie Algebra and representation theory,

Springer Verlag. 3. N. Jacobson, Lie algebras,Wiley Inter Science, New York, 1972. 4. G. Edwood, Introduction to Lie Algebra Queen’s Papers. No.23, Kingston,

1999. 5. B.P.M. Cohn, Lie Groups Cambridge University Press. 1993.

Course Title: Advanced Functional Analysis Course Code: MATH-624 Credit Hours: 3

Fundamental Theorems: Zorn’s lemma, statement of Hahn-Banach theorem for real vector spaces, Hahn-Banach theorem for complex vector spaces and normed spaces, Uniform boundedness theorem, Open mapping theorem, Closed graph theorem. Spectral Theory: Spectral properties of bounded linear operations on Normed Spaces, Further properties of Resolvent and spectrum, use of complex Analysis in spectral theory, compact linear operations on Normed Spaces. Recommended Books: 1. Kreyszig, E., 1978. Introductory Functional Analysis with applications. 1st ed. John Wiley. 2. Brown, A.L., 1970. A , Elements of Functional Analysis. 1st ed. Von Nostrand and Reinhold Company. 3. Oden,J.T., 1979. Applied Functional Analysis. 1st ed. Prentice-Hall Inc.

Course Title: Galois Theory Course Code: MATH-625 Credit Hours: 3 Finite fields, fields extension, Galois theory, Galois theory of equations, construction with straight-edge and compass, splitting field of polynomials, the galio groups, some results on finite groups, symmetric group as Galois group, constructable regular n-gones, the Galois group as permutation group. Recommended Books:

1. Nicholson, W.K., Introduction to Abstract Algebra. 1st Ed. PWS-Kent Publishing Co. 1993.

2. Ames, D.B., Introduction to Abstract Algebra. 1st Ed., Pennsylvania International text book company. 1968.

3. Northcott, D.D., A First Course of Homological Algebra, 1st Ed. Cambridge University Press. 1973.

4. Jacobson, N., Basic Algebra I, 1st Ed. NY Freeman and Co. 1985.

Course Title: Measure Theory Course Code: MATH-627 Credit Hours: 3

Introduction, outer measure, Measurable sets and Lebesgue measure, a

nonmeasurable set. Measurable functions, the Lebesgue integral and Riemann integral,

the Lebesgue integral of bounded function over a set of finite measure, the Integral of

non-negative tfunction. The general Lebesgue Integral, Convergence in measure.

Recommended Books:

1. H.L.Royden, Real Analysis, 3rd Ed. McMillan Publishing Co. New York. 1988.

2. P.R.Halmos, Measure Theory, 1st Ed. Springer, New York. 1975.

3. W.Rudin, Real & Complex Analysis, 3rd Ed. McGraw Hill Book Company, New

York, 1987.

4. R.G.Bartle, The Elements of Integration and Lebesgue Measure, 1st Ed. Wiley-

Interscience. 1995.

COURSES OF APPLIED MATHEMATICS

Course Title: Fluid Mechanics-I Course Code: MATH-629 Credit Hours: 3 Introduction: Definition of Fluid, basics equations, Methods of analysis, dimensions and units. Fundamental concepts, Fluid as a continuum, velocity field, stress field, viscosity, surface tension, description and classification of fluid motions. Fluid Statics: The basic equation of fluid static, The standard atmosphere, pressure variation in a static fluid, fluid in rigid body motion. Basic equation in integral form for a control volume, basic laws for a system, relation of derivatives to the control volume formulation, conservation of mass, momentum equation for inertial control volume, momentum equation for control volume with rectilinear acceleration, momentum equation for control volume with arbitrary acceleration, the angular momentum principle, the first law of thermodynamics, the second law of thermodynamics. Introduction to differential analysis of fluid motion: conservation of mass, stream function for two dimensional incompressible flow, motion of a fluid element (kinematics), momentum equation. Recommended Books:

1. R.W. Fox & A.T. McDonald., Introduction to Fluid Mechanics 6th Ed. John Wiley & Sons, 2004. (Suggested Text Book)

2. White, F.M., Fluid Mechanics, Mc. Graw Hill, 5th Ed. published in 2006. 3. H. Schichting, Boundary Layer Theory, Mc. Graw Hill 4. Milne-Thomson, L. M., Theoretical Hydrodynamics, 6th Ed. Macmillan New

York 2006. Course Title: Fluid Mechanics-II Course Code: MATH-630 Credit Hours: 3

Incompressible inviscid flow, momentum equation for frictionless flow, Euler’s equations, Euler’s equations in streamline coordinates, Bernoulli equation- Integration of Euler’s equation along a streamline for steady flow, relation between first law of thermodynamics and the Bernoulli equation, unsteady Bernoulli equation-Integration of Euler’s equation along a streamline, irrotational flow. Internal incompressible viscous flow, Part-A Fully developed laminar flow, fully developed laminar flow between infinite parallel plates, fully developed laminar flow in a pipe, Part-B Flow in pipes and ducts, shear stress distribution in fully developed pipe flow, turbulent velocity profiles in fully developed pipe flow, energy consideration in pipe flow. External incompressible viscous flow, Part-A, Boundary layers, the boundary concept, boundary thickness, laminar flat plate boundary layer: exact solution, momentum, integral equation, use of momentum integral equation for flow with zero pressure gradient, pressure gradient in boundary-layer flow. Recommended Books:

1. R.W. Fox & A.T. McDonald, Introduction to Fluid Mechanics 6th Ed. John Wiley & Sons, 2004 (Suggested Text Book).

2. White, F. M. Fluid Mechanics 5th Ed. Mc. Graw Hill, published in 2006. 3. H. Schichting, Boundary Layer Theory, Mc. Graw Hill, New York, 1979. 4. Milne-Thomson, L. M., Theoretical Hydrodynamics 6th Ed. Macmillan New

York, 2006.

Course Title: Quantum Mechanics –I Course Code: MATH-631 Credit Hours: 3

Inadequacy of Classical Mechanics: Black body radiation, Photoelectric effect, Compton

effect, Bohr’s theory of atomic structure, Wave-particle duality, the de-Broglie

postulate.The Uncertainty Principle: Uncertainty of position and momentum, statement

and proof of the uncertainty principle, Energy-time uncertainty. Eigenvalues and

eigenfunctions, Operators and eigenfunctions, Linear Operators, Operator formalism in

Quantum Mechanics, Orthonormal systems, Hermitian operators and their properties,

Simultaneous eigenfunctions. Parity operators. Postulates of quantum mechanics, the

Schrödinger wave equation.

Motion in one Dimension: Step potential, potential barrier, Potential well, and Harmonic

oscillator. Recommended Books:

1. J.G Taylor, Quantum Mechanics, George Allen and Unwin. 1970.

2. T.L Powell and B.Crasemann, Quantum Mechanics, Addison Wesley, 1961.

3. E. Merzdacker, Quantum Mechanics John Wiley and sons.

4. R.M. Eisberg, Fundamental of Modern Mechanics, John Willey and Sons

5. H. Muirhead, The Physics of Elementary Particles, Pergamon Press, 1965.

6. R. Dicke, R & J .P. Witke, Quantum Mechanics, Addison Wesley, 1960.

Course Title: Quantum Mechanics –II Course Code: MATH-632 Credit Hours: 3

Motion in three dimensions, angular momentum, commutation relations between

components of angular momentum, and their representation in spherical polar

coordinates, simultaneous Eigen functions of Lz and L2, Spherically symmetric potential

and the hydrogen atom.

Scattering Theory: The scattering cross-section, scattering amplitude, scattering

equation, Born approximation, partial wave analysis.

Perturbation Theory: Time independent perturbation of non-degenerate and degenerate

cases. Time-dependent perturbations.

Identical Particle: Symmetric and anti-symmetric Eigen function, The Pauli exclusion

principle. Recommended Books:

1. J.G Taylor, Quantum Mechanics, George Allen and Unwin, 1970.

2. T.L Powell and B.Crasemann, Quantum Mechanics, Addison Wesley, 1961.

3. E. Merzdacker, Quantum Mechanics John Wiley and sons.

4. R.M. Eisberg, Fundamental of Modern Mechanics, John Willey and Sons

5. H.Muirhead, The Physics of Elementary Particles, Pergamon Press, 1965.

6. R. Dicke, R & J .P. Witke, Quantum Mechanics, Addison Wesley, 1060.

Course Title: Electromagnetic Theory –I Course Code: MATH-633 Credit Hours: 3 Electrostatics: Coulomb’s Law, Electric field and potential, Lines of force and equipotential surfaces, Gausess’s law and deduction, Conductor and Condensers, Dipoles, Dielectrics, Polarization and apparent charges, Electric displacement, Energy of the field, minimum energy. Magnetostatic Field: The Magnetostatic Law of force, Magnetic Doubles, Magnetic shells, Force on magnetic doublets, Magnetic induction, Para and dia and magnetism. Steady and Slowly Varying Currents: Electric current. Linear conductors, Conductivity, Resistance, Kirchoff’s laws, Heat production, Current density vector, Magnetic field of straight and circular current, Magnetic flux, vector potential forces on a circuit in magnetic field. Recommended Books:

1. Ferraro, Electromagnetic theory, Athlone Press, London, 1963. 2. J.R Reitz & Milford. Foundations of Electromagnetic theory, Addison –Wesley

Press, 1960. 3. Pugh &. Pugh, Electricity & Magnetism.

Course Title: Electromagnetic Theory –II Course Code: MATH-634 Credit Hours: 3 Steady and Slowly Varying Currents: Magnetic field energy. Law of Electromagnetic induction, Co-efficient of self and mutual induction, Alternating current and simple I. C.R. Circuit in series and parallel, power factor, Potential problems. The Equations of Electromagnetism: Maxwell’s equations in free space and material media, Solution of Maxwell’s equations: Place electromagnetic waves in homogeneous and isotropic media, Reflection and Refraction of plane waves, Wave guides, Laplace

equation in plane, polar and cylindrical coordinates, Simple introduction to Legendre polynomials, Method of images, Images in a plane, Images with spheres and cylinders. Recommended Books:

1. Ferraro, Electromagnetic Theory. 2. J.R Reitz & Milford. Foundations of Electromagnetic Theory, Addison –Wesley

Press, 1960. 3. Pugh &. Pugh, Electricity & Magnetism.

Course Title: Special Relativity Course Code: MATH-635 Credit Hours: 3+0 Historical background and fundamental concepts of special theory of relativity, Galilean transformations, Lorentz transformations (for motion along one axis), length contraction, time dilation and simultaneity, velocity addition formulae.3-dimensional, Lorentz transformations, introduction to 4-vector formalism. Lorentz transformations in the 4-vector formalism, the Lorentz and Poincare groups, introduction to classical mechanics, Minkowski space-time and null cone, 4-velocity and 4-momentum and 4-force, application of special relativity to Doppler shift and Compton effect, aberration of light, particle scattering, binding energy, particle production and decay, special relativity with small acceleration. Recommended Books:

1. Qadir, 1989. An introduction to the Special Relativity theory. 1st ed. World scientific.

2. D’Inverno R., 1992. Introducing Einstein’s Relativity. 1st ed. Oxford University Press.

3. Rindler W., 1977. Essential Relativity. 2nd ed. Springer Verlag.

Course Title: Elasticity Theory Course Code: MATH-636 Credit Hours: 3 Homogeneous Isotropic Bodies, Elastic Moduli of Isotropic Bodies. Equilibrium equation for an isotropic elastic solid, Dynamical equation of an isotropic elastic solid, Strain- energy function and its connection with Hook’s law, Uniqueness of the solution of the boundary value problems of elasticity, Saint-Verant’s principle extension, Torsion and Flexure of Homogeneous bears, Variational methods. Recommended Books:

1. Sokolinikoff, Mathematical Theory of Elasticity, 2nd Ed. McGraw Hill, New York, 1950.

2. Funk Y. C., Foundations of Solid Mechanics, Prentice – Hall, Englewood Cliffs, 2007

3. A.E.H. Love, A treatise on the Mathematical theory of elasticity. 1944. 4. Allan F. Bower, Applied Mechanics of Solids, CRS Press, 2010.

Course Title: Analytical Dynamics Course Code: MATH-637

Credit Hours: 3

Generalized coordinates, Constraints, Degree of freedom, D’Alembert principle,

Holonomic and non-Holonomic systems, Hamilton’s principle, Derivation of Lagrange

equation from Hamilton’s principle, and Derivation of Hamilton’s equation from a

variational principle. Equations and Examples of Gauge transformations, Equations and

examples of canonical transformations, Orthogonal Point transformations, The Principle

of Least Action, Applications of Hamilton’s equation to central force problems,

Applications to Harmonic oscillator, Hamiltonian formulism, Lagrange bracket and

Poisson brackets with application, The Hamilton Jacobi theory, Hamilton Jacobi Theorem,

The Hamilton Jacobi equation for Hamilton characteristic functions, Bilinear co-variant,

Quasi coordinates, transpositional relations for Quasi coordinates, Lagrange’s equation

for Quasi coordinates, Appel’s equation for quasi coordinates, Whittaker equation with

applications, Chaplygian system and Chaplygian equation. Recommended Books:

1. D.T. Greenwood, Classical Dynamics, Prentice-Hall, Inc. 1965. 2. Chorlton F. Textbook of Dynamics, Van Nostrand. 3. Chester W. Mechanics, George Allen and Unwin Ltd, London. 4. Goldstein H. Classical Mechanics, Cambridge, Mass Addison-Wesely. 5. L.A. Pars, Treatise of Analytical Dynamics, Heimann Press, London. 6. K. Sankara Rao, Classical Mechanics 7. P.V. Panaf, Classical Mechanics, Narosa Publishing House Delhi, 2005.

Course Title: Astronomy-I Course Code: MATH-639 Credit Hours: 3+0 Course Contents: Introduction, The great and small circles, spherical angle and spherical triangle, applications to the Earth, longitude and latitude, basics of spherical trigonometry, the celestial sphere, horizontal and equatorial systems of coordinates, observer’s meridian and diurnal motion, circumpolar stars, right ascension, the equation of time.

Recommended Books: 1. Smart W. M., 1977. Textbook on Spherical Astronomy. 1st ed. Cambridge

University Press. 2. Roy A. E. ,1982. Astronomy: Principles and Practice. 1st ed. Bristor: Adam Hilger

Ltd. 3. Wooland E. W. & Clemence G. M., 1966. Spherical Astronomy, 1st ed. Boston:

Academic Press.

Course Title: Astronomy-II Course Code: MATH-640

Pre Requisite: Astronomy-I Credit Hours: 3+0 Course Contents:

Introduction to celestial navigation on earth; celestial sphere; time-keeping system; refraction; parallax and triangulation, aberration; precession, nutation; tropical measurements, magnitude systems; Naked Eye Observations; Observational techniques; optics and telescopes; Radio telescopes and Doppler imaging.

Recommended Books: 1. Roy A. E., 1982. Astronomy: Principles and Practice. 1st ed. Adam Hilger Ltd. 2. Roy A. E., 1989. Astronomy: Structure of the Universe. 1st ed. Adam Hilger Ltd.,

Bristol.

Courses of Computational Mathematics

Course Title: Operations Research-I

Course Code: MATH-641 Credit Hours: 3 Course Contents: Linear Programming: formulation and graphical solution, simplex method, M-technique and two-phase technique, special cases sensitivity analysis, the dual problem, primal dual relationship, the dual simplex method, sensitivity and post optimal analysis, transportation model, Northwest corner, least cost and Vogel’s approximation methods, the method of multipliers, the assignment model, the transshipment model, network minimization, shortest route algorithms for variables. Recommended Books:

1. Hamdy A. T., 2006 Operations Research an Introduction. 6th ed. NY: Macmillan 2. Gillet B.E., 1979. Introduction to Operations Research. 1st ed. New Delhi:

McGraw Hill.

3. HARVY C.M., 1979.Operations Research. 1st ed. North Holland. 4. Hillier F.S. & Liebraman G.J., 2000. Operations Research. 8th ed., CBS.

Course Title: Operations Research-II Course Code: MATH-642 Credit Hours: 3 Course Contents: Algorithm for cyclic network, maximal flow problems, matrix definition of LP- problems, revised simplex methods, bounded variables decompositions algorithm, parametric linear programming, application of integer programming, cutting plane algorithm, mixed fractional cut algorithm, branch and bound methods, zero-one implicit enumeration, element of dynamics programming, problems of dimensionality, solutions of linear program by dynamics programming, Recommended Books:

1. Hamdy A. T., 2006 Operations Research an Introduction. 6th ed. NY: Macmillan 2. Gillet B.E., 1979. Introduction to Operations Research. 1st ed. New Delhi:

McGraw Hill.

3. Harvy C.M., 1979.Operations Research. 1st ed. North Holland. 4. Hillier F.S. & Liebraman G.J., 2000. Operations Research. 8th ed., CBS

Course Title: Methods of Optimization-I Course Code: MATH-643 Credit Hours: 3 Course Contents: Introduction to optimization and review of related mathematical concepts, unconstrained optimization, conditions for local minimizers, one dimensional search methods, gradient methods, Newton’s method (analysis and modifications), conjugate direction methods, Quasi Newton method, application to neural network, Single Neuron Training, Linear integer programming, introduction, Genetic algorithms, Real number genetic algorithm.

Recommended Books: 1.Sundaram R. K., 1996. A first course in optimization theory. 3rd ed.

CambridgeUniversity Press. 2. Edwin K. P Chong and Stanislaw H. Zak, 2012. An Introduction to Optimization.

4th ed. Wiley Series in Discrete Mathematics and Optimization. 3. Singiresu S. Rao, 1992. Optimization Theory and Applications. 2nd ed. Wiley

Eastern Ltd.

Course Title: Methods of Optimization-II Course Code: MATH-644 Credit Hours: 3 Course Contents: Non-linear constrained optimization, Problems with equality constraints, Introduction, Problem Formulation, Tangent and Normal spaces, Lagrange condition, Second-order conditions. Problems with inequality constraints, Karush-Kuhn-Tucker Condition, Second-order conditions. Convex optimization problems, Introduction, convex functions. Algorithms for constrained optimization, Lagrangian algorithms.

Recommended Books: 1. Sundaram R. K., 1996. A first course in optimization theory. 3rd ed. Cambridge

University Press. 2. Edwin K. P Chong and Stanislaw H. Zak, 2012. An Introduction to Optimization.

4th ed. Wiley Series in Discrete Mathematics and Optimization. 3. Singiresu S. Rao, 1992. Optimization Theory and Applications. 2nd ed. Wiley

Eastern Ltd.

Course Title: Theory of Splines-I Course Code: MATH-645 Credit Hours: 3 Course Contents: Euclidean Geometry: basic concepts of Euclidean geometry, scalar and vector functions, bar centric coordinates, convex hull, matrices of affine maps, translation, rotation, scaling,

reflection and shear. Approximation using Polynomials: curve fitting, least squares line fitting, least squares power fit, data linearization method for exponential functions, nonlinear least-squares method for exponential functions, transformations for data linearization, linear least squares, Polynomial fitting. Interpolation: basic concepts of interpolation, Lagrange’s method, error terms and error bounds of Lagrange’s method, divided differences method, Newton polynomials, error terms and error bounds of Newton polynomials, central difference interpolation formulae, Gauss’s forward interpolation formula, Gauss’s backward interpolation formula, Hermite’s methods. Recommended Books:

1. Sudaran R.K., 1996. A first course in optimization theory. 3rd ed. CUP. 2. Chang E.K.P and Zak, S.I.I, 2004. An Introduction to Optimization. 3nd ed.

Wiley. 3. Rao S.S., 1992. Optimization Theory and Applications. 2nd ed. Wiley Eastern

Ltd.

Course Title: Theory of Splines-II Course Code: MATH-646 Credit Hours: 3 Parametric curves (scalar and vector case), algebraic form, Hermite form, control point form, Bernstein Bezier form, matrix forms of parametric curves, algorithms to compute B.B. form, convex hull property, affine invariance property, variation diminishing property, rational quadratic form, rational cubic form, tensor product surface, B.B. cubic patch, quadratic by cubic B.B. patch, B.B. quartic patch. Spline Functions: splines, cubic splines, end conditions of cubic splines: clamped conditions, natural conditions, second derivative conditions, periodic conditions, Not a knot conditions, general splines, natural splines, periodic splines, truncated power function, representation of spline in terms of truncated power functions, odd degree interpolating splines. Recommended Books:

1. Farin G., 2002. Curves and Surfaces for Computer Aided Geometric Design A Practical Guide. 5th ed. Academic Press

2. Faux I.D. 1979.Computational Geometry for Design and Manufacture. 1st ed. Ellis Horwood

3. Bartle H.R, Beatly C.J., 2006. An Introduction to Spline for use in Computer Graphics and Geometric Modeling. 4th ed. Morgan Kaufmann.

4. Boor C.D., 2001. A Practical Guide to Splines. Revised ed. Springer Verlag.

Course Title: Graph Theory Course Code: MATH-647 Credit Hours: 3 Graphs and digraphs, Degree sequences, paths, cycles, cut-vertices, and blocks, Eulerian graphs and digraphs, Trees, incidence matrix, cut-matrix, circuit matrix and

adjacency matrix. Orthogonality relation, Decomposition, Euler formula, planer and non-planer graphs, Mengers theorem, Hamiltonian’s graphs. Books Recommended:

1. Chartrand and Lesniak , Graphs and Digraphs 5th Ed. Chapman and Hall, 2010. 2. Robin J. Wilson, Introduction to Graph Theory, 4th Ed. Addison Wesley, 1996.

Course Title: Theory of Automata Course Code: MATH-648 Credit Hours: 3 Regular expressions and Regular Languages Finite Automata, Context-free Grammars and Context-free languages, Push down automata. Decision Problems, Parsing, Turing Machines. Recommended Books:

1. Martin, Introduction to Languages and Theory of compution, Mc Graw Hill, 4th Ed. 2010. 2. Cohen. Introduction to Computer theory, Wiley, 2nd Ed. 1996.

Course Title: Control Theory Course Code: MATH-649 Credit Hours: 3 System dynamics and differential equations, some system equations, System Control, Mathematical methods and differential equations, The classical and modern control theory, Transfer functions and block diagram, Review of Laplace Transforms, Applications to differential equations, Transfer functions and Block diagrams, State space formations, State space forms, using transfer functions to define state variables, direct solution of the state equation, Solutions of the state equation by Laplace transforms, the transformation from companion to the diagonal state form, The transform function from the state equation, Transient and steady state response analysis, Response of first order system, Response of second order system, Response of higher order systems, Steady state error, Feedback control. Stability: The concept of stability, Routh stability criterion, Introduction to Liapunor’s method, Quadratic form, Determination of liapunov’s function, the Nyquist stability criterion, the frequency response, An introduction to conformal mapping, Applications of conformal mappings to the frequency response, Controllability and Observability; Controllability, Observability, Decomposition of system state, A transformation into the companion form, Multivariable Feedback and pole location: State feedback of SISO system, Multivariable system observations. Books Recommended:

1. Burghes D. and Graham A. Introduction to Control Theory including optimal control, Ellis Horwood, 1980.

2. Barnett S. and Camron R.G. Introduction to Mathematical Control Theory (2nd Ed.) Oxford V.P, 1985.

Course Title: Applied Matrix Theory

Course Code: MATH-650 Credit Hours: 3 Review of the Theory of Linear System: Eigen values and Eigen vectors, The Jordan canonical forms, Bilinear and quadratic forms, Matrix analysis of differential equations, Variational principles and perturbation theory, the Courant minimax theorem, Weyl’s inequalities, Gershgorin’s theorem, perturbations of the spectrum, vector norms and related matrix norms, the condition number of a matrix. Recommended Books:

1. Strang G. Linear Algebra and its Applications, Academic Press, 2005. 2. William G. Linear Algebra with Applications, Allyn and Bacon, Inc., 7th Ed. 2009. 3. Stewart G.W. Introduction to Matrix Computations, Academic Press, INC, New

York, 1973. Course Title: Finite Element Analysis Course Code: MATH-651 Credit Hours: 3

Objectives: The objective of finite element method is to discretize the domain into finite element for which the governing equations are algebraic equations. Solution of these algebraic equations gives the approximate solution of the non linear differential equations. The convergence is judged by the refinement of mesh. Course Contents Rational Bezier curves, properties of rational Bezier curves, Marsden identity, construction of FEM basis function, the de Boor algorithm, dual functional, error approximation by orthogonal functional, cubic Hermite interpolation, natural spline interpolation, quasi interpolant, Schoenberg scheme, error of quasi interpolation, Lagrangian function for interpolation, interpolation error, curves on uniform grid and their properties, interpolation with curves on uniform grid, geometric Hermite interpolation, non-uniform rational B-splines, construction of finite element basis on multidimensional space, Box splines, recursion for Box splines, approximation on multidimensional space, ellipticity of approximation, Cea’s lemma, approximation theorems for FEM. Recommended Books: 1. Introduction to the Mathematics of Subdivision Surfaces by Lars-Erik Andersson, SIAM,

2010. 2. Numerical Models for Differential Problems by Quarteroni A., Springer, 2009. 3. Finite Element Method by Klaus-Jürgen Bathe, John Wiley & Sons, 2007. 4. Splines and Variational Methods by Prenter, P. M., AWiley-Interscience Publication, 2006.

Annexure-VIII

The BOS Approved to add the following courses in scheme of M. Phil/Ph. D Mathematics each of credit hours 03.

MATH-769 BCK Algebra/BCI Algebra

MATH-770 Minimal Surfaces

MATH-771 Symmetries and Exact Solutions of Differential Equations

MATH-772 Riemannian geometry

MATH-906 APPLICATIONS OF INEQUALITIES

MATH-772: Riemannian geometry

Definition and examples of manifolds; Submanifolds; Tangents; Coordinate vector fields;

Tangent spaces; Dual spaces; Algebra of tensors; Vector fields; Tensor fields; Integral

curves; Affine connections and Christoffel symbols; Covariant differentiation of tensor

fields; Geodesics equations; Curve on manifold; Parallel transport; Lie transport; Lie

derivatives and Lie Brackets; Geodesic deviation; Differential forms; Introduction to

integration theory on manifolds; Riemannian Curvature tensor; Ricci tensor and Ricci

scalar; Killing equations and Killing vector fields.

Books Recommended: 1. Bishop, R.L. and Goldberg, S.I.,. Tensor Analysis on Manifolds. 1st Ed. NY: Dover Publications. 1980. 2. Carmo M.P., Riemannian Geometry. 1st Ed. Boston:Birkhauser. 1992. 3. Lovelock, D. and Rund, H. Tensors., Differential Forms and Variational Principles, John-Willey, 1975. 4. Langwitz, D., Differential and Riemannian Geometry, Academic Press, 1970. 5. Abraham, R., Marsden, J.E. and Ratiu, T., Manifolds, Tensor Analysis and Applications, Addison-Wesley, 1983. MATH-906: APPLICATIONS OF INEQUALITIES Gruüss type inequalities, Chebychev’s type inequalities, Ostrowski’s inequalities, Applications of inequalities involving gamma and beta functions, Introduction to Lp-spaces. Boundedness of integral operators involving some special functions, Hardy-Type Inequalities, Miscellaneous inequalities,

RECOMMENDED BOOKS:

1. Dragomir, SS: A generalization of Grüss inequality in inner product spaces and

applications. J. Math.Anal. Appl. 237,74-82 (1999). 2. Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006). Theory andApplications

of Fractional Differential Equations. North-Holland. Mathematical Studies, 204,

Elsevier B. V., Amsterdam.B.G. Pachpatte, Mathematical inequalities, (North-Holland Mathematical Library,Vol.67), Elsevier, 2005.

3. Constantin Niculescu and Lars-Erik Persson, Convex Functions and their Applications: A Contemporary Approach (CMS Books in Mathematics), Springer, 2005.

4. J. Pečarić, F. Proschan and Y. C. Tong, Convex Functions, Partial Orderings and Statistical Applications, vol. 187 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1992.

5. D. S. Mitrinovic, J. Pečarić and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, The Netherlands, 1993.

6. Mitrinovic, DS, Peharic, JE, Fink, AM: Inequalities for Functions and Their Integrals and Derivatives. Kluwer Academic,Dordrecht (1994).

7. Okikiolu, G. O. (1971). Aspects of the theory of bounded linear operators.Academic Press.

8. Related Research Papers.