SIR PADAMPAT SINGHANIA UNIVERSITY Udaipur Sc _Mathematics_ 2017-19.pdf · SIR PADAMPAT SINGHANIA...

SPSU/SOE/Mathematics/M.Sc./2017 Ver. 1.0 1

SIR PADAMPAT SINGHANIA UNIVERSITY

Udaipur

SCHOOL OF ENGINEERING

Course Curriculum of 2-Year M.Sc. Degree Programme in

Mathematics (Batch: 2017-2019)

Credit Structure

M. Sc. Core M. Sc. Elective

Category Credits Category Credits

Departmental Core Subjects 65 Departmental Electives 15

Minor Subjects 08

Total 73 Total 15

Grand Total 88


Distribution of Total Credits & Contact Hours in all Semesters

S. No. Semester Number Credits / Semester Contact Hours/week

1 I 20 21

2 II 21 23

3 III 26 29

4 IV 21 31

Total 88 --


Course Structure: M.Sc. 2017-19

Semester - I

Semester-II

S. No. Course Code

Course Title L T P Credit(s)

1 MA-564 Probability & Statistics 4 0 0 4

2 MA-565 Complex Analysis 4 0 0 4

3 MA-567 Modern Algebra 4 0 0 4

4 MA-568 Numerical Analysis 4 0 1 5

5 CS-5XX Data Structures & Algorithms 3 0 1 4

Total Credits 21

Total Contact Hours/week 23

S. No. Course Code

Course Title L T P Credit (s)

1 MA-560 Linear Algebra 4 0 0 4

2 MA-561 Multivariable Calculus 4 0 0 4

3 MA-562 Real Analysis 4 0 0 4

4 MA-563 Ordinary Differential Equation 4 0 0 4

5 CS-5XX Computer Programming 3 0 1 4

Total Credits 20



Semester-III

Semester-IV

S. No.

Course Code


1 MA-569 Partial Differential Equations 4 0 0 4

2 MA-570 Topology 4 0 0 4

3 MA-571 Measure & Integration 4 0 0 4

4 MA-5XX Departmental Elective – I 5 0 0 5

5 MA-572 Mathematical Software & Lab 1 0 2 3

6 MA-5XX Departmental Elective – II 5 0 0 5

7 MA-580 Seminar 0 0 0 1

Total Credits 26


S. No.

Course Code


1 MA-5XX Departmental Elective – III 5 0 0 5

2 MA-580A Dissertation Seminar 0 0 0 1

3 MA-580B Dissertation 0 0 0 12

4 MA-580C Dissertation (Viva) 0 0 0 3

Total Credits 21



List of Departmental Electives-I

S. No. Subject Code

Subject L T P Credit(s)

1 MA-573 Special functions 5 0 0 5

2 MA-574 Introduction to General theory of Relativity

5 0 0 5

3 MA-575 Discrete Mathematics 5 0 0 5

4 MA-576 Linear Programming & extensions 5 0 0 5

5 MA-577 Numerical Linear Algebra 5 0 0 5

List of Departmental Electives-II

S. No. Subject Code


1 MA-578 Functional Analysis 5 0 0 5

2 MA-579 Integral Transforms & Integral Equations

5 0 0 5

3 MA-581 Numerical solutions of PDE 5 0 0 5

4 MA-582 Graph Theory 5 0 0 5

5 MA-583 Non Linear programming 5 0 0 5


List of Departmental Electives-III

S. No. Subject Code


1 MA-584 q-Hypergeometric Functions 5 0 0 5

2 MA-585 General Relativity 5 0 0 5

3 MA-586 Classical Mechanics & Calculus of variations

5 0 0 5

4 MA-587 Differential Geometry 5 0 0 5

5 MA-588 Number theory 5 0 0 5


Detailed Syllabus for M.Sc. Degree Programme in

Mathematics

Semester-I

(Departmental Core Subject)

MA-560 L – T – P - C Linear Algebra 4 – 0 – 0 – 4 Objective: The objective of this course is to depict the understanding & importance of Linear Algebra to improve ability to think logically, analytically & abstractly.

Course Content

Finite dimensional vector spaces over a field; linear combination, linear dependence &

independence, basis & dimension, inner-product spaces, linear transformation; matrix

representation of linear transformation; linear functional; dual spaces; eigen values &

eigen vectors; rank & nullity, inverse & linear transformation, Cayley-Hamilton Theorem,

norms of vectors & matrices, Transformation of matrices, adjoint of an operator, normal,

unitary, hermitian & skew-hermitian operators; quadratic forms

Text/Reference Books

1. Linear Algebra. Homan. K. & Kunze R. 2nd Edition. PHI. 2. Linear Algebra, Undergraduate Texts in Mathematics. Lang S. Springer-Verlag.

New York. 1989. 3. Topics in Algebra. Herstein. I. N. 2nd Edition. John Wiley & Sons, (Vikas

Publishing House). 1988. 4. Foundations of Linear Algebra. Golan. J. S. Kluwer Academic Publishers. 1995.



Mathematics

Semester-I


MA-561 L – T – P - C Multivariable Calculus 4 – 0 – 0 – 4 Objective: The objective of this course is to provide in-depth study in multivariable calculus including the Green's theorem, Gauss divergence theorem & stokes theorem.

Course Content

Differential Calculus: Functions of several variables, Open sets, Limits & continuity,

Derivatives of a scalar field with respect to a vector, Directional derivatives, Partial

derivatives, Total derivative, Gradient of a scalar field, Level sets & Tangent planes,

Derivatives of vector fields, Chain rules for derivatives, Mean value theorem, Derivatives

of functions defined implicitly. Applications of Differential Calculus: Maxima, Minima,

Saddle points, Stationary points, Lagrange’s multipliers, Inverse function theorem,

Implicit function theorem. Line Integrals: Paths & line integrals, Fundamental theorems

of calculus for line integrals, vector fields & gradients. Multiple Integrals: Double & triple

integrals, Iterated integrals, Change of variables formula, Applications to area & volume,

Green's theorem, Two-dimensional vector fields & gradients. Surface Integrals:

Parametric presentation of a surface, Fundamental vector product & normal to a

surface, Stokes' theorem, Curl & divergence of a vector field, Gauss' divergence

theorem.

Text/Reference Books 1. Calculus Vol. II. Apostol T. M. 2nd Edition. John Wiley & Sons. 2003. 2. Mathematical Analysis. Apostol T. M. 2nd Edition. Narosa Publishing House.

1997. 3. Advanced Calculus. Widder D. V. 2nd Edition. PHI . 1987. 4. Vector Analysis. Spiegel M. R. Schaum's Outline Series. McGraw-Hill. 1959. 5. Advanced Calculus-A Differential Forms Approach. Edwards H. M.


Birkhauser. 1994.



Mathematics

Semester-I


MA-562 L – T – P - C Real Analysis 4 – 0 – 0 – 4 Objective: The objective of this course is to expose student to basic concepts & proofs of important theorems of Real analysis.

Course Content

Real number system & its order completeness, sequences & series of real numbers.

Metric spaces: Basic concepts, continuous functions, completeness, contraction

mapping theorem, connectedness, Intermediate Value Theorem, Compactness, Heine-

Borel Theorem. Differentiation, Taylor's theorem, Riemann Integral, Improper integrals.

Sequences & series of functions, Uniform convergence, power series, Fourier series,

Weierstrass approximation theorem, equicontinuity, Arzela-Ascoli theorem.

Text/Reference Books 1. Principles of Mathematical Analysis. Rudin W. McGraw-Hill. 1976. 2. Real Mathematical Analysis. Pugh C. C. Springer. 2002. 3. Mathematical Analysis. Apostol T. M. 2nd Edition. Narosa Publishing House.

1997. 4. Topology & Modern Analysis. Simmons. G. F. Krieger Publishing House. 2003.



Mathematics

Semester-I


MA-563 L – T – P - C Ordinary Differential Equation 4 – 0 – 0 – 4 Objective: To develop strong background on finding solutions of ordinary differential equations of various types & to study existence & uniqueness of the solutions.

Course Content

Existence-Uniqueness: Review of exact equations of first order, The method of

successive approximations, Lipschitz condition, Convergence of successive

approximations, Existence & Uniqueness of solutions for first order initial value

problem, Non-local existence of solutions, Existence & uniqueness of solutions to

systems, Existence & uniqueness for linear systems, Equations of order n. Second

Order Equations: General solution of homogeneous equations, Non-homogeneous

equations, Wronskian, Method of variation of parameters, Sturm comparison theorem,

Sturm separation theorem, Boundary value problems, Green's functions, Sturm-Liouville

problems. Series Solution of Second Order Linear Equations: ordinary points, regular

singular points, Legendre polynomials & properties, Bessel functions & properties.

Systems of Differential Equations: Algebraic properties of solutions of linear systems,

The eigenvalue-eigenvector method of finding solutions, Complex eigenvalues, Equal

eigenvalues, Fundamental matrix solutions, Matrix exponential, Nonhomogeneous

equations, Variation of parameters.

Text/Reference Books 1. An Introduction to Ordinary Differential Equations. Coddington E. A. PHI. 1999. 2. Differential Equations with Applications & Historical Notes. Simmons G.F. 2nd

Edition. McGraw-Hill. 1991.


3. Essentials of Ordinary Differential Equations. Agarwal R. P. & Gupta R. C. McGraw-Hill, 1993.

4. Differential Equations & Their Applications. Braun M. 3rd Edition. Springer-Verlag. 1983.

5. Differential & Integral Equations. Collins P. J. Oxford University Press. 2006. 6. Differential Equations: Theory, technique & practice. Simmons G. F. & Krantz S.

G. Tata McGraw-Hill. 2007. 7. Elementary Differential Equations & Boundary Value Problems. Boyce W. E. &

Di-Prima R. C. 7th Editions. John Wiley & Sons. 2001. 8. An Introduction to Ordinary Differential Equations. Agarwal R. P. & O'Regan D.

Springer-Verlag. 2008.



Mathematics

Semester-I

(Minor Subject)

CS-5XX L – T – P - C Computer Programming 3 – 0 – 1 – 4 Objective: To develop awareness about the computer programming.

Course Content

Introduction - the von Neumann architecture, machine language, assembly language,

high level programming languages, compiler, interpreter, loader, linker, text editors,

operating systems, flowchart; Basic features of programming (Using C) - data types,

variables, operators, expressions, statements, control structures, functions; Advance

programming features - arrays & pointers, recursion, records (structures), memory

management, files, input/output, standard library functions, programming tools, testing &

debugging; Fundamental operations on data - insert, delete, search, traverse & modify;

Fundamental data structures - arrays, stacks, queues, linked lists; Searching & sorting -

linear search, binary search, insertion-sort, bubble-sort, selection-sort; Introduction to

object oriented programming.

Programming laboratory will be set in consonance with the material covered in lectures.

This will include assignments in a programming language like C & C++ in GNU Linux

environment.

Text/Reference Books 1. A Book on C. Kelly A. & Pohl I. 4th Edition. Pearson Education. 1999. 2. C: The Complete Reference. Schildt H. 4th Edition. Tata McGraw-Hill. 2000. 3. The C Programming Language. Kernighan B. & Ritchie D. 2nd Edition. PHI.

1988. 4. Programming With C. Gottfried B. & Chhabra J. Tata McGraw-Hill. 2005.



Mathematics

Semester-II


MA-564 L – T – P - C Probability & Statistics 4 – 0 – 0 – 4 Objective: The objective of this course are to introduce axiomatic approach to probability theory, to study some statistical characteristics, discrete & continuous distribution functions & their properties, characteristic function & basic limit theorems of probability.

Course Content

Axiomatic definition of probability, Theorems on probability. Conditional probability & independence. Baye’s theorem, Geometric probability. Random variables, their properties. Some standard discrete & continuous variables. Mathematical expectation, variance, moments, moment generating function. Chebyshev's inequality. Functions of a r.v., their distributions & moments. Joint, marginal & conditional distributions, independence of random variables. Law of large numbers, Central limit theorem, Correlation & regression: Simple, multiple & partial. Sampling distributions. Estimation of parameters : Maximum likelihood & method of moments. Properties of best estimates. Testing of hypotheses, Neyman-Pearson Lemma, standard tests for one & two sample problems.

Text/Reference Books 1. An Introduction to Probability & Statistics. Rohatgi V. K. & Md. E. Saleh A. K. 2nd

Edition. Wiley Eastern. 2002. 2. A First course in Probability. Ross Sheldon M. 6th Edition. PHI. 2001. 3. Introduction to Probability. Hoel P. G., Port S. C. & Stone C. J. Universal Book

Stall. New Delhi. 1998. 4. A Course in Probability Theory. Chung Kai Lai. 3rd Edition. Academic Press.

2001. 5. Introduction to Probability Theory & its Applications (Volume 1). Feller W. John

Wiley & Sons. 1968. 6. Modern Probability Theory: an introductory textbook. Bhat B. R. 2nd Edition. Wiley

Eastern. 1989.



Mathematics

Semester-II


MA-565 L – T – P - C Complex Analysis 4 – 0 – 0 – 4 Objective: The objective of this subject are to expose student to understand the importance of complex variables & study Cauchy integral formula, local properties of analytic functions, general form of Cauchy’s theorem & evaluation of definite integral & harmonic functions.

Course Content

Topology of the complex plane, Riemann sphere, limits, continuity & differentiability,

Analytic functions, harmonic functions & multi-valued functions. Convergence of

numerical series, Radius of convergence of power series, & power series as an analytic

function, Laurent series. Cauchy's integral theorem, Cauchy integral formula, Morera's

theorem, Taylor`s theorem, Laurent's theorem, Liouville's theorem, Schwarz lemma;

Maximum Modulus Principle. Conformal mappings, linear fractional transformations,

Classification of singularities, Cauchy's residue theory & evaluation of real integrals.

Text/Reference Books 1. Complex analysis. Ahlfors L. V. McGraw-Hill Book Company. 1966. 2. Functions of one complex Variable. Conway John. B. 2nd Edition. Narosa

Publishing House. 1978. 3. Complex Analysis. Gamelin T. W. Springer International Edition. 2001. 4. Theory of Complex Functions. Remmert R. Springer Verlag. 1991. 5. Complex Variables with Applications. Ponnusamy S. & Silverman H. Birkhauser.

Boston. 2006. 6. Foundations of Complex Analysis. Ponnusamy S. 2nd Edition. Narosa Publishing

House. 2005. 7. An Introduction to Complex Analysis. Shastri A. R. Macmilan India,

New Delhi. 1999.



Mathematics

Semester-II


MA-567 L – T – P - C Modern Algebra 4 – 0 – 0 – 4 Objective: The objective of this subject is to expose student to understand several important concepts in abstract algebra, including group, ring, field, homomorphism, isomorphism, & quotient structure, & to apply some of these concepts to real world problems.

Course Content

Normal subgroup, quotient group; homomorphism, fundamental theorem of

homomorphism; permutation group, Cayley's theorem; direct product of group.

Cummutative ring with identity: Axioms, examples, integral domain; field, ideals,

quotient ring, prime & maximal ideal, principal ideal domain; Euclidean domain, the field

of quotients of an integral domain; polynomial ring over a field. Roots of polynomials,

extension of fields, splitting fields.

Text/Reference Books 1. First Course in Abstract Algebra. Fraleigh. J. B. 7th Edition, Addison Wesley.

2003. 2. Abstract Algebra. Dummit D. S. & Foote R. M. 2nd Edition, John Wiley & Sons.

1999. 3. Algebra. Lang S. 3rd Edition. Addison-Wesley. 1999. 4. Contemporary Abstract Algebra. Gallian J. A. 4th Edition. Narosa Publishing

House. 1999. 5. Algebra. Artin M. Prentice Hall inc. 1994. 6. Topics in Algebra. Herstein I. N. John Wiley & Sons. 1995. 7. Algebra, Graduate Texts in Mathematics. Hungerford T. A. Vol. 73. Springer-

Verlag. 1980.



Mathematics

Semester-II


MA-568 L – T – P - C Numerical Analysis 4 – 0 – 1 – 5 Objective: The objective of this subject is to expose student to understand the importance of Numerical methods, the analysis behind it & to practice the numerical methods on computer.

Course Content

Errors: Floating-point approximation of a number, Loss of significance & error

propagation, Stability in numerical computation.

Linear Systems: Gaussian elimination with pivoting strategy, LU factorization, Residual

corrector method, Solution by iteration (Jacobi & Gauss-Seidal with convergence

analysis), Matrix norms & error in approximate solution, Eigenvalue problem (Power

method), Gershgorin’s theorem (without proof).

Nonlinear Equations: Bisection method, Fixed-point iteration method, Secant method,

Newton's method, Rate of convergences, Solution of a system of nonlinear equations,

Unconstrained optimization.

Interpolation by Polynomials: Lagrange interpolation, Newton interpolation & divided

differences, Error of the interpolating polynomials, Piecewise linear & cubic spline

interpolation, Trigonometric interpolation, Data fitting & least-squares approximation

problem.

Differentiation & Integration: Difference formulae, Some basic rules of integration,

Adaptive quadratures, Gaussian rules, Composite rules, Error formulas.

Differential Equations: Euler method, Runge-Kutta methods, Multi-step methods,

Predictor-Corrector methods Stability & convergence, Two point boundary value

problems.



This will include assignments in SCILAB.

Text/Reference Books 1. Applied Numerical Analysis. Gerald G. E. & Wheatley P. O. Addison Wesley.

1988. 2. Numerical Methods for Mathematics, Science & Engineering. Mathews J. H. PHI.

1994. 3. Elementary Numerical Analysis. Atkinson K. John Wiley & Sons. 1978. 4. Numerical Methods for Scientific & Engineering Computation. Jain M. K., Iyengar

S. R. K. & Jain R.K. Wiley Eastern. 2003. 5. Numerical Analysis. Burden R. L. & Faires J. D. 7th Edition. Thomson Brooks

Cole. 2001. 6. Numerical Methods for Scientists & Engineers. Rao K. S. PHI. 2001.



Mathematics

Semester-II

(Minor Subject)

CS-5XX L – T – P - C Data Structures & Algorithms 3 – 0 – 1 – 4 Objective: To develop familiarity with the physical concepts & facility with the mathematical methods of classical mechanics.

Course Content

Asymptotic notation; Sorting - merge sort, heap sort, priortiy queue, quick sort, sorting in

linear time, order statistics; Data structures - heap, hash tables, binary search tree,

balanced trees (red-black tree, AVL tree); Algorithm design techniques - divide &

conquer, dynamic programming, greedy algorithm, amortized analysis; Elementary

graph algorithms, minimum spanning tree, shortest path algorithms.


This will include assignments in a programming language like C & C++ in GNU Linux

environment.

Text/Reference Books 1. Introduction to Algorithms. Cormen T. H., Leiserson C. E., Rivest R. L. & Stein C.

2nd Edition. PHI. 2007. 2. Data Structures & Algorithms in Java. Goodrich M. T. & Tamassia R. John Wiley

& Sons. 2006. 3. Data Structures & Algorithms. Aho A.V. & Hopcroft J. E. Addison-Wesley, 1983. 4. Data Structures, Algorithms & Applications in C++. Sahni S., 2nd Edition.

Universities Press. 2005.



Mathematics

Semester-III


MA-569 L – T – P - C Partial Differential Equations 4 – 0 – 0 – 4 Objective: To introduce the main topics of partial differential equations, theory, various methods & their applications to related areas.

Course Content

First order partial differential equations: Linear, quasi-linear & fully nonlinear equations-

Lagrange & Charpit methods. Second order partial differential equations: Classification

& Canonical forms of equations in two independent variables, One dimensional wave

equation- D'Alembert's solution, Reflection method for half-line, Inhomogeneous wave

equation, Fourier Method. One dimensional diffusion equation: Maximum Minimum

principle for the diffusion equation, Diffusion equation on the whole line, Diffusion on the

half-line, inhomogeneous equation on the whole line, Fourier method. Laplace equation:

Maximum -Minimum principle, Uniqueness of solutions; Solutions of Laplace equation in

Cartesian & polar coordinates-Rectangular regions, circular regions, annular regions;

Poison integral formula Diffusion & wave equations in higher dimensions.

Text/Reference Books 1. Partial Differential equations- an introduction with mathematica & maple.

Stavroulakis I. P. & Tersian S. A. World –Scientific. Singapore. 1999. 2. Introduction to partial Differential equations with matlab. Cooper J. Birkhauser.

1998. 3. Techniques in partial Differential equations. Chester C. R. McGraw-Hill. 1971. 4. Introduction to partial Differential equations. Rao K. S.PHI. 1997. 5. Elements of partial Differential equations. Sneddon I. N. McGraw-Hill. New

York.1986. 6. Partial Differential equations. Williams W. E. Clarendon Press. Oxford. 1980.



Mathematics

Semester-III


MA-570 L – T – P - C Topology 4 – 0 – 0 – 4 Objective: The objective of this course is to expose students with the concept & the understanding of topological spaces & compact spaces.

Course Content

Topological Spaces, Basis for a topology, Subspace topology, Closed sets & Limit

points, Nets & convergence, Continuous Functions & homeomorphisms, Product

Topology, Quotient Topology. Connected spaces, Components & Local

Connectedness, Path connectedness, Compact spaces, Local compactness,

Compactifications. The Countability & Separation axioms, The Urysohn Lemma, The

Urysohn Metrization Theorem, The Tietze Extension Theorem, Tychonoff Theorem.

Text/Reference Books 1. Topology. Munkres J. R. 2nd Edition. Pearson Education India. 2001. 2. Introduction to General Topology. Joshi K. D. New Age International. 2000. 3. Introduction to Topology. Deshpande J. V. Tata McGraw-Hill. 1988. 4. Topology. Dugundji J. Allyn & Bacon Inc. 1966. 5. General Topology. Kelley J.L. Van Nostrand. 1955. 6. General Topology. Murdeswar M. G. New Age International. 1990. 7. Introduction to Topology & Modern Analysis. Simmons G. F. Tata McGraw-Hill.

2016. 8. General Topology. Willard S. Addison Wesley. 1970.



Mathematics

Semester-III


MA-571 L – T – P - C Measure & Integration 4 – 0 – 0 – 4 Objective: To study Riemann Stieltjes Integral, Lebesgue measurability–integrability & the Lp – spaces on general measure spaces.

Course Content

Review of Riemann Integral, Riemann-Stieltjes Integral. Lebesgue Measure; Lebesgue

Outer Measure; Lebesgue Measurable Sets. Measure on an arbitrary sigma -Algebra;

Measurable Functions; Integral of a Simple Measurable Function; Integral of Positive

Measurable Functions. Lebesgue's Monotone Convergence Theorem; Integrability;

Dominated Convergence Theorem; Lp - Spaces. Differentiation & Fundamental theorem

for Lebesgue integration. Product measure; Statement of Fubini's theorem.

Text/Reference Books 1. Measure & Integration. Barra G. D.John Wiley & Sons. 1981. 2. Real Analysis. Royden H. L. 3rd Edition. PHI. 1995. 3. Real & Complex Analysis. Rudin W. 3rd Edition, McGraw-Hill.1987. 4. An Introduction to Measure & Integration. Rana I. K. 2nd Edition, Narosa

Publishing House. 2005. 5. Measure Theory. Cohn D. L. Birkhauser. 1997. 6. Lebesgue Measure & Integration. Jain P. K. & Gupta V. P. New Age

International. 2006.



Mathematics

Semester-III


MA-572 L – T – P - C Mathematical Software & Lab 1 – 0 – 2 – 3 Objective: This course aims to practice the students in Mathematics document preparation & utilizing the software facility available for tedious computations.

Course Content

Creating a document using latex : title creation; page layout ; formatting ; fonts ; list structures; tables; bibliography management.

Creating equations using latex / mathtype

Scilab / matlab basics : Algebra & Arithmetic; Calculus, Graphics & Linear algebra; Scilab / matlab programming.

Symbolic computation using sage / mathematica to solve the problems from algebra, calculus & differential equations.


This will include assignments in the Mathematical Software like Latex, Scilab & SAGE.

Text/Reference Books 1. Sage for Undergraduates. Bard G. V. American Mathematical Society. 2015. 2. SAGE Manual. www.sagemath.org. 3. SCILAB Manual. www.scilab.org/. 4. Numerical & statistical methods with SCILAB for science & engineering. Urroz

G. E. University of California. 2011. 5. MikTex Manual. https://miktex.org. 6. More Math Into LaTex. Gatzer G. Springer. 2016.



Mathematics

Semester-III

(Departmental Elective Subject)

MA-573 L – T – P - C Special Functions 5 – 0 – 0 – 5 Objective: “Special functions” arose as solutions of differential equations or integrals of transcendental functions & have found several applications in different areas of science & engineering. This course provides a thorough study of this field.

Course Content

Gamma & Beta functions: Weierstrass’s & Euler’s product definitions of (z), Euler’s

integral for (z) & Beta function, Factorial function, Legendre’s duplication & Gauss’s

multiplication formulae

Gauss’s Hypergeometric Function 2F1(z), Euler’s integral representation of 2F1(z) &

applications, Contiguous function relations, Series manipulations, Hypergeometric

summations & transformations.

Generalized Hypergeometric Functions; Saalschütz’s, Whipple’s & Dixon’s theorems.

Confluent Hypergeometric Functions; Kummer’s first & second formula.

Generating Functions, Orthogonal Polynomials: Legendre, Hermite, Laguerre,

Gegenbaur, Ultraspherical & Jacobi; Recurrence relations. Sister-Celine's technique for

pure recurrence relations.

Text/Reference Books 1. Special Functions. Rainville E. D. MacMillan Co.: New York. 1967. 2. Special Functions. Andrews G. E., Askey R. & Roy R. Cambridge University

Press: Cambridge. 1999. 3. Generalized Hypergeometric Functions. Slater L. J. Cambridge University Press:

Cambridge. 1966. 4. The Special Functions & their Approximations: Vol. I. Luke Y. L. Academic Press:

London. 1969.


5. Generalized Hypergeometric Series. Bailey W. N. Cambridge University Press, Cambridge, reprinted by Hafner: New York. 1964.



Mathematics

Semester-III


MA-574 L – T – P - C Introduction to general theory of relativity 5 – 0 – 0 – 5 Objective: This course provides fundamentals of general theory of relativity which provide important background to study the various cosmological models.

Course Content

Introduction Special Relativity: Space-time approach, 4 vectors, geometric construction.

Introduction to General Relativity: Gravitational red shift; Principle of Equivalence;

Matter & space time curvature. Curved Space-time Principle of Covariance, coordinate

transformations; Tensor Analysis, geometry of curved space. How matter moves in

curved space-times: connection & Geodesic equations. How curvature of space-time

shows up: curvature & geodesic deviations. Einstein's Equation: Vacuum solutions;

Relativistic matter; Dynamics of space-time & matter

Text/Reference Books 1. A First Course in General Relativity. Schutz B. F. 2nd Ed. Cambridge University

Press. 2009. 2. Gravitation & Cosmology. Weinberg S. 1st Ed. John Wiley & Sons. 1972. 3. Problem Book in Relativity & Gravitation. Lightman A.P., Press W.H., Price R.H.

& Teukolsky S. A. Princeton University Press. 1975. 4. Exploring Black Holes: Introduction to General Relativity. Taylor E. F. &. Wheeler

J. A. 1st Ed. Addison Wesley. 2000. 5. General Relativity. Wald R. M. 1st Ed. University of Chicago Press. 1984.



Mathematics

Semester-III


MA-575 L – T – P - C Discrete Mathematics 5 – 0 – 0 – 5 Objective: Computer science uses an area of Mathematics as ‘Discrete Mathematics’. The purpose of this course is to provide knowledge of topics from Discrete Mathematics such as Graph Theory, Counting techniques & logic algebra.

Course Content

Set Theory - sets & classes, relations & functions, recursive definitions, posets, Zorn’s

lemma, cardinal & ordinal numbers; Logic - propositional & predicate calculus, well-

formed formulas, tautologies, equivalence, normal forms, theory of inference.

Combinatorics - permutation & combinations, partitions, pigeonhole principle, inclusion-

exclusion principle, generating functions, recurrence relations. Graph Theory - graphs &

digraphs, Eulerian cycle & Hamiltonian cycle, adjacency & incidence matrices, vertex

colouring, planarity, trees.

Text/Reference Books 1. Discrete Mathematical Structures with Applications to Computer Science.

Tremblay J. P. & Manohar R. McGraw-Hill Book Co. 1997. 2. Finite Mathematics. Lipschutz S. International edition. McGraw-Hill Book

Company, New York. 1983. 3. Elements of Discrete Mathematics. Liu C. L. McGraw-Hill Book Co. 1977. 4. Graph Theory with Applications to Engineering & Computer Sciences Deo

N. PHI. 1974.



Mathematics

Semester-III


MA-576 L – T – P - C Linear programming & extensions 5 – 0 – 0 – 5 Objective: The objective of this course is to expose student to understand the importance of optimization techniques & the theory behind it.

Course Content

Linear Models: Formulation & Examples, Basic Polyhedral Theory- Convexity, Extreme

points, Supporting hyperplanes etc, Simplex Algorithm- Algebraic & Geometrical

approaches, Artificial variable technique, Duality Theory: Fundamental theorem, Dual

simplex method, Primal-dual method, Sensitivity Analysis, Bounded Variable L.P.P.

Transportation Problems: Models & Algorithms, Network Flows: Shortest path Problem,

Max-Flow problem & Min-cost Flow problem, Dynamic Programming: Principle of

optimality, Discrete & continuous models.

Text/Reference Books 1. Linear & Combinatorial Programming. Murty K. G. John Wiley & Sons. 1976.

Revised in 2007. 2. Introduction to Operations Research. Hillier F. S. & Lieberman G. J. 8th Edition.

McGraw- Hill. 2005. 3. Linear Programming & Network flows. Bazara M. S., Jarvis J. J. & Sherali H. D.

John Wiley & Sons, New York. 1990. 4. Mathematical Programming Techniques. Kambo N. S. Affiliated East-West Press

New Delhi. 1984.



Mathematics

Semester-III


MA-577 L – T – P - C Numerical Linear Algebra 5 – 0 – 0 – 5 Objective: The objective of this course is to expose student to understand the importance of numerical solutions to Linear Algebra problems & its applications to science & engineering.

Course Content

Fundamentals - overview of matrix computations, norms of vectors & matrices, singular

value decomposition (SVD), IEEE floating point arithmetic, analysis of roundoff errors,

stability & ill-conditioning; Linear systems - LU factorization, Gaussian eliminations,

Cholesky factorization, stability & sensitivity analysis; Jacobi, Gauss-Seidel &

successive over relaxation methods; Linear least-squares - Gram- Schmidt orthonormal

process, rotators & reflectors, QR factorization, stability of QR factorization; QR method

linear least-squares problems, normal equations, Moore- Penrose inverse, rank

deficient least-squares problems, sensitivity analysis. Eigenvalues & singular values -

Schur's decomposition, reduction of matrices to Hessenberg & tridiagonal forms; Power,

inverse power & Rayleigh quotient iterations; QR algorithm, implementation of implicit

QR algorithm; Sensitivity analysis of eignvalues; Reduction of matrices to bidiagonal

forms, QR algorithm for SVD.

Text/Reference Books 1. Numerical Linear Algebra. Trefethen L. N. & Bau D. SIAM. 1997. 2. Fundamentals of Matrix Computation. Watkins D. S. 2nd Edition. John Wiley &

Sons. 2002. 3. Applied Numerical Linear Algebra. Demmel J. W. SIAM. 1997. 4. Numerical Linear Algebra & Applications. Datta B. N. 2nd Edition. SIAM. 2010.


5. Matrix Computation. Golub G. H. & Van Loan C. F. 3rd Edition. Hindustan book agency. 2007.



Mathematics

Semester-III


MA-578 L – T – P - C Functional Analysis 5 – 0 – 0 – 5 Objective: To study the details of Banach & Hilbert Spaces & their algebra.

Course Content

Normed linear spaces, Banach spaces; Continuity of linear maps, Hahn-Banach

theorem, open mapping & closed graph theorems, uniform boundedness principle;

Duals & transposes, weak & weak* convergence, reflexivity; Spectra of bounded linear

operators, compact operators & their spectra; Hilbert spaces, bounded linear operators

on Hilbert spaces; Adjoint operators, normal, unitary, self-adjoint operators & their

spectra, spectral theorem for compact self-adjoint operators.

Text/Reference Books 1. Functional Analysis. Limaye B. V. 2nd Edition. Wiley Eastern. 1996. 2. Introduction to Functional Analysis with Applications. Kreyszig E. John Wiley &

Sons. 1978. 3. A Course in Functional Analysis. Conway J. B. Springer. 1990.



Mathematics

Semester-III


MA-579 L – T – P - C Integral Transforms & Integral Equations 5 – 0 – 0 – 5 Objective: The objective of this course is to expose student to understand the importance of transform techniques & various types of integral equations & their solutions.

Course Content

Basic integral transforms: Fourier transform, Fourier sine & cosine transforms, Laplace

transform, Hankel transform, Mellin transform, Radon transform. Finite Fourier sine &

cosine transforms, finite Hankel transforms. Construction of the kernels of integral

transforms: kernels on a finite interval, circular region, a semi-finite interval & radially

symmetric interval, kernels for discrete to continuous spectrum. Applications to ODEs,

hyperbolic PDEs.

Occurrence of integral equations, Regular integral equations: Volterra integral

equations, Fredholm integral equations, Volterra & Fredholm equations with regular

kernels. Symmetric kernels & orthogonal systems of functions. Singular integral

equations: weakly singular integral equations, Cauchy singular integral equations,

hypersingular integral equations. Bernstein polynomials: properties & its use in solving

integral equations. Green's function in integral equations.

Text/Reference Books 1. Integral Transforms & Their Applications. Debnath L. & Bhatta D. D. Book World

Enterprises. 2006. 2. Applied Integral Transforms. Antimirov M. Ya., Kolyshkin A. A. & Valliancourt R.

CRM Monograph Series, American Mathematical Society. 2007. 3. The Transforms & Applications Handbook. Poularikas A. D. CRC Press. 1996. 4. Integral Equations. Tricomi F. G. Dover Publications Inc. New York. 1985.


5. Integral Equations: A Practical Treatment from Spectral Theory to Applications. Porter D. & Stirling D. S. G. Cambridge University Press. 1990.

6. Singular Integral Equations. Muskhelishvili N. I. Dover Publications Inc., New York. 2008.



Mathematics

Semester-III


MA-581 L – T – P - C Numerical solutions of PDE 5 – 0 – 0 – 5 Objective: The objective of this course is to expose student to understand the importance of finite difference methods for solving partial differential equations.

Course Content

Finite differences: grids, derivation of difference equations. Elliptic equations, discrete

maximum principle & stability, residual correction methods (Jacobi, Gauss-Seidel &

SOR methods), LOD & ADI methods. Finite difference schemes for initial & boundary

value problems: Stability (matrix method, von-Neumann & energy methods), Lax-

Richtmyer equivalence Theorem. Parabolic equations: explicit & implicit methods

(Backward Euler & Crank-Nicolson schemes) with stability & convergence, ADI

methods. Linear scalar conservation law: upwind, Lax-Wendroff & Lax-Friedrich

schemes & CFL condition.

Text/Reference Books 1. The Finite Difference Methods in Partial Differential Equations. Mitchell R. &

Griffiths S. D. F. John Wiley & Sons, NY. 1980. 2. Numerical Solutions of Partial Differential Equations. Smith G. D. 3rd Edition,

Calrendorn Press, Oxford. 1985. 3. Finite difference Schemes & Partial Differential Equations. Strikwerda J. C.

Wadsworth & Brooks/ Cole Advanced Books & Software, Pacific Grove, California. 1989.

4. Numerical Partial Differential Equations : Finite Difference Methods. Texts in Applied Mathematics, Vol. 22. Thomas J. W., Springer Verlag, NY, 1999.

5. Numerical Partial Differential Equations: Conservation Laws & Elliptic Equations. Texts in Applied Mathematics, Vol. 33. Thomas J. W. Springer Verlag, NY, 1999.



Mathematics

Semester-III


MA-582 L – T – P - C Graph Theory 5 – 0 – 0 – 5 Objective: The objective of this course is to expose student to advanced topics in Graph theory.

Course Content

Basic definitions. Blocks. Ramsey Numbers. Degree sequences. Connectivity. Eulerian

& Hamiltonian Graphs. Planar graphs & 5-colour theorem. Chromatic numbers.

Enumeration. Max-Flow Min-Cut Theorem. Groups & graphs. Matrices & graphs.

Matchings & Hall’s Marriage Theorem. Eigen values of graphs.

Text/Reference Books 1. Graph Theory & Applications. Bondy J. A. & Murthy U. S. R. Macmillan, London.

1976. 2. A First look at Graph Theory. Clark J. & Holton D. A. Allied Publishers, New

Delhi. 1995. 3. Graph Theory. Gould R. Benjamin/Cummings, Menlo Park. 1989. 4. Algorithmic Graph Theory. Gibbons A. Cambridge University Press, Cambridge.

1989. 5. Graphs : An Introductory Approach. Wilson R. J. & Watkins J. J. John Wiley &

Sons, New York. 1989. 6. Introduction to Graph Theory. 4th Edition. Wilson R. J. Pearson Education. 2004. 7. A First Course in Graph Theory. Choudum S. A. MacMillan India Ltd. 1987.



Mathematics

Semester-III


MA-583 L – T – P - C Non Linear programming 5 – 0 – 0 – 5 Objective: This course introduces advanced topics in non-linear programming & operation research. Course Content

Nonlinear programming Problems: One variable unconstrained optimization, multi-

variable unconstrained optimization, Karush-Kuhn-Tucker(KKT) conditions for

constrained optimization, quadratic programming, separable programming, convex &

nonconvex programming. Dynamic Programming: Characteristics of dynamic

programming problems, deterministic dynamic programming & probabilistic dynamic

programming, Network Analysis, Including PERT-CPM : Shortest path problem,

minimum spanning tree problem, maximum flow problem, minimum cost flow problem,

network simplex method, project planning control with PERT-CPM. Sequencing

Problems: N jobs through two & three machines, N jobs through M machines, Inventory

Models: Generalized inventory models, deterministic models & probabilistic models.

Decision Analysis: Decision making with & without experimentation, decision tree, utility

theory, Forecasting method, Markov chains & Markov Decision Processes.

Text/Reference Books 1. Nonlinear Programming. Bazarra M. & Shetty C. Theory & Algorithms. John

Wiley & Sons. 1979. 2. Optimization: Theory & Practice. Joshi M. C. & Moudgalya K. Narosa Publishing

House. 2004. 3. Introduction to Optimization. Beale E. M. L. John Wiley & Sons. 1988.



Mathematics

Semester-IV


MA-584 L – T – P - C q-Hypergeometric Functions 5 – 0 – 0 – 5 Objective: This course provides a thorough study of the generalized basic (or q-) hypergeometric series.

Course Content

Basic (or q-) analogue of hypergeometric functions, The q-binomial theorem, Heine's

transformation formulas for 2¢1 series, Heine's q-analogue of Gauss' summation

formula, Jacobi's triple product identity, theta functions, & elliptic numbers, The Bailey's

transform, its q-analogue & applications, A q-analogue of Saalschütz's summation

formula, The Bailey-Daum summation formula, q-analogues of the Karlsson-Minton

summation formulas, The q-gamma & q-beta functions, The q-integral, Summation,

transformation, & expansion formulas, Well-poised, nearly-poised, & very-well-poised

hypergeometric & basic hypergeometric series, A summation formula for a terminating

very-well-poised 4¢3 series, A summation formula for a terminating very-well-poised 6¢5

series, Watson's transformation formula for a terminating very-well-poised 8¢7 series,

Jackson's sum of a terminating very-well-poised balanced 8¢7 series, Some special &

limiting cases of Jackson's & Watson's formulas: the Rogers-Ramanujan identities,

Bailey's transformation formulas for terminating 5¢4 & 7¢6 series, Bailey's transformation

formula for a terminating 10¢9 series

Text/Reference Books 1. Basic Hypergeometric Series. Gasper G. & Rahman M. Cambridge University

Press: Cambridge. 2004. 2. Generalized Hypergeometric Functions. Slater L. J. Cambridge University Press:

Cambridge. 1966.


3. q-Hypergeometric Functions & Application. Exton H. Ellis Horwood Ltd.: Chichester. 1983.

4. Special Functions. Andrews G. E., Askey R. & Roy R. Cambridge University Press: Cambridge. 1999.

5. Generalized Hypergeometric Series. Bailey W. N. Cambridge University Press, Cambridge, reprinted by Hafner: New York. 1964.



Mathematics

Semester-IV


MA-585 L – T – P - C General Relativity 5 – 0 – 0 – 5 Objective: This course provides advanced concepts of general theory of relativity which provide important background to study the various cosmological models.

Course Content

Linearised Theory: Gravitational radiation; Einstein's equation for weak fields;

Generation, propagation & detection of gravitational waves; Conservation of energy

momentum & angular momentum.

Relativistic Astrophysics: Spherically symmetric space-times; Schwarzschild metric;

Stellar models & gravitational collapse; Trajectories around a compact object. Black

holes.

Relativistic Cosmology: Cosmological principles; Standard Model: Robertson-Walker

metric & Friedman solution; De Sitter space-time & inflationary cosmology; The early

universe.

Text/Reference Books 1. Cosmology: The origin & evolution of cosmic structures. Coles P. & Lucchin F.

2nd Ed. John Wiley & Sons. 2002. 2. Gravitation & Cosmology. Weinberg S. 1st Ed. John Wiley & Sons.1972. 3. Problem Book in Relativity & Gravitation. Lightman A. P., Press W. H. Price R. H.

& Teukolsky S. A. Princeton University Press. 1975. 4. Exploring Black Holes: Introduction to General Relativity. Taylor E. F. &. Wheeler

J. A. 1st Ed. Addison Wesley. 2000. 5. Introduction to Cosmology. Narlikar J. V. 3rd Ed. Cambridge University

Press.2002.



Mathematics

Semester-IV


MA-586 L – T – P - C Classical Mechanics & Calculus of variations 5 – 0 – 0 – 5 Objective: The aim of the course is to provide the concept of calculus of variations & its applications. The classical mechanics portion provides mathematical techniques needed in quantum mechanics & modern physics.

Course Content

Generalized coordinates, Lagrange’s equations, Hamilton’s canonical equations,

Hamilton’s principle & principle of least action, Two-dimensional motion of rigid bodies,

Euler’s dynamical equations for the motion of a rigid body about an axis, theory of small

oscillations.

Variation of a functional, Euler-Lagrange equation, Necessary & sufficient conditions for

extrema. Variational methods for boundary value problems in ordinary & partial

differential equations.

Text/Reference Books 1. Methods of Mathematical Physics. Vol I. Curant R. & Hilbert D. Interscience

Press. 1953. 2. Calculus of Variations. Elsgolc L.E. Pergamon Press Ltd. 1962. 3. Calculus of Variations with Applications to Physics & Engineering. Robert W.

Dover. 1974. 4. Classical Dynamics. Greenwood D. T. PHI. 1985. 5. Classical Mechanics. 3rd Edition. Goldstein H. Narosa Publishing House. 2011. 6. Classical Mechanics. Rane N.C. & Joag P. S. C. Tata McGraw Hill. 1991. 7. Principles of Mechanics. 3rd Edition. Synge J. L. & Griffth B. A. McGraw Hill Book

Co., New York. 1970.



Mathematics

Semester-IV


MA-587 L – T – P - C Differential Geometry 5 – 0 – 0 – 5 Objective: To introduce the curves & surfaces in E3 & to study their nature with the help of vector calculus.

Course Content

Graphs & level sets of functions on Euclidean spaces, vector fields, integral curves of

vector fields, tangent spaces.

Surfaces in Euclidean spaces, vector fields on surfaces, orientation, Gauss map.

Geodesics, parallel transport, Weingarten map.

Curvature of plane curves, arc length & line integrals.

Curvature of surfaces.

Parametrized surfaces, local equivalence of surfaces.

Gauss-Bonnet Theorem, Poincare-Hopf Index Theorem

Text/Reference Books 1. Differential Geometry of Curves & Surfaces. doCarmo M. Prentice Hall. 1976. 2. Elementary Differential Geometry. O'Neill B. Academic Press, New York. 1966. 3. Differential Geometry. Stoker J. J. Wiley-Interscience. 1969. 4. Elementary Topics in Differential Geometry. Thorpe J. A. Springer (India). 2004.



Mathematics

Semester-IV


MA-588 L – T – P - C Number Theory 5 – 0 – 0 – 5 Objective: To introduce the students to the fascinating world of numbers. There are so many open problems such as Goldbach Conjecture. In fact there are plenty of properties of integers which are yet to be explored.

Course Content

Divisibility, Primes, Congruences, Residue systems, Primitive roots; Quadratic

reciprocity, Some arithmetic functions, Farey fractions, Continued fractions, Some

Diophantine equations, Bertrands postulate & the partition function.

Text/Reference Books 1. Introduction to the Theory of Numbers. Adams W. W. & Goldstein L. J. 3rd

Edition. Wiley Eastern. 1972. 2. A Concise Introduction to the Theory of Numbers. Baker A. Cambridge

University Press, Cambridge. 1984. 3. An Introduction to the Theory of Numbers. Niven I. & Zuckerman H. S. 4th

Edition. John Wiley & Sons, New York. 1980.

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