PG Syllabus

Department of Mathematics

Assam University, Silchar (To be implemented from session 2013-14)

Structure of PG-Syllabus

Department of Mathematics

Assam University, Silchar

Semester-I

Semester-II

Marks

Internal

External Total

Paper No.

Name of the paper

M.M. P.M. M.M. P.M.

L P T C

M-101 Algebra-I 25 10 75 30 100 4 1 5

M-102 Real Analysis-I 25 10 75 30 100 4 1 5

M-103 Ordinary Differential

Equations

25 10 75 30 100 4 1 5

M-104 Numerical Analysis 25 10 75 30 100 4 1 5

M-105

(Theory)

Computer

Programming: C &

C++

25 10 45 18 4 1 5

M-105

(Practical-I)

Practical-I (External) 30

(5+5+20)

Notebook+

Viva-voce +

Experiment

70+30=

100 3

Total 125 50 375 150 500 20 3 5 25

Marks

Internal

External Total

Paper No.

Name of the paper

M.M. P.M. M.M. P.M.

L P T C

M-201 Topology 25 10 75 30 100 4 1 5

M-202 Algebra-II 25 10 75 30 100 4 1 5

M-203 Discrete Mathematics 25 10 75 30 100 4 1 5

M-204 Calculus of

Variations and

Integral Equations

25 10 75 30 100 4 1 5

M-205

(Theory)

Partial Differential

Equations

25 10 45 18 4 1 5

M-205

(Practical-II)

Practical-II (External) 30

(5+5+20)

Notebook+

Viva-voce+

Experiment

70+30=

100 3

Total 125 50 375 150 500 20 3 5 25

Semester-III

Semester-IV

In external examinations, each unit of a paper containing 75 (45) marks carries 15(09)

marks.

L=Lecture, T=Tutorial, P=Practical, C=Credit, MM=Maximum Marks, PM=Pass Marks

Marks

Internal

External Total

Paper No.

Name of the paper

M.M. P.

M.

M.M. P.M.

L P T C

M-301 Real Analysis-II 25 10 75 30 100 4 1 5

M-302 Fluid mechanics 25 10 75 30 100 4 1 5

M-303 Complex Analysis 25 10 75 30 100 4 1 5

M-304 Classical Mechanics and

Tensors

25 10 75 30 100 4 1 5

M-305

(Theory)

Operations Research and

Optimization Techniques-I

25 10 75 30 100 4 1 5

Total 125 50 375 150 500 20 3 5 25

Marks

Internal

External Total

Paper

No.

Name of the paper

M.M. P.M. M.M. P.M.

L P T C

M-401 Functional Analysis 25 10 75 30 100 4 1 5

M-402 Linear Algebra 25 10 75 30 100 4 1 5

M-403 Operations Research and

Optimization Techniques-II

25 10 75 30 100 4 1 5

M-404 Optional Paper (any one)

a. Relativity

b. Differential

Geometry

c. Mathematical

Modeling

d. CFD

e. Operator Theory

f. Groups and

Representations

g. Mathematical

Statistics

25 10 75 30 100 4 1 5

M-405

Project 25 10 75

50+25

(Dissertation+

Viva-voce)

30 100

4 1 5

Total 125 50 375 150 500 20 5 25

M-101Algebra-I

UNIT-I

Relation and function, equivalence relation, partition, binary operation, semigroups, groups, abelianand non-abelian groups, subgroups, cyclic subgroups, definition and example of cyclic groups, propertiesof a cyclic group (application in solving problems).

UNIT-II

Permutation group, cycles and cycle notations, decomposition of a permutation as a product of disjointcycles, even and odd permutations, alternating groups, conjugate elements in a group, conjugacy classesin Sn, left and right cosets, groups of cosets, Lagranges theorem (application in solving problems).

UNIT-III

Normal subgroups (normal subgroups of to be determined), automorphisms and inner automorphisms,factor groups, homomorphism of groups, kernel, fundamental theorem on group homomorphism, iso-mormorphism theorems and applications, simplicity of An for n ≥ 5 , commutator subgroups, Cayleystheorem (application in solving problems).

UNIT-IV

Rings, integral domain, fields, characteristic of a ring, Fermats and Eulers theorems, ring homomor-phism and isomorphism, kernel, quotient fields of an integral domain (thrust is on solving problems).

UNIT-V

Subrings, ideals (left and right ideals), quotient rings, algebra of ideals, prime and maximal ideals,factorization in rings (associates, irreducible, prime elements, gcd etc.), Euclidean domain, principalideal domain, unique factorization domain (thrust is on solving problems).

Recommended Text:

1. Gallien, Joseph, Contemporary Abstract Algebra, Narosa Publishing House.

References:

1. Hungerford, T.W., Algebra, Springer (Unit I-III)

2. Fraleigh, J.B., A First Course in Abstract Algebra, Narosa Publishing House (Unit-I-V)

3. Dumit, D.S. and Foote, R.M., Algebra, John Wiley and Sons

1

M-102Real Analysis-I

UNIT-I

Basic review of mathematical logic (statements, negation, contra positive, truth table), relations andfunctions, finite, infinite, countable, uncountable sets and related concepts of notions f(∪An), f−1(∩An)etc., real number system as a complete order field, Archimedean property, supremum, infimum.

UNIT-II

Open and closed subsets, limit points, interior points in R, Bolzano-Weirestrass theorem on infinite sets,Heine-Borel theorem in , sequences and series of real numbers and related theorems on convergence,limsup, liminf, Bolzano-Weirestrass theorem for sequences.

UNIT-III

Continuity, uniform continuity, differentiability, mean-value theorems of real-valued functions of a realvariable.

UNIT-IV

Sequences and series of functions, pointwise and uniform convergence, power series, radius of conver-gence.

UNIT-V

Monotone functions, types of discontinuities, functions of bounded variations.

Recommended Text:

1. Sohrab, Hausang H., Basic Real Analysis, Birkhauser

References:

1. Bartle, Robert G., The Elements of Real Analysis, John Wiley and Sons

2. Rudin, Walter, Primciples of Mathematical Analysis, McGraw-Hill

2

M-103Ordinary Differential Equations

UNIT-I

Existence and uniqueness of initial value problems for first order ODEs, singular solutions of 1st orderODEs, simultaneous linear 1st order ODEs , reduction of higher order linear ODEs to a system of1st order equations, non-linear autonomous system, phase plane analysis, critical points, stability,linearization, Liapunov stability.

UNIT-II

BVP for ODE, Strum-Liouville problem, orthogonality of eigen functions, Gram-Schimidt method oforthonormality, adjoint and self-adjoint equations, solutions of equation from the solution of its adjointequation, linear operator system, Greens function and its applications.

UNIT-III

Ordinary and Regular Singular Power series solutions, Legendre equation, Legendre Polynomial, Ro-drigues formula, Legendre-Fourier series, Associated Legendre equation and its functions, Bessels equa-tion, Bessels functions, Orthogonality of Bessels functions, Fourier- Bessel Series, Modified Bessels equa-tions, reduction of equations to Bessels form. Hyper-geometric functions, Confluent hyper-geometricfunctions, Integral representation of hyper-geometric and confluent hyper-geometric functions. Hyper-geometric equation and its solution.

UNIT-IV

Integral Transform: Laplace transform of functions, existence of Laplace transform, properties ofLaplace transform, inverse Laplace transform, Heavisides expansion formula, convolution theorem, ap-plication of Laplace transform to solve initial and boundary value problems. Fourier transform fofunctions, half-range Fourier transform, application of Fourier transform to solve conditional ODEs.

UNIT-V

Numerical methods in BVP: Shooting method, Finite difference estimations of derivatives, finite differ-ence method, weighted residual methods, Cubic Spline method.

Recommended Text:

1. Coddington, E.A. Levinson, N., Theory of Ordinary Differential equation, Tata McGraw Hill

2. Simmons,G.F., Differential Equations with Applications, Tata McGraw Hill

3. Ross,Shephly L.,Differential Equations, John Wiley and Sons

References:

1. Somasundaran, D., Ordinary Differential Equation, Narosa Publications

2. Kreyzig,E., Advanced Engineering Mathematics, John Weily and Sons

3. Ahsan, Z., Differential Equations and their Applications, Prentice Hall of India

4. Spiegel,M.R., Laplace Transforms: Tata McGraw-Hill Publications

5. Mondol, C.R., Text book of Ordinary Differential Equations, Prentice Hall of India

6. Carlaw , H.S.. and Jager,J.E., Operational Methods in Applied Mathematics, Dover Publishers

3

7. Harper, C., Introduction Mathematical Physics, PHI

8. Froberg, C.F., Introduction to Numerical Analysis, AddisonWesley Reading

9. Burden and Faires, Numerical Analysis

10. Hildebrand,F.B., Introduction to Numerical Analysis, Tata McGraw-Hill

4

M-104Numerical Analysis

UNIT-I

Number systems and errors, types and sources of error, propagation of error, error in Numerical Anal-ysis, Hermite interpolation,central difference interpolating formula, piecewise interpolation, inverse in-terpolation, estimation of error in different interpolation formulae, spline interpolation cubic splines ,least square approximation to discrete data,

UNIT-II

Numerical differentiation, Error in numerical differentiation, Numerical integration- the Trapezoidalrule, Simpsons rules, Booles and Weddles rules, Romberg integration, Gauss quadrature rules ,singularintegrals. Approximation of function : Chebyshev approximation , Lanczos economisation, least squareapproximation and orthogonal polynomials,min-max polynomial approximation.

UNIT-III

Numerical solution of algebraic and transcendental equations in one unknown, bi-section method,regula-falsi method, fixed point iteration method, secant method, Newton-Raphson method, rate ofconvergence of iterative methods, generalized Newtons method,system of linear equations-Direct anditerative methods.

UNIT-IV

Numerical solution of ordinary differential equations, Picards method, Eulers method, modified Eulersmethod, Runge-Kutta method, solution of boundary value problems for linear second order equations.

UNIT-V

Difference equations-Defination of difference equation ,order of difference equation ,general and particu-lar solution of difference equation, Linear difference equations, Homogenious linear difference equationswith constant coefficients,linear independent solutions, solution of non linear homogeneous differenceequations by different methods,simultaneous difference equations,partial difference equations.

Recommended Text:

1. Atkinson, Kendall,E.: An Introduction to Numerical Analysis(John Wiely Sons (Asia) Pvt.Ltd.

2. Acharya , B.P. and Das, R.N.: A Course on Numerical Analysis, Kalyani Publishers

3. Murray R Spiegel, Theory and Problems of calculus of finite differences and difference equations

References:

1. Sarborough, J.B., Numerical Mathematical Analysis, Oxford and Intl

2. Conle Boor, Elementary Numerical Analysis, McGraw Hill

3. Boehm, W, Numerical Methods, University Press and Prautzsc

4. Sastry, S.S., Introduction to Methods of Numerical Analysis, Prentice Hall of India

5. Jain, M.K.,Numerical Methods- Problems and Solutions, New Age Intl.

5

M-105 (Theory)Computer Programming: C and C++

UNIT-I

The C character set, variables, data types, constants, arithmetic expressions, logical operators, condi-tional operators, control structures in C, if, nested if, if....else, switch...case, for, while, do while, break,continue. Arrays, one dimensional array, two dimensional array, array processing searching, sorting,merging, matrix multiplication.

UNIT-II

Functions, user defined functions, function declaration and prototypes, recursive functions, pointersarrays of pointers, pointers to function, Evaluation of definite integrals by summation method. struc-tures declaration, arrays of structures, pointers to structures, structure as function arguments Datafiles opening, closing file processing.

UNIT-III

Object oriented programming techniques, C++ programming basics basic program construction com-ments, variables, constants, expressions, statements, cin and cout, manipulators, type conversion, arith-metic operators, library functions. Objects and classes simple class, specifying the class, using the class.C++ objects as data types constructors, destructors objects as function arguments, returning objectsfrom functions and classes.

UNIT-IV

Arrays, multidimensional arrays, passing arrays to functions, arrays as class member data arrays ofobjects strings arrays of strings, strings as class members, a user defined string type. Operatoroverloading - overloading unary operators, overloading binary operators arithmetic operators, multipleoverloading data conversion. Inheritance derived class and base class derived class constructorsoverriding member functions class hierarchies public and private inheritance levels of inheritancemultiple inheritance, class within classes.

UNIT-V

Virtual functions and polymorphism, Templates, Exception handling. Files and streams stream classhierarchy, stream class, header files string I / O character I/O I/O with multiple objects fstreamclass and open function file pointers tellg and seekg functions .Recommended Text:

1. Balaguruswamy, E., Programming in C, TMH

2. Gollfried, Byron S.,Programming with C, S. Series TMH

3. Kanetkar, Y, Let us C, BPB Publications

4. Budd, T., An introduction to object oriented programming Timothy Budd, Addison-Wesleypublishing company

5. Object oriented programming in Turbo C , Robert Lafore, Galgotia

6. Grady Booch, Object oriented analysis and design, Pearson Education

1. Programming with C (Tata McGraw Hill) By : D. Ravichanoran.

2. Object-Oriented programming with C, Tata McGraw Hill

3. Liberty, Jess Keosh, Jim : An Introduction to programming C, Prentice-Hall India

4. Nancy, Berry, Network Programming in C, PHI

6

M-105Computer Programming: C and C++ (Practical-I)

Problems may be solved using C and C++ programming languages.

7

M-201Topology

UNIT-I

Metric spaces, examples, open and closed sets, neighbourhoods, closure, dense subsets, separable metricspaces, boundaries, interiors, sequences in metric spaces, convergence of sequences and continuity,homeomorphism.

UNIT-II

Complete metric spaces, cantors intersection theorem, completion of a metric space, isometry, isometricisomorphism, Bourbakis Mittag-Leffler theorem, Baires category theorem, compactness in metric spaces,Heine-Borel theorem.

UNIT-III

Topological spaces, definition and examples, open and closed sets, metrizable and Hausdorff spaces,neighbourhoods, basis, first countable spaces, closure, Kuratowski closure operation, dense subsets,separable spaces, boundary and interiors, continuity and homeomorphism.

UNIT-IV

Compactness, product topology, Tychonoffs theorem, local compactness, one-point compactification,paths, path-connected ness, connectedness, intermediate value theorem, components, totally discon-nected spaces, local connectedness.

UNIT-V

Separation properties, T0, T1-spaces, regular, normal, completely regular spaces, Urysohns metrizationtheorem, Tietzes extension theorem.

Recommended Text:

1. Volker Runde, A Taste of Topology, Springer, 2005 (Unit-I, page 23-40: Unit-II, page 41-58:Unit-III, page 61-78: Unit-IV, page 79-100: Unit-V, page 100-116)

References:

1. Munkers, J.K., Topology, Pearson Prentice Hall

2. Kelley, J., Topology, Springer

3. Dugundgi, J., Topology, Allyn and Bacon

8

M-202Algebra-II

UNIT-I

Direct and semidirect product of groups, group actions, class equations, automotphism groups, innerautomorphism groups (applications through permutation groups and the general linear groups etc.).

UNIT-II

Sylows theorems and applications, structure theorems for finite groups, solvable groups.

UNIT-III

Polynomial rings, Euclidean algorithm, irreducible polynomials, Gauss lemma, Eisensteins irreducibil-ity criterion and applications, elements of vector spaces (vector spaces, linear dependence and indepen-dence, linear combination, linear transformation, basis, dimension etc.).

UNIT-IV

Field extension, degree of an extension, algebraic extension, splitting field, normal extensions.

UNIT-V

Separable extensions, finite field, perfect fields, Galois extension, fundamental theorem of Galois theory.

Recommended Text:

1. Bhattacharjee, P.B., Jain, S.K., Nagpaul, S.R., Basic Abstract Algebra, Cambridge UniversityPress

References:

1. Hungerford, T.W., Algebra, Springer

2. Dumit, D.S. and Foote, R.M., Algebra, John Wiley and Sons

9

M-203Discrete Mathematics

UNIT-I

Divisibility, greatest common division, least common multiple, prime numbers, fundamental theorem ofarithmetic, Eulers Phi function, Diophantine equations ax+ by = c, congruences and elementary prop-erties, residue systems, theorems of Euler, Fermat and Wilson, chinese remainder theorem, polynomialcongruences; applications.

UNIT-II

Primitive roots and indices, quadratic residues, Legendre symbol, Jacobi symbol, law of quadraticreciprocity, arithmetic functions, multiplicative arithmetic functions, Mobius inversion formula; appli-cations.

UNIT-III

Graph, Subgraph, Varities of graphs, degree and incidence, isormorphism, intersection graph, operationson graph, walks and connectedness, trees, Forest, spanning trees, cycles and cocycles.

UNIT-IV

Traversability : Eulerian and Hamiltomian graphs, plane and planner graphs, Kuratowskis theorem.

UNIT-V

Poset and lattices, Boolean lattices and Boolean algebras, Boolean functions, applications, basics ofautomata theory.

Recommended Text:

1. Burton, D.M. : Elementary Number Theory (universal Book Stall)

2. Niven,l I.H.S. Zuckerman, H.L. Montgomery. An Introduction to the Theory of Numbers (JohnWiley LPE)

3. Harary, F. : Graph Theory (Narosa Publishing House)

4. Liu, C.L.: Elements Discrete Maths (Tata McGraw Hill)

References

1. Ireland, K. Rosen, M., A classical Introduction to Modern Number Theory. (Springer LPE)

2. West, Introduction to Graph Theory (Prentice Hall of India)

3. Kolman,B., Busby, R.C. Ross, S., Discrete Mathematical Structures (Prentice Hall of India)

10

M-204Calculus of Variation and Integral Equations

UNIT-I

Integral equations of the first, second and third kinds, examples, finite difference approximations, theFredholm alternative, Hadamards inequality, Hilbert spaces.

UNIT-II

Fixed point theorem, elementary existence theorems, Volterra equations, kernels with weak singularities,degenerate kernels, Volterra equations of the first kind.

UNIT-III

Compact operators, self-adjoint compact operators, applications to differential equations, Positive op-erators, approximation of eigen values, Fredholm equations with self-adjoint compact operators, theFredholm alternative.

UNIT-IV

Linear functionals, ordinary differential operators, partial differential operators, partial differential equa-tions, Fourier transforms, applications of Fourier transforms, Laplace transforms, application of Laplacetransforms, application of Laplace transform.

UNIT-V

Function spaces, Functionals, variation of a functional, the fundamental lemmas of the calculus of vari-ation, a necessary condition for the differentiable functional to have an extremum, Eulers equation,related results (theorems and applications), Eulers equation of several variables, invariance of Eulersequation, isometric problems, brachistochrone problem.

Recommended Text:

1. Harry Hochstadt, Integral Equations, a Wiley Inter-science publications, 1973

2. I.M. Gelfand, S.V. Fomin, A Calculus of Variations, Dover Publications

References

1. Kikhlin, S.G., Linear Integral Equations, Hudson Book Agency

2. Kanwal, R.P., Linear Integral Equations, Academic Press, New York, 1998

3. Gupta, A.S., Calculus of Variations with Applications

11

M-205(Theory)Partial Differential Equations

UNIT-I

First order partial differential equations (linear and nonlinear) and their solutions by inspection, La-granges and Charpits methods, characteristic curves and characteristic surfaces, linear PDE with con-stant coefficient-homogeneous equation to find CF PI; non-homogenous linear equation with constantcoefficients-reducible irreducible cases equation reducible to linear homogeneous form.

UNIT-II

Second order PDE with variable coefficients: Introduction, Monges method for solving the equationsof the form : Rr + Ss + Tt + U(rts)2 = V (x, y), reduction of Rr + Ss + Tt + f(x, y, z, p, q) = 0 tocanonical forms, method of variation of parameter.

UNIT-III

Application of PDE in Physical Problems : (a) Laplace equations (upto two dimensions) occurrencesolution in different coordinate systems, boundary value problems, separation of variables. (b) Thewave equation- occurrence, solution of one-dimensional wave equation; the Riemann Volterra solution,vibrating membranes, transverse vibrations of a stretching string. (c)The diffusion equation- occurrenceand elementary solution, heat equation in one dimension, separation of variables, the use of integraltransforms (separation of variables for heat equation).

UNIT-IV

Integral transform methods (Laplace and Fourier transforms ) to solve conditional PDE, Browmwitchintegral.

UNIT-V

Numerical solutions of elliptic, parabolic and hyperbolic PDEs (2 independent variable case) by finitedifference method, Liebmanns iterative method for elliptic equations, Bender- Schmidt method, Crank-Nicholson method for solving parabolic equation.

Recommended Text:

1. Partial Differential Differential Equation : E. T. Copson, Oxford University Press

2. Numerical Methods : Vedamurthy and Iyenger (VPH Pub.)

3. Differential Equation by Piaggio(CBS. Publ.)

4. Theory of differential equation Fersyth

5. Partial Diff. Equations : P. Prasad R. Ravindran (New Age)

References:

1. Elements of PDE : I. N. Sneddon (Oxford Univ. Press)

2. An Elementary course in Partial Differential Equation : T Amarnath, narosa Pub.

3. Numeric al Solution of Differential Equation : M.K. Jain, New Age Pub.

4. Method of Mathematical Physics Magenal Murphy

5. Applied Mathematics for Engineers Physicist: Pipes Harvill.

6. Schaums outline of theory and problems of Diff. Equation. Metric Units : Frank Ayres.

7. Partial Differential equations : Epstein.

12

M-205Partial Differential Equations (Practical-II)

Problems from M-205 (Theory) may be solved with the help of softwares like MATLAB, Mathematica.

13

M-301Real Analysis-II

UNIT-I

Basic concepts of normed linear spaces and metric spaces (basic examples as Rn and space of continuousfunctions), functions of several variaaables, total and directional derivatives, equality of mixed partialderivatives, Schwarz lemma, Taylors theorem.

UNIT-II

Extremum problem with/ without constraints, implicit and inverse function theorem.

UNIT-III

Reimann integration and its properties, set functions (additive, countably additive), regular set function,construction of the Lebesgue measure, measurable sets (Borel sets, Cantor set as example).

UNIT-IV

Measurable function, simple function, Lebesgue integration.

UNIT-V

Lebesgues monotone convergence theorem, Fatous theorem, Lebesgues dominated convergence theo-rem, comparison with Reimann integral.

Recommended Text:

1. Rudin, Walter, Principles of Mathematical Analysis, McGraw-Hill

References:

1. Royden, H.L., Real Analysis, Macmillan Pub. Company

2. Apostol, T.M., Mathematical Analysis, Narosa Publishing House (for Unit-I,II)

14

M-302Fluid Mechanics

UNIT-I

Governing equations of fluid motion : Lagrangian and Eulerian Methods of description, Stream line,path line, Vorticity and circulation, Equation of continuity in Fluid Motion (in Lagrangian and Eule-rian Methods). Equivalence of the two forms of equations of continuity, Boundary conditions, Eulersequations of motion, for perfect fluids, integrals of Eulers equations of motion. Lagranges equations ofmotion, Cauchys integrals, Equation of Energy.

UNIT-II

Motion in two dimensions : Two-dimensional motions, stream function, complex potential, source, sinkand doublet; Image, Image in two-dimensions ; Images of a source with regard to a plane, a circle anda sphere; Image of a doublet, Milne-Thomson circle theorem, Theorem of Blasius.

UNIT-III

Motions in three-dimensions : (a) Uniform motion of a sphere in a liquid, axisymmetric motion (b)Vortex Motion : Helmholtz properties of varities, velocity in a vortex field, Motion of a circular vortex,Infinite rows of vortices, Karmans vortex street.

UNIT-IV

Viscous fluids : Navier Stokes equations for viscous flows-some solutions, diffusion of vorticity, Dissipa-tion of energy, Reynolds Number, Steady motion of a viscous fluid between two parallel plates, steadyflow through circular cylindrical pipe and annulus.

UNIT-V

Boundary layer theory : - Dynamical similarity, Parandtls boundary layer equations in two dimensions,Blasius solution. Boundary layer thickness. Displacement thickness, Karman integral equations.

Recommended Text:

1. Text book of Fluid Dynamics F. Chorlton. (Van Nostrand Reinhold Co)

2. Fluid Dynamics D.E. Rutherford. (oliver Boyd)

3. Theoretical Hydrodynamics L.M. Milne Thomson.

4. A treatise of Hydromechanics by W.H. Besant and A.S. Ramsey.

5. Ideal and Incompressible Fluid Dynamics M.E. DNeill and F. Chorlton

References:

1. Hydrodynamics Shantiswarup ( Krishna Prakashan )

2. Theoratical Hydrodynamics Bansilal

3. Hydrodynamics H. Lamb

4. Modern Fluid Dynamics : N. Curle H.J. Davies (Van Nostrand Reinhold Co,)

5. Principles of Ideal Fluid Aerodynamics : Karmacheti Krishna Murti (John Wiley Sons)

15

M-303Complex Analysis

UNIT-I

Extended Complex Plane, Stereographic projections, arguments, complex logarithms, Power series,Holomorphic and Analytic Functions, Cauchy-Riemann equations, the exponential functions, the loga-rithmic functions, complex trigonometric functions.

UNIT-II

Line integrals, differential forms, homotopy and simple connectivity, winding number, Cauchys theo-rem, Goursats theorem, Cauchys integral formula, Power series expansion of holomorphic functions,Moreras theorem, Cauchys inequalities, Liouvilles theorem, Fundamental theorem of algebra, cyclesand homology.

UNIT-III

Counting zeros of holomorphic function, open mapping theorem, Maximum modulus principle, Schwarzslemma, singularities and their classification, Laurent series expansions, Casorati-Weierstrasss theorem,meromorphic functions.

UNIT-IV

Residues, residue theorem, Evaluation of definite integrals, Argument principle, Rouches theorem,harmonic conjugates, Poissons formula.

UNIT-V

Examples of images of regions under elementary holomorphic functions, conformal maps, Mobius trans-formation, cross-ratio, orientation and symmetry principles.

Recommended Text:

1. Gilman, Jane P., Kra, Irwin Rodriguez, Rubi E. : Complex Analysis, In the Spirit of LipmanBers, GMT 245, Springer - Verlag

2. Conway, John B.: Functions of One Complex Variable, Second Edition, Narosa Publishing House.

References:

1. Cartan, H., Elementary Theory of Analytic Functions of One or Several Complex Variable, DoverPublishing House

2. Shastri, Anant R., An Introduction to Complex Analysis, Macmillan India Limited.

3. Ahlfors, L.V., Complex Analysis, McGraw-Hill Book Co.

4. Choudhary,B., The Elements of Complex Analysis, New Age International (P) Limited, Publishers

5. Ponnusamy, S., Foundations of Complex Analysis, Narosa Publishing House.

16

M-304Classical Mechanics and Tensors

UNIT-I

Generalised coordinates, holonomic non-holonomic systems, constraints, DAlemberts principle La-granges equations, Calculus of variations, Euler-Lagrange equation, application of calculus of variationsin dynamical problems, Hamiltons principle, Lagranges equations from Hamiltons principle, extensionof Hamiltons principle to non-conservative and non-holonomic systems, conservation theorems andsymmetry properties.

UNIT-II

Two dimensional motion of rigid bodies, Eulers dynamical equations of motion for a rigid body, Motionof a rigid body about an axis, motion about revolving axis, Eulerian angles, Eulers theorem on themotion of a rigid body, infinitesimal rotations, rate of change of a vector, Coriolis force, Eulers equationsof motion, force free motion of a rigid body.

UNIT-III

Hamilton canonical equations, Hamiltons equations of motion, conservation theorems and physicalsignificance of Hamiltonian, Hamiltons equations from variatioinal principle, principle of least action,Equations of canonical transformation, integral invariants of Poincare, Lagrange and Poisson bracketsas canonical invariants, equations of motion in Poisson bracket notation.

UNIT-IV

Summation convention, dummy and free suffix, Kroneeker delta, definition of tensor, Invariance of tensorequation, covariant and contravariant tensors, addition, subtraction, outer product, Inner productof tensors, line element, The fundamental tensor in cartesian, cylindrical and spherical coordinates,Christoffel symbols of the first and second kind, transformation of Christoffel symbols, formula forsecond-order partial derivative in terms of Christoffel symbols.

UNIT-V

The covariant derivative of a covariant vector, contravariant vector a mixed tensor of second order, ruleof covariant differentiation, velocity gradient tensor in cylindrical and spherical co-ordinates. curvaturetensor, Riemmann Christoffel tensor, Riccei tensor, equation of geodesic, geodesic coordinates, bianchiidentities, Einstein tensor.

Recommended Text:

1. Goldstein, H., Classical Mechanics,Addison Wisley pub. Co.

2. Spain, B., Tensors

References:

1. R.G. Takwala P.S. Puranik, Introduction to Classical Mechanics

2. N.C. Rana P.S. Joag, Classical Mechanics

17

M-305Operations Research and Optimization Techniques-I

UNIT-I

Revised Simplex Method, Post-Optimal Analysis, dynamic programming Sequencing Problem- Process-ing n-job through two machines, three machines, k-machines, processing of 2 job through k-machines

UNIT-II

Game theory, games without saddle point, mixed strategy, algebraic method, graphical method, domi-nance property, solution of a game by L.P. method.

UNIT-III

Integer programming problems, Gomorys all integer cutting plane method, Gomorys mixed integercutting plane method, Branch and bound technique.

UNIT-IV

Deterministic Inventory control Models-advantage of carrying inventory, techniques of inventory controlfor economic lot size models with constant demand/different rates of demand in different cycles finitereplenishment rate ; deterministic inventory models with shortages, techniques of inventory control withshortages for economic lot size models with constant demand and variable order cycle time/constantdemand and fixed reorder cycle time/finite replenishment rate, EOQ models with quantity discounts.

UNIT-V

Replacement problems, Replacement of items when value of money remains constant/changes with con-stant rate, replacement of items that fails completely, group replacement policy, project management.PERT and CPM techniques, activities, Network diagram, forward pass method, float of activity andevent, critical path.

Recommended Text:

1. Wagner, H.M., Principles of Operations Research,Prentice Hall

2. Sharma. J.K., Operations Research : Theory and Application, Mcmillan

3. Man Mohan, Gupta, P.K., Swarup Kanti, Operation Research, S. Chand Sons

4. Taha, H.A, Operation Research, An Introduction, Prentice Hall

5. Sharma,S. D., Operations Research, Kedar Nath Ram Nath

References:

1. Mustafi C.R., Operations Research Methods Practice, New Age Int.

2. Shenoy, L.V., Linear Programming : Methods Applications.

3. Mittal K.V., Optimization Methods : In O.R. and systems Analysis, New Age Int

4. Vohra, N.D., Quantitative Techniques in Management, (Tata McGraw Hill)

18

M-401Functional Analysis

UNIT-I

Normed linear spaces and Banach spaces, completion of normed linear spaces, finite-dimensional normedlinear spaces and subspaces, equivalent norms, compactness and finite dimension,

UNIT-II

Linear operators, Inverse operators, Bounded and continuous linear operators, finite-dimensional do-main and boundedness of operators, norm of a bounded linear linear operator, Bounded linear exten-sion, Linear functionals, continuity and boundedness of linear functionals, algebraic dual and algebraicreflexivity, normed linear space of bounded linear operators, topological dual space and examples.

UNIT-III

Zorns lemma, Hahn-Banach theorem, Hahn-Banach theorem for complex linear spaces and normedlinear spaces, consequences of Hahn-Banach theorems, natural embedding into second topological dualspaces, topological reflexive spaces, topological reflexive spaces and separability and other related re-sults.

UNIT-IV

Adjoint of a bounded linear operator, uniform Boundedness theorem, open mapping theorem, closedgraph theorem and their applications.

UNIT-V

Inner product spaces, Hilbert spaces, properties of inner product spaces, orthogonal complements anddirect sums, orthonormal sets and sequences, series related to orthonormal sequences and sets, totalorthonormal sets and sequences, Parsevals identity, examples of total orthonormal sets, presentationof functionals on Hilbert spaces, Riesz representation theorem, Hilbert-adjoint operators, self-adjoint,unitary and normal operators.

Recommended Text:

1. Kreyszig, Erwin: Introductory Functional Analysis With Applications, John Wiley Sons.

References:

1. Limaye, B.V., Functional Analysis, New Age Internation Publications

2. Bachman, George; and Narici, Lawrence : Functional Analysis, Dover Publications, Inc.

3. Simmons, G.F. : Introduction to Topology and Modern Analysis, McGraw- Hills.

4. Lahiri, B.K. : Elements of Functional Analysis, World Press, Kolkata

5. Jain, P.K; Ahuja, O.P.; and Ahmed, K : Functional Analysis, New Age International

19

M-402Linear Algebra

UNIT-I

Linear Transformations, Rank Plus Nullity Theorem, Representation of Linear transformations by Ma-trices, Change of Basis Matrices, Algebra of Linear Transformations, Algebra isomorphism between thealgebra of Linear Transformations and Algebra of Matrices Change of Bases for Linear Transforma-tions, Equivalence of Matrices, Similarity of Matrices, Quotient spaces, Isomorphism Theorems, LinearFunctionals, Dual Space, dual Bases, annihilators.

UNIT-II

Characteristic roots, Characteristic vectors, Characteristic Polynomials, relation between characteristicpolynomial and Minimal Polynomial of an Operator, Eigenvalues, Cayley-Hamilton Theorem (proof tobe given later), Diagonalizablility, necessary and sufficient condition for diagonalizability, Projectionsand their relation with direct sum decomposition of vector spaces, Invariant Subspaces, Direct sum De-compositions, Invariant Direct Sums, The Primary Decomposition Theorem, Geometric and Algebraicmultiplicities.

UNIT-III

Cyclic subspaces, companion matrices, a proof of Cayley-Hamilton theorem, Triangulability, Canonicalforms of nilpotent transformations, Diagonal Forms, Triangular Forms, Rational Canonical Forms.

UNIT-IV

Trace and Transpose, Inner product spaces, Linear functionals and Adjoints, Orthogonality, Orthonor-mality, Projection Theorem, Gram-Schmidt Orthogonalization, Orthonormal Basis, Riesz Represen-tation Theorem, Adjoint of Operators, Orthogonal Diagonalizability, Self-Adjoint Operators, Unitaryand Normal Operators, Orthogonal Diagonalization, Orthogonal Projection.

UNIT-V

Bilinear Forms, Correspondence between bilinear forms and matrices, Rank of a Bilinear Form, non-degenerate bilinear form, Quadratic forms, reduction and classification of quadratic forms, Symmetricand Skew-symmetric bilinear forms.

Recommended Text:

1. Herstein, I.N., Topics in Algebra,Wiley Eastern Limited/New Age International Second Edition.

2. Hoffman and Kunze, Linear Algebra, Prentice Hall of India Private Limited

3. Roman, Steven, Advanced Linear Algebra, Graduate Texts in Mathematics 135,Springer- Verlag

References:

1. Bhattacharjee, Jain Nagpaul, First Course in Linear Algebra,New Age International.

2. Halmos, Paul R., Finite-Dimensional Vector Spaces, Sringer.

20

M-403Operations Research and Optimization Techniques-II

UNIT-I

Basic concept of probability and probability distributions- Binomial- Poission- Negative ExponentialNormal Emperical distributions, Generation of random numbers Expectation of a random variable,mean and variance of a random variable and joint Random variables.

UNIT-II

Simulation Definition and types of simulation, limitation of simulation technique, Monte-Carlo simu-lation, application of simulation, introduction to forecasting models.

UNIT-III

Goal programming introduction, formulation of linear goal programming problem, solution of goalprogramming by graphical method, simplex method, classical optimisation, different methods.

UNIT-IV

Queuing theory: Introduction, queuing system, classification and solutions of queuing models. Markovchain, brand switching models, transient and steady state.

UNIT-V

Probabilistic inventory control models : Instantaneous demand inventory control models without setupcost different models with uncertain demand for single period, optimal order point marginal analysisapproach, with shortages and discrete replenishment, with shortages and continuous replenishment,Re-order lead time; Decision analysis : Introduction, decision making environment, decision under un-certainty, decision under risk, decision tree analysis.

Recommended Text:

1. Wagner, H.M, Principles of Operations Research, Prentice Hall

2. Sharma, J.K., Operations Research : Theory and Application, Mcmillan

3. Man Mohan, Gupta, P.K., Swarup Kanti : Operations Research, S. Chand Sons

4. Taha, H. A., Operations Research: An Introduction, Prentice Hall

References:

1. Mustafi C.R., Operations Research Methods Practice, New Age Int.

2. Shenoy, L.V., Linear Progarmming : Methods and Applications, New Age Int.

3. Mittal, K.V., Optimization Methods : In O.R. and System Analysis, New Age Int.

4. Vhora, N.D., Quantitative Techniques in Management, Tata McGraw Hill

21

M-404 (a)Relativity

UNIT-I

Physical Meaning of Geometrical Propositions, The System of Co-ordinates, Space and Time in ClassicalMechanics, The Galileian System of Co-ordinates, The Principle of Relativity.

UNIT-II

On the Idea of Time in Physics, The Relativity of Simultaneity, On the Relativity of the Conceptionof Distance, The Lorentz Transformation, Minkowski’s Four-dimensial Space.

UNIT-III

Special and General Principle of Relativity, The Gravitational Field, Euclidean and non-EuclideanContinuum, Gaussian Co-ordinates, The Solution of the Problem of Gravitation on the Basis of theGeneral Principle of Relativity.

UNIT-IV

The electromagnetic field equations of Maxwell, invariance of Maxwells equations, Biot-Savart law,Faradays law, transformation of electric and magnetic field components, electromagnetic wave equation,principles of covariance and equivalence, energy momentum tensor, field equations of general relativity,inertial and gravitational mass, Poissons equation as approximation.

UNIT-V

Schwarzschilds exterior solution, planetary orbits, advance of perihelion, gravitational shift of spectrallines, schwarzschilds interior solution, bending of light, cosmological models, Einsteins model, De-sittersmodel, comparison of the models the big bang models, the expanding universe.

Recommended Text:

1. Introduction to special Relativity Robert Resnick (New Age)

2. Special Relativity A.P. French (ELBS/Van Nostrand Reinhold (UN)

3. Introduction to the Theory of Relativity P.G. Bergman (Prentice Hall)

4. Theory of Relativity (Special and General) M. Ray (S. Chand Co. Delhi )

5. The Mathematical Theory of Relativity A.S. Eddington.

6. The Theory of Relativity C. Moller.

References:

1. Relavivistic Mechanics Satya Prash, Pragati Prakashan Meerut; U.P

2. An Introduction to The Special Theory of Relativity R. Latz (Van Nostran ,Princeton, N.J.)

3. The Theory of Relativity R.K. Pathria (Hindustan Publishing Co., Delhi)

4. Relativity, Thermodynamics and Cosmology R.C. Tolman

22

M-404 (b)Differential Geometry

UNIT-I

Curves with Torsion : Space curves, their curvature and torsion, Fundamental theorem of space curves,tangent, principal normal, curvature, bi-normal, torsion, Serret-Frenet formulae, locus of center ofcurvature, examples I , spherical curvature, locus of center of spherical curvature, theorem of curvedetermined by its intrinsic equations, helices, spherical indicatrix of tangent etc., involutes, evolutes,Betrand curves, examples II.

UNIT-II

Envelopes Developable Surfaces : Surface, tangent plane, normal; one-parameter family of surfaces;envelope, characteristics, edge of regression; developable surfaces; osculating developable; polar de-velopable, rectifying developable; two parameter family of surfaces, envelope, characteristic points.,examples III.

UNIT-III

Curvilinear co-ordinates on a surface, fundamental magnitudes, curves on surfaces, first and secondfundamental forms, Gaussian curvature, curvilinear coordinates : first order magnitudes ; directionsof a surface, the second order magnitudes, derivatives of N, curvature of normal section, Mausnierstheorem.

UNIT-IV

Curves on a surface and lines of curvature : Principal directions and curvatures, first and secondcurvatures, Eulers theorem, Dupins indicatrix, the surface X=f(x,y), surface of revolution, examples ofasymptotic lines, curvature and torsion.

UNIT-V

Geodesics , Fundamental equations of surface theory, Geodesic property, equation of geodesics, surfaceof revolution, torsion of a geodesic.

Recommended Text:

1. Weather burn ,C.E., Differential Geometry of three Dimensions, Cambridge University Press

2. Bansi Lal, Three Dimentional Differential Geometry ,S. Chand

References:

1. Guggenheimer ,H., Differential Geometry, McGraw Hill

23

M-404(c)Mathematical Modelling

UNIT-I

Mathematical modelling introduction, techniques, classifications, some illustrations :mathematicalmodelling through geometry/algebra/trigonometry/calculus, mathematical modelling through ODE offirst order: linear growth and decay model, non-linear growth and decay model, compartment modelsmathematical modelling of dynamics, geometrical problem.

UNIT-II

Mathematical modeling through systems of ordinary differential equations of first order: in populationdynamics, epidemics, economics, medicine, dynamics, mathematical modelling through ODE of secondorder : of planetary motions and motion of satellites, modelling through linear ordinary differentialequations of second order in electrical circuits, catenary.

UNIT-III

Mathematical modelling through difference equations with constant coefficients : in population dynam-ics and genetics, mathematical modelling through PDE : mass-balance equations, momentum balanceequations, variational principles, model for traffic on a highway.

UNIT-IV

Mathematical modelling through graphs : in terms of directed graphs in terms of signed graphs, interms of weighted diagraphs and in terms of unoriented graphs.

UNIT-V

Mathematical modelling through linear programming : of different industrial oriented problems, math-ematical modelling through calculus of variations : on geometrical problems, problems of mechan-ics/bioeconomics.

Recommended Text:

1. Kapur, J.N., Mathematical modelling, New Age International

References:

1. Burghes, D.N., Mathematical modelling in social, management and life sciences, Ellios Horwoodand John Wiley

2. Giordano, F.R.,and Weir, M.D. : A first course in Mathematical Modelling, Brooks Cole

3. Kapur, J.N., Insight into mathematical modeling, Indian National Science academy

4. Bellomo and Preziosi, Modelling Mathematical methods and Scientific computation, CRC

24

M-404(d)Computational Fluid Mechanics

UNIT-I

Irrotational flow, Potential function, Full potential equation, 2-D incompressible fluid flow, Stream func-tion, conservation principles, Steady and unsteady fluid flow, vortex motion, vorticity equation,vorticitytransport equations, viscous fluid flow, boundary layer approximation to viscus fluid motions.

UNIT-II

Nature of problems, Finite difference formulations for elliptic problems. Simple, general and higherorder derivative, mixed derivatives. Higher order accuracy schemes, accuracy of F.D. solutions. Iterativesolution method, FD methods for 2-D and 3-D elliptic BVP with 2nd and 4th order and their application.FD approximations to Poissons equation in cylindrical and spherical co-ordinates, Alternation directionmethod.

UNIT-III

Two and three level explicit and implicit F.D. approximations to parabolic equations. Stability analysis(matrix and Von-Neumann method). The method of factorization, fractional step methods, solutionof 1-D non-linear parabolic equations. compatibility, consistency and convergence of the differencemethod, FD approximations to heat conduction equation in cylindrical and spherical co- ordinates.

UNIT-IV

Explicit and implicit schemes, Von-Neumann stability analysis, Multi step methods, solution of 1-Dnon-linear hyperbolic problems, solution of non-linear wave equation, explicit and implicit schemes forBungers equation.

UNIT-V

Finite element formulations, the construction of finite elements, convergence rates for FEM, stabilityof FEM, Galerpins method, elementary ideas of finite volume method, one-dimensional computationsby FVM, conversion of FVM to FDM, simple problem.

Recommended Text:

1. 1. C.A.J. Eletcher, Computational techniques for Fluid Dynamics,Springer-Verleg

2. C.Y. Chow, Introduction to Computation Fluid Dynamics, John Willey

3. D.A. Anderson, J.C. Tasnechill, R.H. Pletcher, Computational Fluid Dynamics and Heat Trans-formation, McGraw Hill

References:

1. T.J. Chung : Computational Fluid Dynamics, Cambridge University Press

2. Peyret and T.D. Taylor : Computational Fluid Dynamics, Springer-Verleg

3. P. Wesseling : Principles of Computational Fluid Dynamics, Springer-Verleg

25

M-404(e)Operator Theory

UNIT-I

Banach algebras, inverse of an element, Spectrum and Resolvent, Compact linear operators on normedlinear spaces. The ideal of compact operators, the separability of the range and spectral properties ofa compact linear operator, operator equations involving compact linear operators.

UNIT-II

Fredholm Type Theorems, the Fredhalm Alternative, Spectral properties of bounded selfadjoint linearoperators. Positive operators. Square roots of a positive operator. Projection operators. Spectralfamilies. Spectral family of a bounded self-adjoint linear operators.

UNIT-III

Extension of the spectral theorem to continuous functions. Properties of the spectral family of abounded self-adjoint linear operator. Unbounded linear operators and their Hilbert-adjoint operators.Symmetric and self adjoint linear operators. Closed linear operators closable operators and their clo-sures.

UNIT-IV

Spectral properties of self-adjoint linear operators. Spectral representation of unitary and selfadjointlinear operators. Multiplication operator and differentiation operator. Functional calculus and spectralmapping theorem for analytic function. The Riesz decomposition theorem.

UNIT-V

Semigroups of bounded linear operators. Exponential growth property and the resolvent, generationof semigroups, dissipative semigroups and compact semigroups. Elementary examples of semigroups :Cauchy problem.

Recommended Text:

1. Kreyszig,E., Introductory Functional Analysis with Applications, John Filey and Sons

2. Rajdavi, H. and Rosenmthal, P., Invarian subspaces, Springer-Verlag (Chapter 2)

3. Balakrishnan,A.V., Applied Functional Analysis, Springer-Verlag

References:

1. Lahiri,B.K., Elements of Functional Analysis by , World Press, Kolkata

2. Limayce, B.V., Functional Analysis, New Age Int

3. Dunford N., Schwartz, J.P., Linear Operators, Pt-I II : Intesciences Publishers

4. Taylor A.E.,Ley,D.C., Introduction to Functional Analysis by John Wiley and Sons

26

M-404(f)Groups and Representations

UNIT-I

(Brief review of basic concepts of group theory ), isomorphism theorems, group presentations, symmetricgroups, dihedral groups, general linear groups, group actions, Sylows theorems.

UNIT-II

Minimal and maximal normal subgroups, automorphism groups, commutators, composition series, solv-able groups, Jordan-Holder theorem, nilpotent groups, supersolvable groups, Fitting and Frattini sub-groups.

UNIT-III

Semidirect product, central product, wreath product, p-groups, extra-special p-groups, complements ofsubgroups, the Schur- Zassenhaus theorem.

UNIT-IV

Modules and representations, Maschkes theorem, Wedderburn theory.

UNIT-V

Characters, character table, theorems of Burnside and Hall, induced characters.

Recommended Text:

1. J. L. Alperin Rowen B. Bell, Gropus and Representations, GTM 162, Springer (1995)

References:

1. Derik J. S. Robinson, A course in the theory of Groups, GTM 80, Springer (1996)

2. David S. Dummit Richard M. Foote, Abstract Algebra, John Wiley Sons, Inc (1999)

3. Joseph J. Rotman, An Introduction to the Theory of Groups, third edition, Allyn Bacon, Inc(1984)

4. M. J. Collins, Representations and characters of finite groups, Cambridge studies in advancedmathematics 22, Cambridge University Press (1990)

27

M-404 (g)Mathematical Statistics

UNIT-I

The postulates of probability, Some elementary theorems, Addition and multiplication rules, Bayes ruleand future Bayes theorem, Random variables, mathematical expectation, probability mass functionsand probability density function, distribution function and its properties.

UNIT-II

Uniform, Bernoulli and binomial distribution, Hypergeometric and geometric distribution, Negative bi-nomial and Poisson distribution, Uniform and exponential distribution, Gamma and beta distributions,Normal distribution, Log-normal distribution.

UNIT-III

Moments and moment generating functions, Probability Generating Functions and Characteristic func-tion Moments of binomial, Hypergeometric, Poisson, Gamma, Beta and normal distributions, Trans-formation of variables: One variable, Several variables.

UNIT-IV

The distribution of sample moments, The distribution of differences of means and variances, The Chi-Square distribution, the t distribution and the F distribution.

UNIT-V

Multiple linear regression, estimation of parameters using method of least square, Using of dummyvariables, Binomial logestic regression and multinomial logestic regression- estimation of the regressioncoefficients and their interpretation.

Recommended Text:

1. J. E. Freund, Mathematical Statistics, (Prentice Hall Inc., 1992)

2. Bhattacharjee D. and Das K K, A Treatise on Statistical Inference and Distributions¸(AsianBooks, 2010)

References:

1. Hogg and Craig, Introduction to Mathematical Statistics, (Collier Macmillan, 1958)

2. Mood, Greyill and Boes, Introduction to the Theory of Statistics, (McGraw Hill)

3. R. E. Walpole, Introduction to Statistics, (Macmillan Publishing Company, 1982)

4. M. R. Spiegel and L. J. Stephens, Statistics, (McGraw Hill Book Company, 1984)

28

M-405Project

Here each student has to prepare a project report under a supervisor and to present his/her workbefore external expert. The work may be a survey based report or may be a new finding or may besolving exercises from any standard text covering the topic which is not taught in the PG course in thisdepartment.

29