Syllabus of

M. Sc Mathematics

Name of the School: School of Basic and Applied Sciences

Division: Mathematics

Year: 2017-18

Curriculum

Master of Science (Mathematics)

2017-19 Employability

Semester-I Skill Development

Sl. No Course

Code

Name of the Course

L T P C

1 MSCM5001 Linear Algebra 4 1 0 4

2 MSCM5002 Real Analysis 4 1 0 4

3 MSCM5003 Mathematical Statistics 4 1 0 4

4

MSCM5004

Ordinary Differential

Equations

4 1 0 4

5

CENG5001

Professional and

Communication Skills

0 0 4 2

Semester-II

Sl. No Course

Code

Name of the Course

L T P C

1 MSCM5005 Abstract Algebra 4 1 0 4

2 MSCM5006 Complex Analysis 4 1 0 4

3 MSCM5007 Partial Differential Equations 4 1 0 4

4 MSCM5008 Continuum Mechanics 4 1 0 4

5 MSCM5009 General Topology 4 1 0 4

6 MSCM5010 Computer Programming 3 0 0 3

7 MSCM5011 Computer Programming Lab 0 0 2 1

Semester-III

Sl. No Course

Code

Name of the Course

L T P C

1 MSCM6001 Functional Analysis 4 1 0 4

2 MSCM6002 Differential Geometry 4 1 0 4

3

MSCM6003

Integral Equations & Calculus

of Variations

4 1 0 4

4 MSCM* Elective I

4 0 0 4

5 MSCM* Elective II

4 0 0 4

6 MSCM9998 Project(Stage I)

0 0 8 4

Semester-IV

Sl. No Course

Code

Name of the Course

L T P C

1 MSCM6012 Operations Research

4 1 0 4

2 MSCM6013 Applied Numerical Analysis

3 1 0 3

3

MSCM6014

Applied Numerical Analysis

Lab

0 0 2 1

4 MSCM* Elective III

4 0 0 4

5 MSCM* Elective IV

4 0 0 4

6 MSCM9999 Project (Final)

0 0 16 8

Total Credits 90

Elective-I

Sl. No Course

Code

Name of the Course

L T P C

1 MSCM6004 Module Theory 4 0 0 4

2 MSCM6005 Measure and Probability Theory 4 0 0 4

3 MSCM6006

Analytical Number Theory

4 0 0 4

4 MSCM6007 Harmonic Analysis 4 0 0 4

Elective-II

Sl. No Course

Code

Name of the Course

L T P C

1 MSCM6008 Algebraic Topology 4 0 0 4

2 MSCM6009 Dynamical systems 4 0 0 4

3 MSCM6010 Fluid Mechanics 4 0 0 4

4 MSCM6011 Discrete Structures 4 0 0 4

Elective-III

Sl. No Course

Code

Name of the Course

L T P C

1 MSCM6015 Manifolds and Applications 4 0 0 4

2 MSCM6016 Mathematical Modelling 4 0 0 4

3 MSCM6017 Financial Mathematics 4 0 0 4

4 MSCM6018 Coding Theory 4 0 0 4

Elective-IV

Sl. No Course

Code

Name of the Electives

L T P C

1 MSCM6019 Finite Element method 4 0 0 4

2a. MSCM6020 Computational Fluid Dynamics 3 0 0 3

2b.

MSCM6021

Computational Fluid Dynamics

Lab

0 0 2 1

3 MSCM6022 Stochastic Processes 4 0 0 4

4 MSCM6023 Automata & Formal Languages 4 0 0 4

5 MSCM6024 Cryptography 4 0 0 4

Detailed Syllabus

First Semester

Course Objectives: To use computational techniques and algebraic skills essential for the

study

of systems of linear equations, vector spaces, inner product spaces, eigen values and eigenv

ectors, orthogonality and diagonalization, Jordan & Bilinear forms.

Course Outcomes

CO1 Apply Gauss Elimination method to solve system of linear equations

CO2 Ability to understand linear transformation and its matrix representation, duality

and transpose.

CO3 Develop understanding and knowledge of eigen value and eigen vector,

diagonalization and Jordan canonical form.

CO4 Exposure to Gram-Schmidt orthonormalization, linear functional, normal operators,

Rayleigh quotient, Min-Max Principle

CO5 Get familiar with Bilinear forms, real quadratic forms, Sylvester's law of inertia

Text Book (s)

1. Hoffman, K. and R. Kunze, Linear Algebra, 2nd ed.,Pearson Education (India), 2003.

2.Artin, M., Algebra, Prentice Hall of India, 1994.

3.Lax, P., Linear Algebra, John Wiley & Sons, New York, Indian Ed. 1997.

Reference Book (s)

3. Rose,H.E., Linear Algebra, Birkhauser, 2002.

4.Lang, S., Algebra, 3rd ed., Springer (India), 2004.

5. Zariski, O. and P. Samuel, Commutative Algebra, Vol. I, Springer, 1975

6.Ramachandra, A.R. and P. Bhimasankaram, Linear algebra, Tata McGraw-hill,1992.

7.Gilbert Strang, Linear Algebra and Its Applications, Thomson/Brooks Cole (Available

in a Greek Translation)

Name of The Course Linear Algebra

Course Code MSCM5001

Prerequisite Basic concepts of matrices

Corequisite

Antirequisite

L T P C

4 1 0 4

Unit-1 10 Hours

Review of Matrices, Vector spaces and Linear Transformation: Systems of linear

equations, matrices, rank, Gaussian elimination. Basics of fields, Vector spaces over fields, subspaces, bases and dimension. Linear transformation, representation of linear

transformation by matrices, rank-nullity theorem, duality and transpose.

Unit-2 10 Hours

Diagonalization and Canonical Forms: Eigenvalues and eigenvectors, characteristic polynomials, minimal polynomials, Cayley-Hamilton Theorem, triangulation, diagonalization, rational canonical form, Jordan canonical form.

Unit-3 10 Hours

Inner product spaces and Introduction to operators: Inner product spaces, Gram-

Schmidtorthonormalization, orthogonal projections, linear functionals and adjoints, Hermitian, self-adjoint, unitary and normal operators, Spectral Theorem for normal

operators, Rayleigh quotient, Min-Max Principle.

Unit-4 10 Hours

Bilinear and Quadratic forms: Bilinear forms, symmetric and skew-symmetric bilinear forms,real quadratic forms, Sylvester's law of inertia, positive definiteness.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Real Analysis

Course Code MSCM5002

Prerequisite Calculus

Corequisite

Antirequisite

L T P C

4 1 0 4

Course Objectives: To make students understand real numbers, least upper bounds, and

the triangle inequality, set theory, convergence and divergence of series, metric spaces,

continuity and differentiability of functions.

Course Outcomes

CO1 Explain fundamental properties of the metric spaces that lead to the formal

development of real analysis.

CO2 Illustrate function of several variables as a linear transform from Rn to Rm and their

properties.

CO3 Apply the concept of improper integrals and Explain the theory of Riemann-

Stieltjes.

CO4 Apply different theorems to find the convergence of series of arbitrary terms.

CO5 Solve the problems related to uniform convergence of sequence and series of

functions and explain power series.

Text Book (s)

1. Rudin, W., Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1983.

2. Royden, H.I., Real Analysis,4rth ed.,Pearson’s Education ISBN-13: 978-0131437470.

3. Apostol, T., Mathematical Analysis, 2nd ed., Narosa Publishers, 2002.

Reference Book (s)

4. Ross, K., Elementary Analysis: The Theory of Calculus, Springer Int. Edition, 2004.

5. Malik, S.C., Savita Arora, Mathematical Analysis ,2nd ed., New age publication,1999.

Unit-1 10 Hours

Real number System: Real numbers as a complete ordered field, Dense subsets, Baire

Category theorem. Separable, second countable and first countable spaces. Continous

functions. Extension Theorem. Uniform continuity Isometry and homeomorphism.

Equivalent metrices. Compactness. Sequential compactness. Totally bounded spaces.

Finite intersection property.

Unit-2 10 Hours

Functions of several variables: Derivative of functions in a open subset of ℜn into ℜm as a linear transformation. Chain rule. Partial derivatives. Taylor’s theorem. Inverse function theorem. Implicit function theorem. Partitions of unity, Differential forms, Stokes Theorem.Definition and existence of Riemann-Stieltjes integral, Conditions for R-S integrability. Properties of the R-S integral, R-S integrability of functions of a function.

Unit-3 10 Hours

Numerical Sequence and Series: Series of arbitrary terms. Convergence, divergence and

oscillation, Abel’s and Dirichilet’s tests. Multiplication of series. Rearrangements of terms

of a series, Riemann’s theorem.

Unit-4 10 Hours

Sequences and series of functions: Point wise and uniform convergence, Cauchy’s criterion

for uniform convergence. Weierstrass M-test, Abel’s and Dirichlet’s tests for uniform

convergence, uniform convergence and continuity, uniform convergence, uniform

convergence and differentiation. Weierstrass approximation theorem. Power series.

Uniqueness theorem for power series, Abel’s and Tauber’s theorems.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Mathematical Statistics

Course Code MSCM5003

Prerequisite Basic concepts of probability

Corequisite

Antirequisite

L T P C

4 1 0 4

Course Objectives: To provide the students foundational introduction to the fundamental

concepts in statistics.

Course Outcomes

CO1 Summarize the basic concepts of probability, random variable and probability

distributions

CO2 Summarize the concept of bivariate distribution and correlation and regression.

CO3 Explain the concepts of sampling distributions and apply it to estimate the confidence

intervals.

CO4 Explain the concepts of estimator and estimates.

CO5 Identify the type of statistical test and Apply it to solve the hypothesis testing

problems.

Text Book (s)

1.R.V. Hogg, A. Craig, Probability and Statistical Inference, 6th.Ed.,Pearson Education.

2006.

2.I. Miller, M. Miller, “Mathematical Statistics with Applications”, Pearson Education.

2006

Reference Book (s)

3.W. H. William, C. M. Douglas, D. M. Goldman, C. M. Borror, Probability and

Statistics in

Engineering”, John Wiley. 2003

4. S.C. Gupta and V.K. Kapoor, Fundamental of Mathematical Statistics, S. Chand Pub.

Unit-1 12 Hours

Random Variables & Distributions Discrete, continuous and mixed random variables,

probability mass, probability density and cumulative distribution functions, mathematical

expectation, moments, moment generating function, Chebyshev’s inequality. Special

Distributions: Discrete uniform, binomial, geometric, negative binomial, hypergeometric,

Poisson, uniform, exponential, gamma, normal, beta, lognormal, Weibull, Laplace, Cauchy,

Pareto distributions.

Unit-2 8 Hours

Bivariate Data – Correlation & Regression: Bivariate random variables, joint and

marginaldistributions, covariance, correlation and regression analysis, transformation of

variables product moments, correlation, independence of random variables, bivariate

normal distribution, simple, multiple and partial correlation, regression.

Unit-3 10 Hours

Sampling Distribution: Law of large numbers, Central Limit Theorem, Distributions of

thesample mean and the sample variance for a normal population, Random sampling and

sampling distribution, fundamental distributions derived from normal distribution viz. ,t, F, χ 2 and Z (central) distributions. The method of moments and the method of maximum

likelihood estimation, properties of best estimates, confidence intervals for the mean(s) and

variance(s) of normal populations.

Unit-4 10 Hours

Testing of Hypothesis: Statistical Inference: Baysian inference, estimation-point an

interval, testing of hypothesis, Neyman-Pearson Lemma. Some tests based on t, χ 2 and F

distributions. Testing of Hypothesis: Null and alternative hypotheses, the critical and acceptance regions,

two types of error, power of the test, the most powerful test and Neyman-Pearson

Fundamental Lemma, Standard tests for one and two sample problems for normal

populations.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Ordinary Differential Equations

Course Code MSCM5004

Prerequisite Basic Calculus

Corequisite

Antirequisite

L T P C

4 1 0 4

Course Objectives: To impart existing knowledge of calculus and apply it towards the

construction and solution of mathematical models in the form of differential equations .

Course Outcomes

CO1 Know the behaviour of differential equations about the existence and uniqueness of

solutions.

CO2 Learn about the solution of linear ODE's and their general solutions.

CO3 Learn about the solutions of 2nd order or higher order ODE’s

CO4 Learn the different kinds of singularity behaviour in ODE’s

CO5 Know the series solution of ODE’s

Text Book (s)

1. Raisinghania, M.D.,Ordinary and Partial Equations,18th ed.,S.Chand.

2. Coddington, E.A.,An Introduction to Ordinary Differential Equations, Prentice-Hall

of India Private Ltd., New Delhi.

3. Simmon, G.F., Differential equations with applications and Historical notes,2nd ed.,

McGraw- Hill, 1991

4. Ross, S.L., Differential Equations,3rd ed.,Wiley.

Reference Book (s)

5. Martain, W.T. and E. Relssner, Elementary Differential Equations, 3rd ed., Addison

Wesley Publishing Company, inc., 1995.

6. Codington, E.A. and N. Levinson,Theory of Ordinary Differential Equations ,

TataMc Graw hill Publishing Co. Ltd. New Delhi, 1999.

7. Braun, M., Differential Equations and Their Applications, Springer-Verlag, New York

Heidelberg, Berlin.

Unit-1 10 Hours

Lipschitz condition, Existence and uniqueness of solution of ordinary differential equation

of first order, Existence theorem in complex plane, Existence and uniqueness theorem for

simultaneous differential equations of first order, The method of successive

approximations, convergence of successive approximations, Existence and uniqueness of

solution Initial value problem, Non-local existence of the solution, Existence and

uniqueness of solutions to linear systems, Equations of order n.

Unit-2 10 Hours

Second order equations: General solutions of homogeneous equations, Non- Homogeneous

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

Strum separation theorem, Boundary value problems, Green’s function, Strum-Liouville

problems.

Unit-3

Linear equations with regular singular points – introduction; Euler equation, second order equations with regular singular points – example and the general case, convergence proof, exceptional cases, Bessel equation, regular singular points at infinity.

Unit-4 10 Hours

Series Solution: Ordinary point and singularity of a second order linear differential

equation in the complex plane, Fuch’s theorem, solution about an ordinary point, solution of Hermite equation as an example, Regular singularity, Frobenius’ method – solution

about a regular singularity, solutions of hypergeometric, Legendre, Laguerre and Bessel’s equation as examples.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Professional and Communication Skills

Course Code CENG5001

Prerequisite

Corequisite

Antirequisite

L T P C

0 0 4 2

Course Objectives: To develop the professional and communicational skills of learners in a

technical environment, acquire functional and technical writing skills, and acquire

presentation skills to technical and non-technical audience.

Course Outcomes

CO1 Develop the understanding into the communication and language as its medium

CO2 Develop the basic understanding of spoken English

CO3 Improve their reading fluency skills through extensive reading

CO4 Use and assess information from academic sources, distinguishing between main

ideas and details

CO5 Compare and use a range official support through formal and informal writings

Text Book(s)

1.Rajendra Pal and J.S.Korlahalli. Essentials of Business Communication. Sultan

Chand & Sons. New Delhi.

Reference Book(s)

2. Kaul. Asha. Effective Business Communication.PHI Learning Pvt. Ltd. New

Delhi.2011.

3.Murphy, Essential English Grammar, CUP.

4.J S Nesfield, English Grammar: Composition and Usage

5. Muralikrishna and S. Mishra, Communication Skills for Engineer

Unit-1 9 Hours

Aspects of Communication; Sounds of syllables; Past tense and plural endings;

Organizational techniques in Technical Writing; Paragraph Writing, Note taking,

Techniques of presentation

Unit-2 9 Hours

Tense, Voice, conditionals, Techno-words; Basic concepts of pronunciation; word stress;

Business letters, email, Techniques for Power Point Presentations; Dos and don’ts of

Group Discussion

Unit-3 9 Hours

An introduction to Modal and Phrasal verbs; Expansion; Word formation; Technical

Resume; Company Profile Presentation; Interview Skills

Continuous Assessment Pattern

Internal Assessment (IA)

End Term Test

(ETE)

Total Marks

50

50 100

Second Semester

Name of The Course Abstract Algebra

Course Code MSCM5005

Prerequisite Abstract Algebra

Corequisite

Antirequisite

L T P C

4 1 0 4

Course Objectives: This course aims to provide a first approach to the subject of algebra,

which is one of the basic pillars of modern mathematics. The focus of the course will be the

study of certain structures called groups, rings, fields and some related structures.

Course Outcomes

CO1 Ability to understand advanced group structures like Sylow p-subgroups, Normal

series and free abelian groups

CO2 Develop understanding and knowledge of various ring structures like Euclidean domain PID, UID and polynomial rings

CO3 Exposure to various field structures like finite fields, separable and inseparable

extensions, Splitting fields and Cyclotomic fields

CO4 Get familiar with fundamental concepts of Galois theory

Text Book (s)

1. I. N. Herstein, Topics in Algebra, Wiley & Sons publications 1975.

2. D.S. Malik, J. N. Mordeson and M. K. Sen, Introduction to Abstract Algebra,

USA, 2007

3. N. Jacobson, Basic Algebra I, 2nd Ed., Hindustan Publishing Co., 1984.

Reference Book (s)

4. J. A. Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa, 1999.

5. J. S. Milne, Fields and Galois Theory, 2017

6. J. B. Fraleigh, A first course in Abstract Algebra, 3rd Ed., Narosa Publishing, 1986

7. M. Artin, Algebra, Prentice Hall of India, 1994.

8. D.S. Dummit and R. M. Foote, Abstract Algebra, 2nd Ed., John Wiley, 2002.

Unit-1 10 Hours

Sylow’s theorems, Sylow p-subgroups, Direct product of groups, Simple groups and

solvable groups, nilpotent groups, simplicity of alternating groups, Normal and subnormal series, composition series, Jordan-Holder Theorem. Semidirect products. Free groups, free

abelian groups.

Unit-2 10 Hours

Rings, Rings of fractions, Chinese Remainder Theorem for pairwise co maximal ideals. Euclidean Domains, Principal Ideal Domains and Unique Factorizations Domains. Polynomial rings over UFD's.

Unit-3 10 Hours

Fields, Characteristic and prime subfields, Field extensions, Finite, algebraic and finitely

generated field extensions, Classical ruler and compass constructions, Splitting fields and normal extensions, algebraic closures. Finite fields, Cyclotomic fields, Separable and

inseparable extensions.

Unit-4 10 Hours

Galois groups, Fundamental Theorem of Galois Theory, Composite extensions, Examples (including cyclotomic extensions and extensions of finite fields). Norm, trace and

discriminant. Solvability by radicals, Galois' Theorem on solvability. Cyclic extensions,

Abelian extensions, Trans-cendental extensions.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Complex Analysis

Course Code MSCM5006

Prerequisite Real Analysis

Corequisite

Antirequisite

L T P C

4 1 0 4

Course Objectives: To introduce the fundamental ideas of the functions. of complex variables

and developing a clear understanding of the holomorphic functions and its various features.

Course Outcomes

CO1 Determine continuity/differentiability/analyticity and integral of a complex

function.

CO2 Apply the concepts of Cauchy Integral theorem and formula to solve complex

integration.

CO3 Classify singularities of an analytic function and to find the Laurent’s and Taylor’s

series of a complex function.

CO4 Compute the residue of a function and use the residue theory to evaluate a contour

integral or an integral over the real line.

CO5 Understand the concept of transformation in a complex space (linear and non-

linear) and sketch their associated diagrams.

Text Book (s)

1. Churchill & Brown, Complex Variables and Applications, McGraw-Hill Higher

Education; 8th edition (1 October 2013)

2. S. Ponuswamy, Foundation of Complex Analysis (Second edition). Publisher: Alpha

Science Int Ltd, 2006, ISBN 10: 1842652230 ISBN 13: 9781842652237

3. Murray Spiegel, Seymour Lipschutz, Complex Variables (Schaum’s Outlines), 2nd

ed., McGraw-Hill Profesional

Reference Book (s)

4. A.R. Shastri, An Introduction to Complex Analysis, Macmilan India, New Delhi,

1999

5. J.B. Conway, Functions of One Complex Variable, 2nd ed., Narosa, New Delhi

6. Walter Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill International Editions

Mathematics Series.

Unit-1

10 Hours

Functions of Complex Variable: Introduction, Limit, Continuity, Differentiability of

function, Analytic function, Cauchy-Riemann Equations in Cartesian and Polar form, Necessary and sufficient conditions for a function to be analytic Harmonic functions and

simple application to flow problems. Complex Integration: Integration of complex valued functions, Cauchy theorem, Cauchy-Goursat theorem, Cauchy Integral formula,

Generalized Cauchy Integral formula.

Unit-2 10 Hours

Conformal Mapping :Introduction, conformal transformation, sufficient condition for

w=f(z) to represent a conformal mapping, necessary condition for w=f(z) to represent a

conformal mapping, superficial transformations, some special transformation, power,

special power, the inverse mapping, the mapping w=ez , the mapping w=logz , the mapping

w= zn. Zeroesand Singularities of complex valued functions, Taylor's and Laurent's series,

radius and circle of convergence.

Unit-3 10 Hours

Calculus of Residues: Residues, Residue theorem and it’s application in evaluation of real

integrals around unit and semi circle, Cauchy Residue theorem, evaluation of the real

definite integral, case of poles on real axis, evaluation of the integral when the integrand

involves multiple valued functions, uses of rectangular contours, summation of infinite

series.

Unit-4 10 Hours

Analytic Continuation: Determination of a given function, analytic in a domain by a

function elements, extension of a function by power series with a finite non-zero radius of

convergence, analytical continuation of a function, analytic continuation to a point, complete

analytic function, natural boundary, continuation by power series, Mittag –Leffler theorem.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Partial Differential Equations

Course Code MSCM5007

Prerequisite

Corequisite

Antirequisite

L T P C

4 1 0 4

Course Objectives: Students will gain knowledge about the partial differential equations

(pde’s) and how they can serve to model many physical processes such as mechanical

vibrations, transport phenomena including diffusion, heat transfer, advection and

electrostatics etc. They will learn about heat and wave equations in 1D, 2D and 3D using the

divergence Theorem.

Course Outcomes

CO1 Define the partial diff. equations and their solutions by some well known methods

Like Lagrange’s, Charpit and Monge’s method.

CO2 To classify the 2nd order pde’s and define canonical forms.

CO3 To know and solve elliptic, hyperbolic and Laplace equations and wave equations

by using separations of variables.

CO4 Establish the properties of solutions like existence, uniqueness, weakness and

strongness etc.

CO5 Define Green’s functions of heat, wave and Laplace equations.

Text Book (s)

1. I. N. Sneddon, Elements of partial differential equations, Dover Publications, New

York, 2006.

2. F. John, Partial Differential Equations, Springer-Verlag, New York, 1985.

3. C. Constanda, Solution techniques For elementary partial Differential Equations,

Chapman and Hall/CRC, New York,2002.

4. S.J Farlow, Partial Differential Equations for scientist and Engineers, Birkh, auser,

New York, 1993.

5. E. DiBenedetto, Partial Differential Equations, Birkhauser, Boston, 1995.

Reference Book (s)

6. L.C. Evans, Partial Differrential Equations, Graduate Studies in Mathematics, Vol.

19,

AMS, Providence, 1998.

7. E. Zauderer, Partial Differential Equations of Applied Mathematics, 2nd ed., John

Wiley and Sons, New York, 1989.

8. K. Sankara Rao, Introduction to Partial Differential Equations, 3rdedition,PHI, ISBN-

13: 9788120342224

Unit-1 10 Hours

First order partial differential equations, linear and quasi-linear first order equations, method of characteristics, general first order equations, Cauchy problem for second order

p.d.e. characteristics, canonical forms, Cauchy problem for hyperbolic equations, one dimensional wave equation, Linear second order PDE with variable coefficients,

characteristic curves of the second order PDE, Monge’s method of solution of non-linear

PDE of second order.

Unit-2 10 Hours

The solution of linear hyperbolic equations, Separation of variables in a PDE, Laplace equation: mean value property, weak and strong maximum principle, Green's function, Poisson's formula, Dirichlet's principle, existence of solution using Perron's method (without proof).

Unit-3 10 Hours

Wave equation: uniqueness, D'Alembert's method, method of spherical means and

Duhamel's principle, elementary solutions of one-dimensional wave equation, vibrating membranes, three dimensional problems. Green’s function for wave equation

Unit-4 10 Hours

Heat equation: initial value problem, fundamental solution, weak and strong maximum principle and uniqueness results, Diffusion equation, solution of boundary value problems

for diffusion equation, elementary solutions of diffusion equation, separation of variables. Green’s function for diffusion equation.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Continuum Mechanics

Course Code MSCM5008

Prerequisite

Corequisite

Antirequisite

L T P C

4 1 0 4

Course Objectives: The purpose of the course is to expose the students to the basic elements

of continuum mechanics. The students should be able to study a wide variety of advanced

courses in solid and fluid mechanics

Course Outcomes

CO1 To provide the students with a foundation in Continuum Mechanics.

CO2 To learn the conservation principles and derive the equations governing the

mechanics of solids and fluids within the continuum hypothesis.

CO3 To learn the constitutive equations for solid and fluids.

CO4 To develop practical skills in working with tensors.

CO5 To develop problem solving skills, applying the conservation principles and the

constitutive equations to solve practical engineering problems.

Text Book (s)

1. Y.C. Fung : A First Course in Continuum Mechanics, Paulo Silva, 2014

2. W. Prager : Mechanics of Continuous Media, Courier Corporation, 1961

Reference Book (s)

3. Jog, C.S., Continuum mechanics: Foundations and applications of mechanics, Volume

I, Third edition, 2015, Cambridge University Press.

4. Chadwick, P., Continuum mechanics: Concise theory and problems, 1999, Dover

Publications, Inc., New York.

5. Gurtin, M.E., An introduction to continuum mechanics, 1981, Academic press, Inc.

6. Gurtin, M. E. , Fried, E. and Anand, L., The mechanics and thermodynamics of

continua, 2010, Cambridge University Press, New York.

7. Liu, I-S., Continuum mechanics, 2002, Springer, Berlin.

8. Bonet, J., and Wood, R. D., Nonlinear continuum mechanics for finite element

analysis, 1997, Cambridge University Press, Cambridge.

9. Martinec, Z. , Lecture notes on continuum mechanics. (Hyperlink:

http://geo.mff.cuni.cz/vyuka/Martinec-ContinuumMechanics.pdf)

10. D. S. Chandrasekharaiah and L. Debnath, Continuum Mechanics, 1994, Academic

Press Inc., London.

11. E. B. Tadmor, R. E. Miller and R. S. Elliott, Continuum Mechanics and Thermo-

dynamics from Fundamental Concepts to Governing Equations, 2012, Cambridge

University Press, UK.

Unit-1 10 Hours

Principles of continuum mechanics, axioms. Forces in a continuum. The idea of internal stress.Stress tensor. Equations of equilibrium. Symmetry of stress tensor. Stress transformation laws.Principal stresses and principal axes of stresses. Stress invariants. Stress quadric of Cauchy.Shearing stresses. Mohr’s stress circles.

Unit-2 10 Hours

Deformation. Strain tensor. Finite strain components in rectangular Cartesion coordinates. Infinitesimal strain components. Geometrical interpretation of infinitesimal strain components. Principal strain and principal axes of strain. Strain invariants. The compatiability conditions. Compatibility of strain components in three dimensions.

Unit-3 10 Hours

Constitutive equations. Inviscid fluid. Circulation. Kelvins energy theorem. Constitutive equation for elastic material and viscous fluid. Navier and Stokes equations of motion.

Unit-4 10 Hours

Motion of deformable bodies. Lagrangian and Eulerian approaches to the study of motion of continua. Material derivative of a volume integral. Equation of continuity. Equations of motion. Equation of angular momentum. Equation of Energy. Strain energy density function.

Continu/ous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course General Topology

Course Code MSCM5009

Prerequisite

Corequisite

Antirequisite

L T P C

4 1 0 4

Course Objectives: This course aims to teach the fundamentals of point set topology and

constitute an awareness of need for the topology in Mathematics.

Course Outcomes

CO1 Explain a topological space and construct a topology on a set in a number of 4-

Mandate: Course Handout ways so that to make it in to a topological space.(K2)

CO2 Summarize some of the elementary concepts associated with topological spaces,

Continuous spaces and Subspaces.(K2)

CO3 Explain Connectedness and Compactness in topological spaces and apply them to

Construct new topological spaces from the given ones.(K5)

CO4 Classify the countability and separation axioms and prove related theorems.(

CO5 Apply rules of Product space to prove related theorems as well as Urysohn

lemma.(K3)

Text Book (s)

1. J. R. Munkres, Topology A First Course, 2nd ed., Prentice Hall of India Pvt. Ltd., New

Delhi, 2000.

2. T. B. Singh, Elements of topology, CRC press, New Delhi, 2013

Reference Book (s)

3. M. A. Armstrong, Basic Topology, Springer (India), 2004.

4. K.D. Joshi, Introduction to General Topology, New Age - International, New Delhi,

2000.

5. J.L. Kelley, General Topology, Van Nostrand, Reinhold Co., New York, 1995

6. J. Dugundji, Topology, Allyn and Bacon, 1966 (reprinted in India by Prentice Hall of

India Pvt. Ltd.).

Unit-1 12 Hours

Topological space, open and closed sets, Stronger and Weaker topologies, Usual Topology, Co-finite or finite complement topology, Co-countable topology, Upper and Lower limit topology.

Unit-2 8 Hours

Basis and sub-basis for a topology, Neighbourhood of a point, Neighbourhood system,

Interior, Exterior, Closure, Boundary and Derived set of a set, Interior operator and Kuratowski closure operator, Subspace topology, First countable. Second countable and

separable spaces, their relationships and hereditary property.

Unit-3 12 Hours

Definition, examples and characterizations of continuous functions, composition of

continuous functions. Open and closed functions, Homeomorphism, homeomorphic spaces,

Topological invariant property. Tychonoff product topology in terms of standard subbase,

projection maps. Characterisation of product topology as smallest topology with

projections continuous, continuity of a function from a space into a product of spaces,

countability and product spaces

Unit-4 8 Hours

Separation axiom: Kolmogorov,or Frechet’s,or Housdorff, Regular, Completely regular, Tychnoff , Normal spaces. Separated sets, completely normal spaces Urish Schawn

Lemma.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Computer Programming

Course Code MSCM5010

Prerequisite

Corequisite

Antirequisite

L T P C

3 0 0 3

Course Objectives: To learn the fundamental concepts of C programming and develop the

software using the concept of ‘C’ Language.

Course Outcomes

CO1 Understand the basic terminology used in programming and able to write, compile

and debug programs in C programming and acquire knowledge about basic

elements of programming with conditional and control statements to solve problem.

CO2 Learn character concepts in C and understand operators.

CO3 Understand the modular techniques such as functions and difference between call

by value and call by reference methods.

CO4 Implement Matlab programs with object oriented programming concepts Linear

Algebra, numerical integration and differentiation.

Text Book (s)

1. Programming and Problem solvng with Python, Ashok NamdevKamthane,Amit

Ashok Kamthane,McGrawHill.

2. Guttag, John ,Introduction to Computation and Programming using Python, PHI

Publisher

3. Thareja, Reema , Python Programming using problem solving Approach , 1st ed.(10

June 2017)Oxford University, Higher Education Oxford University Press, ISBN-10:

0199480173

Reference Book (s)

4. Lambert, Kenneth A ,Fundamentals of Python first Programmes ,1st ed(6th February

2009) Copyrighted material Course Technology Inc.

5. Budd, T., Exploring Python, 1st ed,Tata Mac Graw Hill, 2011

Unit-1 10 Hours

Fundamental Data Types and Storage Classes: Character types, Integer, short, long,

unsigned, single and double-precision floating point, storage classes, automatic, register,

static and external, Operators and Expressions: Using numeric and relational operators,

mixed operands and type conversion, Logical operators, Bit operations.

Unit-2 10 Hours

Programming in C: Introduction, Basic structures, Character set, Keywords, Identifiers,

Constants, Variable-type declaration, Operators: Arithmetic, Relational, Logical,

assignment, Increment, decrement, Conditional. Operator precedence and associativity,

Arithmetic expression, Evaluation and type conversion, Character reading and writing,

Formatted input and output.

Unit-3 10 Hours

Decision making (branching and looping) – Simple and nested IF, IF – ELSE, WHILE –

DO, FOR. Arrays-one and two dimension, String handling with arrays – reading and

writing, Concatenation, Comparison, String handling function, User defined functions.

Unit-4 10 Hours

Introduction to Matlab Programming – Introduction to the Matlab interface as well as basic

programming techniques. Introduction to Numerical Methods – Linear algebra, numerical

integration and differentiation, solving systems of ODE’s and interpolation of data.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Computer Programming Lab

Course Code MSCM5011

Prerequisite

Corequisite

Antirequisite

L T P C

0 0 2 1

Course Objectives: To learn and practice the fundamental concepts of C programming and

develop the software using the concept of ‘C’ Language.

Course Outcomes

CO1 Able to use C compiler and its operation

CO2 Write simple C code for a given algorithm

CO3 Write and use functions and parameter passing options

CO4 Implement common data structures in C programs — namely arrays, strings, lists

etc.

CO5 Implement algorithm for solving mathematical problems

Text Book (s)

1. Programming and Problem solvng with Python, Ashok NamdevKamthane,Amit

Ashok Kamthane,McGrawHill.

2. Guttag, John ,Introduction to Computation and Programming using Python, PHI

Publisher

3. Thareja, Reema , Python Programming using problem solving Approach , 1st ed.(10

June 2017)Oxford University, Higher Education Oxford University Press, ISBN-10:

0199480173

Reference Book (s)

4. Lambert, Kenneth A ,Fundamentals of Python first Programmes ,1st ed(6th February

2009) Copyrighted material Course Technology Inc.

5. Budd, T., Exploring Python, 1st ed,Tata Mac Graw Hill, 2011

S. No. Experiment

1. Write a program to print “Hello World” message on the screen.

2. Write a program to add two numbers

3. Write a program to check whether the given number is odd or even.

4. Write a program to swap two numbers.

5. Write a program to print larger of the three numbers.

6. Write a program to find the roots of a quadratic equation.

7. Write a program to find factorial of a number.

8. Write a program to find whether the given number is palindrome or not.

9. Write a program to display n natural numbers.

10. Write a program to find sum of n natural numbers.

11. Write a program to find GCD of a number.

12. Write a program to find area of a rectangle.

13. Write a program to print multiplication table of a given number.

14. Write a program to implement string operation.

15. Write a program to implement user defined function.

16. Write a program to find average of four numbers using arrays.

Continuous Assessment Pattern

Internal Assessment (IA) End Term Test

(ETE)

Total Marks

50

50 100

Third Semester

Name of The Course Functional Analysis

Course Code MSCM6001

Prerequisite

Corequisite

Antirequisite

L T P C

4 1 0 4

Course Objectives: To develop with the purpose to cover theoretical needs of Partial

differential equations and mathematical analysis. The Functional Analysis is related to

problems arising in Partial Differential Equations, Measure Theory and other branches of

Mathematics.

Course Outcomes

CO1 Summarize the basic concepts on Normed Spaces and Banach Spaces

CO2 Summarize concept Fundamental Theorems for Normed and Banach Spaces

CO3 Apply the concepts of continuity and reflexivity for Normed and Banach Spaces

CO4 Identify and Apply the concepts of Inner Product Spaces

CO5 Able to use Bessel’s inequality and Parseval’s identity

Text Book (s)

1. E. Kreyszig: Introductory Functional Analysis with Applications: Wiley student Edition 2007

2. B.V. Limaye, Functional Analysis: New Age International Publications, Third Edition 2014

Reference Book (s)

3. J. B. Conway: A course in Functional Analysis, Springer: Second Edition,

2007

Unit-1 10 Hours

Normed Spaces and Banach Spaces :Normed Linear spaces, Quotient space of normed

linear spaces and its completeness, Banach spaces and examples, Bounded linear

transformations, Normed linear space of bounded linear transformations.

Unit-2 10 Hours

Fundamental Theorems for Normed and Banach Spaces-1:Equivalent norms, Basic properties of finite dimensional normed linear spaces and compactness, Reisz Lemma, Open mapping theorem, Closed graph theorem, Uniform boundness theorem.

Unit-3 10 Hours Fundamental Theorems for Normed and Banach Spaces-2: Continuous linear functional, Hahn-Banach theorem and its consequences, Embedding and reflexivity of normed spaces, Dual spaces with examples, Boundedness and Continuity of Linear operators.

Unit-4 10 Hours

Inner Product Spaces: Inner product spaces, Hilbert space and its properties. Orthgonality

in Hilbert spaces, Phythagorean theorem, Projection theorem, Orthonormal sets, Bessel’s

inequality,Complete orthonormal sets, Parseval’s identity, Basic concepts of spectral

theory

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Differential Geometry

Course Code MSCM6002

Prerequisite

Corequisite

Antirequisite

L T P C

4 1 0 4

Course Objectives: To introduce the fundamentals of differential geometry primarily by

focusing on the theory of curves and surfaces in three space, the fundamental quadratic forms

of a surface, intrinsic and extrinsic geometry of surfaces, and the Gauss-Bonnet theorem.

Course Outcomes

CO1 Define the parametric representations and tangent.

CO2 Find the plane of curvatures and torsion.

CO3 Learn differentiability on surface and types of curvatures.

CO4 Explore his/her knowledge about geodesic.

CO5 Learn tensor analysis and tensor differentiation.

Text Book(s)

1. C.E.Wetherburn, An Introduction to Riemannian Geometry and the Tensor Calculus, CUP Cambridge, 1957.

2. T. J. Willmore, An Introduction to Differential Geometry, Oxford University

Press, 1959.

Reference Book(s)

3. U. C. De, A. A. Shaikh, J. Sengupta, Tensor Calculus, Narosa Publications, New Delhi, 2014.

4. Andrew Pressley, Elementary Differential Geometry, Springer, 2001.

5. Barrett O’ Neill, Elementary Differential Geometry, Academic Press, 2006.

6. Manfredo P. Do’Carmo, Differential Geometry of Curves and Surfaces, Prentice

Hall

Inc., New Jersey U.S.A. 1976.

7. S. Montiel and A. Ros, Curves and Surfaces, American Mathematical Society,

2005.

Unit-1 10 Hours

Differentiable curves in R3 and their parametric representations, Vector fields, Tangent vector, Principal normal, Binormal, Curvature and torsion, Serret - Frenet formula, Frame fields, Covariant differentiation, Connection forms, The structural equations.

Unit-2 10 Hours

Surfaces, Differentiable functions on surfaces, Differential of a differentiable map, Differential forms, Normal vector fields, First fundamental form, Shape operator, Normal curvature, Principal curvatures, Gaussian curvature, Mean curvature, Second fundamental form.

Unit-3 10 Hours

Gauss equations, Weingarten equation, Codazzi-Mainardi equations, totally umbilical surfaces, Minimal surfaces, Variations, First and second variations of arc length, Geodesic, Exponential map, Jacobi vector field, Index form of a geodesic, Gauss-Bonnet theorem.

Unit-4 10 Hours

Co-ordinate transformation, Covariant, Contravariant and Mixed tensors, Tensors of higher rank, Symmetric and Skew-symmetric tensors, Tensor algebra, Contraction, Inner product, Riemannian metric tensor, Christoffel symbols, Covariant derivatives of tensors.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Integral Equations & Calculus of Variations

Course Code MSCM6003

Prerequisite

Corequisite

Antirequisite

L T P C

4 1 0 4

Course Objectives: The course is aimed to lay a broad foundation for an understanding of

the problems of the calculus of variations, its many methods and techniques. Also to make

the students familiar with the methods of solving Integral Equations.

Course Outcomes

CO1 To classify integral equation and its relation to ordinary differential.

CO2 To understand the use of Resolvent kernels and Neumann series methods to solve

the integral equation.

CO3 To demonstrate the Abel’s integral equations and tantochrone problem.

CO4 To use of Laplace and Fourier transforms to solve integral equations.

CO5 To solve problems related to calculus of variations.

Text Book(s)

1. R. P. Kanwal, Linear Integral Equations: Theory and technique, Academic Press, NewYork, 1971.

2. A. S. Gupta, Text Book on Calculus of Variation, Prentice-Hall of India, New Delhi.

Reference Book(s)

3. Harry Hochsdedt, Integral Equations, John-Wiley & Sons, Canada, 1973.

4. Murry R. Spiegal, Laplace Transform (SCHAUM Outline Series), McGraw-Hill,

1965.

5. N. Kumar, An Elementary Course on Variational Problems in Calculus, Narosa Publications, New Delhi, 2005.

Unit-1 10 Hours

Classification of integral equations of Volterra and Fredholm types. Conversion of initial and boundary value problem into integral equations. Conversion of integral equations into

differential equations (when it is possible). Volterra and Fredholm integral operators and their iterated kernels.

Unit-2 10 Hours

Resolvent kernels and Neumann series method for solution of integral equations. Banach contraction principle, its application in solving integral equations of second kind by the method of successive iteration and basic existence theorems.

Unit-3 10 Hours

Abel’s integral equations and tantochrone problem. Fredholm-alternative for Fredholm

integral equation of second kind with degenerated kernels. Use of Laplace and Fourier transforms to solve integral equations.

Unit-4 10 Hours

Functionals, Deduction of Euler’s equations for functionals of first order and higher order for fixed boundaries. Shortest distance between two non-intersecting curves. Isoperimetric problems. Jacobi and Legendre conditions (applications only).

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Fourth Semester

Name of The Course Operations Research

Course Code MSCM6012

Prerequisite

Corequisite

Antirequisite

L T P C

4 1 0 4

Course Objectives: Operation research aims to introduce students to use quantitative methods

and techniques for effective decisions–making; model formulation and applications that are

used in solving decision problems.

Course Outcomes

CO1 Understand the concept of optimization.

CO2 Formulate the real life problem into mathematical form and apply various

techniques to get their optimal solution.

CO3 Understand the concept of assignment problem.

CO4 Explain different measures of queues, used to design a service facility.

CO5 Explain the concept of Inventory policy .

Text Book (s) 1. Hamdy A.Taha: Operations Research, Prentice Hall of India, 9th ed. 2010 2. Kanti Swarup, Gupta &Manmohan : Operations Research, S.Chand, 14th ed.

Reference Book (s)

3. Wagner :Principles of Operations Research (PH)

4. Sasievir, Yaspan, Friedman : Operations Research: Methods and Problems (JW)

5. J. K. Sharma : Operations Research – Theory and Applications, Macmillan Publishers

6. Kasana and Kumar :Introduction to Operations research, Springer

7. Schaum’s Outline Series : Operations Research ,Tata McGraw

8. Hillier & Lieberman : Introduction to Operations Research , Tata McGraw Hill Education Private Limited

9. Donald Gross, John F. Shortle, James M. Thompson, Carl M. Harris Fundamentals of

Queueing Theory, 4th Edition, Wiley

10. L. Kleinrock ,Queueing System(Vol 1) Theory, John Wiley and Sons

11. G. Hadly: Linear Programming, Narosa Publishing House

Unit-1 12 Hours

Introduction to Linear Programming: Graphical method , Simplex algorithm, feasible

solution, the artificial basis techniques, Two phase and Big-M method with artificial

variables. General Primal-Dual pair, formulating a dual problem, primal-dual pair in matrix form, Duality theorems, complementary slackness theorem, duality and simplex method,

economic interpretation of duality, dual simplex method. Game Theory: Two-person zero-sum games, maximin, minimax principle, games without saddle points(Mixed strategies), graphical solution of 2*n and m*2 games.

Unit-2 8 Hours

General transportation problem, transportation table, duality in transportation problem, loops in transportation tables, LP formulation, solution of transportation problem, test for

optimality, degeneracy, Transportation algorithm (MODI method). Mathematical formulation of assignment problem, assignment method, typical assignment problem.

Unit-3 12 Hours

Queuing Theory: Introduction, Queuing System, elements of queuing system, distributions of arrivals, inter arrivals, departure and service times. Classification of queuing models, Steady- state solutions of Markovian Queuing Models .Single service queuing model with infinite capacity (M/M/1):( /FIFO), (M/M/1): (N/FIFO), Generalized Model: Birth-Death Process, (M/M/C):( /FIFO), (M/M/C) (N/FIFO), M/G/1.

Unit-4

8 Hours

Inventory Control: The inventory decisions, costs associated with inventories, Classification of Inventories, Advantage of Carrying Inventory, Features of Inventory System factors affecting Inventory control, economic order quantity (EOQ) Deterministic inventory problems with no shortage and with shortages, EOQ problems with price breaks, Multi item deterministic problems.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Applied Numerical Analysis

Course Code MSCM6013

Prerequisite

Corequisite

Antirequisite

L T P C

3 1 0 4

Course Objectives:

1. To develop a numerical method foundation as a tool.

2. To understand error analysis in numerical methods in comparison with analytical

methods.

3. To understand the role of numerical methods in various numerical problems.

4. Analyze solution of partial differential equation by numerical method.

Course Outcomes

CO1 Compute eigen values and eigenvectors for different kind of matrices by

several numerical techniques.

CO2 Summarize various methods and techniques of numerical methods to solve ODE

problems and analysis its error estimation.

CO3 Differentiate between elliptic, parabolic and hyperbolic partial differential

equations and apply various method to solve elliptic PDE problems.

CO4 Describe concept of compatibility, convergence and stability and apply appropriate

methods to solve parabolic PDE problems.

CO5 Apply numerical techniques to solve hyperbolic PDE and assess the reliability of

numerical results through extensive error analysis.

Text Book (s)

1. Jain, M. K., “ Numerical Solution of Differential Equations”, John Wiley (1997).

2. Gerald, C. F. and Wheatly P. O., “Applied Numerical Analysis”, 6th Ed., Addison-Wesley Publishing (2002).

Reference Book (s)

3.Smith, G. D., “ Numerical Solution of Partial Differential Equations”, Oxford

University

Press (2001).

Unit-1

10 Hours

Computations of Eigen Values of a Matrix: Power method for dominant, sub-dominant and smallest eigen-values, Method of inflation Jacobi, Givens and Householder methods for

symmetric matrices, LR and QR methods.

Unit-2

10 Hours

Solution of ODE: Multistep methods; Predictor-corrector Adam-Bashforth, Milne 's method, their error analysis and stability analysis. System of first order ODE, higher order IVPs.

Unit-3 10 Hours

Solution of PDE Elliptic PDE: Five point formulae for Laplacian, replacement for Dirichlet and Neumann’s

boundary conditions, curved boundaries, solution on a rectangular domain, block tri-diagonal form and its solution using method of Hockney, condition of convergence.

Unit-4 10 Hours

Parabolic PDE: Concept of compatibility, convergence and stability, Explicit, full implicit, Crank-Nicholson, du-Fort and Frankel scheme, ADI methods to solve two-dimensional equations with error analysis. Hyperbolic PDE: Solution of hyperbolic equations using FD, and Method of characteristics, Limitations and Error analysis.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Applied Numerical Analysis Lab

Course Code MSCM6014

Prerequisite

Corequisite

Antirequisite

L T P C

0 0 2 1

Course Objective: The objective of this course is to continue with the exploration on facilities

provided by software (C language ) to the computation related to eigenvalue & eigenvectors

and solving Ordinary and Partial differential equations (Laplace, Heat & Wave) in general and

then extending the exploration to solving domain related problems by numerical method

approach.

Course Outcome

CO1 Write a C language code for finding eigen values & eigen vectors by various

numerical methods

CO2 Write a C language code to form a tridiagonal matrix by different numerical methods

CO3 Write a C language program to solve ODE & IVP and study it graphically.

CO4 Write a C language program for the solution of one dimensional heat equations &

Laplace equation by various numerical approach

CO5 Write a C language program for the solution of one dimensional wave equations by

various numerical method.

Text Book (s)

1. Jain, M. K., “ Numerical Solution of Differential Equations”, John Wiley (1997).

2. Gerald, C. F. and Wheatly P. O., “Applied Numerical Analysis”, 6th Ed., Addison-Wesley Publishing (2002).

Reference Book (s)

3.Smith, G. D., “ Numerical Solution of Partial Differential Equations”, Oxford

University

Press (2001).

S. No. Experiment

1. WAP in C to evaluate the smallest eigenvalue using the power method

2. WAP in C to evaluate the largest eigenvalue & eigenvectors using the power

method

3. WAP in C to evaluate the eigenvalue & eigenvectors using the Jacobi’s

method.

4. WAP in C to reduce in tridiagonal form a symmetric matrix by Given’s

method.

5. WAP in C to solve first order ODE by Milne’s method.

6. WAP in C to solve first order ODE by Adams-Bashforth method.

7. WAP in C to solve Laplace Equation by Jacob’s method.

8. WAP in C to solve Laplace Equation by Gauss - Seidal method.

9. WAP in C to solve Heat Equation by Crank –Nicolson method.

10 WAP in C to solve Heat Equation by Du Fort and Frankel method.

11 WAP in C to solve Wave Equation by implicit scheme.

12 WAP in C to solve Wave Equation by explicit scheme.

Continuous Assessment Pattern

Internal Assessment (IA) End Term Test

(ETE)

Total Marks

50

50 100

Elective-I

Name of The Course Module Theory

Course Code MSCM6004

Prerequisite

Corequisite

Antirequisite

L T P C

4 0 0 4

Course Objectives: Students will be able to understand Module theory as linear algebra over

general rings. They will have knowledge of special classes of modules such as free modules,

projective modules, flat modules to name a few. Students will have knowledge of theory of

modules over PID and its application to Jordan and Rational canonical forms.

Course Outcomes

CO1 To Summarize various types of modules and module homomorphism.

CO2 To describe different properties of modules and tensor product.

CO3 To define rings and ideals and understand various properties.

CO4 To explain Noetherian Rings, Primary Decomposition and different theorems

related to them.

Text Book(s)

1. Lang, S., Algebra, Addison-Wesley, 1993. Lam, T.Y., A First Course in Non-Commutative Rings, Springer Verlag. Hungerford, T.W., Algebra, Springer.

2. Jacobson, N., Basic Algebra, II, Hindusthan Publishing Corporation, India.

Reference Book(s)

3. Dummit, D.S., Foote, R.M., Abstract Algebra, Second Edition, John Wiley & Sons, Inc., 1999.

4. Atiyah, M., MacDonald, I.G., Introduction to Commutative Algebra, Addison-Wesley, 1969.

5. Malik, D.S., Mordesen, J.M., Sen, M.K., Fundamentals of Abstract Algebra, The McGraw-Hill Companies, Inc.

6. Curtis, C.W., Reiner, I., Representation Theory of Finite Groups and Associated Algebras, Wiley-Interscience, NY.

Unit-1 10 Hours

Units and Module Homomorphisms, Submodules and Quotient Modules, Operations on submodules, Direct Sum and Product, Finitely Generated Modules, Free Modules.

Unit-2 10 Hours

Tensor Products of modules, Universal Property of the tensor product, Restriction and Extension of Scalars, Algebras. Exact Sequences, Projective, Injective and Flat Modules,

Five Lemma, Projective Modules and HomR(M,-), injective modules and HomR(-,M), Flat modules and M x R - .

Unit-3 10 Hours

Rings and Modules of Fractions, Local Properties, Extended and contracted ideals in rings of fractions. Nilradical and Jacobson radical, Nakayama’s Lemma, Operations on Ideals, Prime Avoidance, Chinese Remainder Theorem, Extension and Contraction of ideals.

Unit-4 10 Hours

Noetherian Rings, Primary Decomposition in Noetherian Rings. Integral Dependence,

Lying-Over Theorem, Going-Up Theorem, Integrally Closed Domains, Going-Down Theorem, Noether Normalization, Hilbert Nullstellensatz. Transcendence Base, Separably

Generated Extensions, Schmidt and Lüroth Theorems.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Measure and Probability Theory

Course Code MSCM6005

Prerequisite

Corequisite

Antirequisite

L T P C

4 0 0 4

Course Objectives: The aim of this course is to learn the basic elements of Measures and

Probability Theory. It provides a foundation for many branches of mathematics such as

harmonic analysis, theory of partial differential equations and probability theory.

Course Outcomes

CO1 Summarizes the basic ideas of measure, Lebesgue measure, probability measure

and random variables with variety of examples.

CO2 Solve challenging problems, develop proofs of theorems on their own and present

those proofs clearly and coherently with appropriate illustrative examples.

CO3 Define and illustrate the concept of measurable functions, Borel and Lebesgue

measurability, the 𝐿𝑝- space.

CO4 Define and illustrate the concept of probability space, limit of events, random

vectors, distribution and expectation.

CO5 Define the concepts of sequence of random variables, moment generating function

and modes of convergence.

CO6 Prove a selection of theorems concerning Weak and strong laws of large number,

continuity theorem and central limit theorem.

Text Book(s)

1. P. Billingsley, Probability and Measure, 3rd ed., John Wiley & Sons, New York, 1995

2. G. De Barra, Measure theory and Integration, New age international publishers, 2012

Reference Book(s)

3. J. Rosenthal, A First Look at Rigorous Probability, World Scientific,

Singapore, 2000.

4. A.N. Shiryayev, Probability, 2nd ed., Springer, New York, 1995.

5. K.L. Chung, A Course in Probability Theory, Academic Press, New York, 1974.

Unit-1 10 Hours

Measure Theory: Measures and outer measures. Measure induced by an outer measure,

Extension of a measure.Uniqueness of Extension, Completion of a measure. Lebesgue outer

measure. Measurable sets. NonLegesgue measurable sets. Regularity. Measurable

functions. Borel and Lebesgue measurability. Integration of non-negative functions. The

general integral. Convergence theorems. Riemann and Lebesgue integrals. The LP -space.

Convex functions. Jensen’s inequality. Holder and Minkowski inequalities.

Unit-2 10 Hours

Probability measure, probability space, construction of Lebesgue measure, extension theorems, limit of events, Borel-Cantelli lemma.

Unit-3 10 Hours

Random variables, Random vectors, distributions, multidimensional distributions,

independence. Expectation, change of variable theorem, convergence theorems.

Unit-4 10 Hours

Sequence of random variables, modes of convergence. Moment generating function

and characteristics functions, inversion and uniqueness theorems, continuity

theorems, Weak and strong laws of large number, central limit theorem.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Course Objectives: Students will develop an understanding of Analytic aspects and

methods used to study the distribution of prime numbers, Arithmetic functions and their

utility in the analytic theory of numbers including the distribution of primes.

Course Outcomes

CO1 Introduction to various special kind of functions like Moebius function, Euler phi

(totient) function, Von Mangoldt function, divisor and sum-of-divisors functions to

name a few.

CO2 Exposure to Riemann zeta function, Partial sums of the Euler phi function and

averages of the Moebius functions.

CO3 Develop understanding and knowledge of Dirichlet series and its analytic

properties.

CO4 Ability to understand the proofs of basic theorems of Analytical Number Theory.

Text Book(s)

1. A.J. Hildebrand : Introduction to Analytic Number Theory Math 531 Lecture Notes, Fall 2005 , http://www.math.uiuc.edu/~hildebr/ant

Reference Books(s)

2. Harold Davenport: Multiplicative number theory, third ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000,

3.T. Apostol : Introduction to analytic number theory, New York: Springer-Verlag,

1976

Unit-1 10 Hours

Review of: Primes and the Fundamental Theorem of Divisibility and primes , The

Fundamental Theorem of Arithmetic , The infinitude of primes , Elementary theory of Arithmetic functions, Introduction and basic examples .Additive and multiplicative

functions. The Moebiusfunction . The Euler phi (totient) function. The von Mangoldt function. The divisor and sum-of-divisors functions.

Name of The Course Analytical Number Theory

Course Code MSCM6006

Prerequisite Basic concepts of number theory

Corequisite

Antirequisite

L T P C

4 0 0 4

Arithmetic functions II: Asymptotic estimates : Big oh and small oh notations, asymptotic

equivalence , Basic definitions, Extensions and Examples, The logarithmic integral, Sums of smooth functions: Euler’s summation formula, Statement of the formula , Partial sums

of the harmonic series, Partial sums of the logarithmic function and Stirling’s formula.

Unit-2 10 Hours

Integral representation of the Riemann zeta function. Removing a smooth weight function

from a sum: Summation by parts, The summation by parts formula, Kronecker’s Lemma ,

Relation between different notions of mean values of arithmetic functions , Dirichlet series

and summatory functions , Approximating an arithmetic function by a simpler arithmetic

function: The convolution method , Description of the method, Partial sums of the Euler

phi function , The number of squarefree integers below x, Wintner’s mean value theorem ,

A special technique: The Dirichlet hyperbola method, Sums of the divisor function,

Distribution of primes I: Elementary results , Chebyshev type estimates, Mertens type

estimates, Elementary consequences of the PNT, The PNT and averages of the Moebius

function.

Unit-3 10 Hours

Arithmetic functions III: Dirichlet series and Euler products , Algebraic properties of

Dirichlet series , Analytic properties of Dirichlet series, Dirichlet series and summatory functions, Mellin transform representation of Dirichlet series , Analytic continuation of the

Riemann zeta function, Lower bounds for error terms in summatory functions , Evaluation

of Mertens’ constant , Inversion formulas.

Unit-4 10 Hours

Distribution of primes II: Proof of the Prime Number Theorem :Introduction , The

Riemann zeta function, I: basic properties , The Riemann zeta function, II: upper bounds ,

The Riemann zeta function, III: lower bounds and zero free region, Proof of the Prime

Number Theorem.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Course Objectives: Harmonic analysis is equated with the study of Fourier series and integrals. In this course we will study Fourier Transforms on the Line, Fourier Analysis on Locally Compact Abelian Groups , Commutative Banach Algebras and Spectral synthesis in regular algebras.

Course Outcomes

CO1 To define Fourier transform in various function spaces

CO2 To explain Fourier Analysis on Locally Compact Abelian Groups and Haar

measure.

CO3 To define Commutative Banach Algebras.

CO4 To perform spectral synthesis in different Algebras.

Text Book(s)

1. Yitzhak Katznelson ., : An Introduction to Harmonic Analysis, Third Edition , Cambridge University Press

2. Henry Helson. - Harmonic analysis, Springer Verlag- 1995.

Reference book(s)

3. T. W. Kroner,: Fourier Analysis, Cambridge University Press.

Unit-1 10 Hours

Fourier Transforms on the Line : Fourier transforms for L 1, Fourier–Stieltjes transforms , Fourier transforms in L p (R) , 1 < p ≤ 2 , Tempered distributions and pseudo-measures ,

Almost-

Periodic functions on the line , The weak-star spectrum of bounded functions , The Paley–

Wiener theorems , The Fourier–Carleman transform, Kronecker’s theorem.

Unit-2 10 Hours

Fourier Analysis on Locally Compact Abelian Groups : Locally compact abelian groups, The Haar measure, Characters and the dual group, Fourier transforms, Almost-periodic functions and the Bohr compactification

Unit-3 10 Hours

Commutative Banach Algebras : Definition, examples, and elementary properties

Name of The Course Harmonic Analysis

Course Code MSCM6007

Prerequisite

Corequisite

Antirequisite

L T P C

4 0 0 4

Maximal ideals and multiplicative linear functionals , The maximal-ideal space and the Gelfand representation , Homomorphisms of Banach algebras , Regular algebras, Wiener’s general Tauberian theorem.

Unit-4 10 Hours

Spectral synthesis in regular algebras, Functions that operate in regular Banach algebras , The algebra M (T) and functions that operate on Fourier–Stieltjes coefficients, The use of tensor products.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Elective-II

Name of The Course Algebraic Topology

Course Code MSCM6008

Prerequisite

Corequisite

Antirequisite

L T P C

4 0 0 4

Course Objectives: This course focuses on the computation of homotopy invariants

of topological spaces, in particular the fundamental group, the homology groups and the co

homology ring.

Course Outcomes

CO1 To define Homotopy of paths, contractibility and the fundamental group of circle.

CO2 To explain different homology groups, their properties and homomorphism induced

by continuous map.

CO3 To define covering projections and homomorphism.

CO4 To describe Singular homology, the Excision Theorem, and Mayer-Vietoris

sequence.

Texts Book(s)

1. S. Deo, Algebraic Topology, Hindustan book agency, India, 2003

2. A. Hatcher, Algebraic Topology, Cambridge Univ. Press, Cambridge, 2002.

References Book(s)

3. W. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, Berlin,

1991

4. J.J. Rotman, An Introduction to Algebraic Topology, Springer (India), 2004.

Unit-1 10 Hours

Homotopy of paths, homotopy equivalence, contractibility, deformation retracts. Fundamental groups and its properties, The fundamental group of circle.

Unit-2 10 Hours

Simplicial complexes and simplicial maps; Homology groups; Barycentric subdivision; The simplicial approximation theorem. Simpilicial homology,simplicial chain complex and homology, Properties of integral homology groups, invariance of homology groups, subdivision chain map, homomorphism induced by continuous map, homotopy invariance, Lefschetz fixed point theorem, The Borsuk-Ulam theorem.

Unit-3 10 Hours

Covering projections and its properties, Application of homotopy lifting theorem, lifting of

an arbitrary map, covering homomorphisms, Universal covering spaces.

Unit-4 10 Hours

Singular homology, singular chain complex, one dimensional Homology, Homotopy axiom

for singular homology, The Excision Theorem, Homology and cohomology theories,

Mayer-Vietoris sequence.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Dynamical Systems

Course Code MSCM6009

Prerequisite Differential Equations

Corequisite NA

Antirequisite NA

L T P C

4 0 0 4

Course Objectives: The course objectives to introduce the main features of dynamical

systems, particularly as they arise from systems of ordinary differential equations as models

in applied mathematics. The topics presented will include phase space, fixed points and

stability analysis, bifurcations, Hamiltonian systems and dissipative systems. Discrete

dynamical systems will also be discussed briefly.

Course Outcomes

CO1 Describe the main features of dynamical systems and their realisation as systems of

ordinary differential equations

CO2 Identify fixed points of simple dynamical systems, and study the local dynamics

around these fixed points, in particular to discuss their stability and bifurcations

CO3 Use a range of specialised analytical techniques which are required in the study of

dynamical systems

CO4 Prove simple theoretical results about abstract dynamical systems

CO5 Analyze the chaotic behaviour of any dynamical system.

Text Book(s)

1. M. W. Hirsch & S. Smale – Differential Equations, Dynamical Systems and Linear Algebra (Academic Press 1974)

2. L. Perko – Differential Equations and Dynamical Systems (Springer – 1991)

Reference Book(s)

3. Ferdinand Verhulst : Nonlinear differential equations and dynamical systems: 2nd Edition, Springer, 1996.

4. D.W. Jordan and P. Smith : Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems, 4th Edition, (Oxford University Press, 2007).

5. J.K. Hale and H. Kocak : Dynamics and Bifurcations: (Springer, 1991).

6. I.P. Glendinning : (Cambridge Stability, Instability and Chaos : University Press

1994).

Unit-1 10 Hours Introduction: Phase variables and Phase space, continuous and discrete time systems,

flows(vector fields), maps (discrete dynamical systems), orbits, asymptotic states, fixed

(equilibrium) points periodic points, concepts of stability and SDIC (sensitive dependence

of initial conditions) chaotic behaviour, dynamical system as a group.

Unit-2 10 Hours Linear systems: Uncoupled Linear Systems, Diagonalization, Fundamental theorem and its application. Properties of exponential of a matrix, Exponential of operators ,linear systems

in R, Complex eigenvalues, multiple eigenvalues generalized eigenvectors of a matrix,

nilpotent matrix, Jordan Canonical Forms , stability theory ,stable, unstable and center subspaces, hyperbolicity, contracting and expanding behaviour. Non-homogeneous Linear

systems.

Unit-3 10 Hours Nonlinear Systems: Local Theory, Fundamental existence theorem dependence on initial conditions and parameters, the maximal interval of existence, Flow defined by a differential equation. Linearization, stable manifold theorem,

Unit-4 10 Hours

Nonlinear Vector Fields : Stability characteristics of an equilibrium point. Liapunov and

asymptotic stability. Source, sink, basin of attraction. Phase plane analysis of simple systems, homoclinic and heteroclinic orbits, hyperbolicity, statement of Hartmann-Grobman theorem and stable manifold theorem and their implications. Liapunov function and Liapunov theorem.Statement of Lienard’s theorem and its application to vander Pol equation, Poineare-Bendixsom theorem (statement and

applications only), structural stability and bifurcation through examples of saddle-node, pitchfork and Hopf bifurcations.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Fluid Mechanics

Course Code MSCM6010

Prerequisite

Corequisite

Antirequisite

L T P C

4 0 0 4

Course Objectives: To give fundamental knowledge of fluid, its properties and behavior under

various conditions of internal and external flows.To develop understanding about hydrostatic

law, principle of buoyancy and stability of a floating body and application of mass, momentum

and energy equation in fluid flow.

Course Outcomes

CO1 Know basic definition, about fluid motion, equation of continuity.

CO2 Study Bernoulli’s equation, the irrotational motion, cyclic motions, Vortex motion,

sources and sinks and some related theorems.

CO3 Study motion of circular and elliptic cylinders, theorem of Kutta and Juokowski,

some special transformation, Source, sinks, doublets and their images with regards

to a plane and sphere.

CO4 Learn about the Vortex motion in detail.

Text Book(s)

1. A. S. Ramsay, “Hydrodynamics: A Treatise on Hydromechanics – Part II ”, Bell,

1913.

2. L. D. Landau and E. M. Lifshitz, “Fluid Mechanics”, Pergamon Press,1959.

Reference Book(s)

3.H. Lamb, “Hydrodynamics”, Cambridge University Press, 1932.

4. L. M. MilneThomson, “Theoretical Hydrodynamics”, MacMillan, 1955.

5.S. Swaroop, “Fluid Dynamics”, Krishna Prakashan, 2000.

Unit-1 10 Hours

Lagrange’s and Euler’s methods in fluid motion. Equation of motion and equation of continuity, Boundary conditions and boundary surface stream lines and paths of particles.

Irrotational and rotational flows, velocity potential. Bernoulli’s equation. Impulsive

action. Equations of motion and equation of continuity in orthogonal curvilinear co-ordinates. Euler’s momemtum theorem and D’Alemberts paradox.

Unit-2 10 Hours

Theory of irrotational motion flow and circulation. Permanence irrotational motion. Connectivity of regions of space. Cyclic constant and acyclic and cyclic motion. Kinetic energy. Kelvin’s minimum. Energy theorem. Uniqueness theorem. Complex potential, sources. sinks, doublets and their images circle theorem. Theorem of Blasius.

Unit-3 10 Hours

Motion of circular and elliptic cylinders. Steady streaming with circulation. Rotation of elliptic cylinder. Theorem of Kutta and Juokowski. Conformal transformation. Juokowski transformation. Schwartz-chirstoffel theorem. Motion of a sphere. Stoke’s stream function. Source, sinks, doublets and their images with regards to a plane and sphere.

Unit-4 10 Hours

Vortex motion. Vortex line and filament equation of surface formed by stream lines and vortex lines in case of steady motion. Strength of a filament. Velocity field and kinetic

energy of a vortex system. Uniqueness theorem rectilinear vortices. Vortex pair. Vortex doublet. Images of a vortex with regards to plane and a circular cylinder. Angle infinite

row of vortices. Karman’s vortex sheet.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Discrete Structures

Course Code MSCM6011

Prerequisite

Corequisite

Antirequisite

L T P C

4 0 0 4

Course Objectives: Discrete structure is the study of mathematical structures that are

fundamentally discrete rather than continuous. The objective of this course is to teach students

how to think logically and mathematically.

Course Outcomes

CO1 Apply advance counting techniques to solve a variety of problems .

Apply the Rules of Inference in solving variety of problems including the validity of

an argument.

CO2 Apply various methods to solve recurrence relations.

CO3 Understand Posets and Lattices and their various types.

CO4 Understand the concept of graph theory and is various applications.

Text Book(s)

1. Kenneth Rosen,” Discrete Mathematics and it’s Applications”, 7th Ed,Mc Graw Hill publications, 2012.

2. Y.N Singh, Discrete Mathematical Structures,Willey India, New Delhi, 1st edition,2010.

Reference Book(s)

3. Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall of India,1984.

4. Liu and Mohapatra,”Elements of Discrete Mathematics”, Mc Graw Hill,

Unit-1 10 Hours

Basic counting principles: Permutations and Combinations (with and without

repetitions), Binomial theorem, Multinomial theorem, Counting subsets, Set-partitions,

Stirling numbers,Principle of Inclusion and Exclusion, Derangements, Inversion

Formulae.

Unit-2 12 Hours

Recurrence Relation and Generating functions: Algebra of formal power series,

Generating function models, Calculating generating functions, Exponential generating

functions, Recurrence relations: Recurrence relation models, Divide and conquer relations,

Solution of recurrence relations, Solutions by generating functions

Unit-3 10 Hours

Lattices: Partial order sets, Hasse diagram, Lattices: definition, properties of lattices, bounded lattices, complemented lattices, modular lattices, modular and complete lattices, morphism of lattices.

Unit-4 8 Hours

Graph Theory: Definitions of different types of graphs, degree, sub-graph, intersection of graphs, homeomorphism and isomorphism of graphs. Computer representation of graphs

and diagraphs. Adjacency and incidence matrices of a graph and a diagraph. Walks, trails and paths, cycles, connectedness. Trees, forests and spanning trees. Euler graph, postman

problem, Moor’s, Bellman’s and Dijkstra’s algorithms for shortest path.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Elective-III

Name of The Course Manifolds and Applications

Course Code MSCM6015

Prerequisite

Corequisite

Antirequisite

L T P C

4 0 0 4

Course Objective: The course will be an introduction to differentiable manifolds with an eye towards Lie groups and Lie bracket. We will start with the basics of differentiable manifolds (tangent spaces, vector fields, Lie brackets, etc.) and come to grips with differential forms and tensors. Course also contains Riemannian manifolds, Submanifolds and Hypersurfaces and Almost Complex manifolds.

Course Outcomes

CO1 Understand basic concepts of manifolds.

CO2 Explain Riemannian manifolds and tensors.

CO3 Understand Submanifolds and Hypersurfaces

CO4 Understand Almost Complex manifolds, Nijenhuis tensor, Contravariant and

covariant almost analytic vector fields

Text Book(s)

1. Serge Lang, Introduction to Differentiable Manifolds, Springer-verlag, 2002.

2. R. S. Mishra, Structures on a differentiable manifold and their applications, Chandrama Prakashan, Allahabad, 1984.

Reference Book(s)

3. R. S. Mishra, A course in tensors with applications to Riemannian Geometry,

Pothishala (Pvt.) Ltd., 1965.

4. K. Yano and M. Kon, Structure of Manifolds, World Scientific Publishing Co. Pvt. Ltd., 1984.

5. B. B. Sinha, An Introduction to Modern Differential Geometry, Kalyani Publishers, New Delhi, 1982.

6. U. C. De and A. A. Shaikh, Differential Geometry of Manifolds, Narosa Publishing House Pvt. Ltd., 2007.

7. Gerardo F. Torres Del Castillo, Differentiable Manifolds: A Theoretical Physics Approach, Birkhauser Boston, 2011.

Unit-1 10 Hours

Definition and examples of differentiable manifolds, Vector fields and Tangent spaces,

Jacobian map, One parameter group of transformations, Lie bracket, Covariant, Lie and Exterior derivative.

Unit-2 10 Hours

Riemannian manifolds, Riemannian connection, Torsion tensor, Curvature tensors, Ricci tensor, scalar curvature, Sectional Curvature, Schur’s theorem, Geodesics in a Riemannian manifold, Projective curvature tensor and conformal curvature tensor.

Unit-3 10 Hours

Submanifolds and Hypersurfaces, Normals, Gauss’ formulae, Weingarten equations, Lines of Curvature, Generalized Gauss and Mainardi-Codazzi equations.

Unit-4 10 Hours

Almost Complex manifolds, Nijenhuis tensor, Contravariant and covariant almost analytic vector fields, F-connection.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Mathematical Modelling

Course Code MSCM6016

Prerequisite Linear algebra & Calculus

Corequisite NA

Antirequisite NA

L T P C

4 0 0 4

Course Objectives: The overall objectives of this course is to enable students to build

mathematical models of real-world systems, analyze them and make predictions about

behaviour of these systems. Variety of modelling techniques will be discussed with examples

taken from physics, biology, chemistry, economics and other fields. The focus of the course

will be on seeking the connections between mathematics and physical systems, studying and

applying various modelling techniques to creating mathematical description of these systems,

and using this analysis to make predictions about the system’s behaviour.

Course Outcomes

CO1 Assess and articulate what type of modelling techniques are appropriate for a given

real world system

CO2 Construct a mathematical model of a given real world system and analyze it,

CO3 Make predictions of the behaviour of a given real world system based on the

analysis of its mathematical model.

CO4 Recognise the power of mathematical modelling and analysis and be able to apply

their understanding to their further studies.

CO5 Apply Sensitivity analysis and find Pitfalls in mathematical models

Text Book(s)

1. Kapur , J.N.,”Mathematical Modelling”,New Age international publisher, 1988.

2. Burghes D.N , “Modelling with differential equations”, Ellis Horwood and

John Wiley,1991

Reference Book(s)

3. Burghes, D.N.,” Mathematical Modelling in the Social Management and Life

Science”,

Ellie Herwood and John Wiley.

4. Charlton, F.,” Ordinary Differential and Difference Equations”, Van Nostrand.

5. Brauer, Castillo-Chavez ,”Mathematical Models in Population Biology and

Epidemiology”.

Unit-1 10 Hours

Need, Techniques and classification:

Linear growth and decay model, Non Linear growth and decay model, compartment

model, some simple models.

Unit-2 10 Hours

Modelling through Ordinary Differential Equations: Basic theory, Models in Economics and finance, Population dynamics and Genetics.

Unit-3 10 Hours

Modelling through Partial Differential Equations: Mass balance approach, Momentum Balance approach, Models for traffic flow on

highway, BOD-DO models

Unit-4 10 Hours

Modelling through Graphs:

Directed graphs, signed graphs, weighted digraphs, Linear programming models in forest

management, Transportation and assignment models.

Analyses of models: Sensitivity analysis, Pitfalls in modelling, Illustrations

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Financial Mathematics

Course Code MSCM6017

Prerequisite

Corequisite

Antirequisite

L T P C

4 0 0 4

Course Objectives:. To make students to understand Interest rates, annuities and mortgages,

bonds and bond market structure.

Course Outcomes

CO1 Summarize the concepts of time value of money using simple interest and

discounting

CO2 Able to apply compound interest model with effect of investment

CO3 Able to apply discounted cash flow techniques in different project appraisal

CO4 Calculate the price of a forward contract

CO5 Applying hedging in the contract

Text Book (s)

1. Suresh Chandra, S. Dharmaraja, Aparna Mehra, R. Khemchandani, Financial

Mathematics: An Introduction, Narosa Publication House, 2012

Reference Book (s)

2. D.G. Luenberger, Investment Science, Oxford University Press, Oxford, 1998.

3. J.C. Hull, Options, Futures and Other Derivatives, 4th ed., Prentice-Hall, New York, 2000.

4. J.C. Cox and M. Rubinstein, Options Market, Englewood Cliffs, N.J.: Prentice Hall, 1985.

Unit-1 10 Hours

Interest rates, Simple interest rates, Present value of a single future payment. Discount

factors, effective and nominal interest rates. Real and money interest rates.

Unit-2 10 Hours

Compound interest rates. Relation between the time periods for compound interest rates

and the discount factor. Compound interest functions. Annuities and perpetuities.

Unit-3 10 Hours

Loans. Introduction to fixed-income instruments. Generalized cash flow model. Net present

value of a sequence of cash flows. Equation of value. Internal rate of return. Investment

project appraisal. Cash flow, present value of a cash flow, securities, fixed income

securities, types of markets.

Unit-4 10 Hours

Forward and futures contracts, options, properties of stock option prices, trading strategies

involving options, option pricing using Binomial trees, Black – Scholes model, Black-

Scholes formula, Risk-Neutral measure, Delta – hedging, options on stock indices,

currency options.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Coding Theory

Course Code MSCM6018

Prerequisite

Corequisite

Antirequisite

L T P C

4 0 0 4

Course Objective: The course aims an introduction to traditional and modern coding theory.

It provides an overview of various encoding and decoding methods and their application.

Course Outcomes

CO1 Explain basic concepts of coding theory.

CO2 Understand various types of coding and decoding techniques .

CO3 Understand the Golay Codes, Codes and Lattices, Weight Enumerators.

CO4 Apply a Double-Error Correcting Decimal Code and Introduce to BCH Codes.

Text Book(s)

1. Raymond Hill: A First Course in Coding Theory, Oxford Applied

Mathematics and Computing Science Series.1990

2. W. Wesley Peterson and E. J. Weldon: Error Correcting Code, 2nd ed., MIT

Press. 1972

Reference Book(s)

3. Mac Williams and Sloane: The Theory and Practice of Error-Correcting

Codes, North Holland Pub Company

4. Van Lint, J. H. Introduction to coding theory, Third edition. Graduate

Texts in Math-ematics, 86. Springer-Verlag, Berlin, 1999.

5. Huffman, W. C. and Pless, V. Fundamentals of error-correcting codes.

Cambridge University Press, Cambridge, 2003.

Unit-1 10 Hours

Introduction to error correcting codes, Minimum distance, types and properties of codes, linear and non linear codes, Repetition Codes, Main coding theorem problem, Shannon's Noisy Channel Coding Theorem. Review of number theory, arithmetics in Finite Fields and Vector Spaces over Finite fields

Unit-2 10 Hours

Introduction to Linear Codes, Encoding and Decoding with a Linear Code, The Dual Code,

the Parity-Check Matrix, and Syndrome Decoding, Bounds on Codes, The Hamming

Codes, Perfect Codes

Unit-3 10 Hours

The Golay Codes, Codes and Lattices, Weight Enumerators and the MacWilliams

Theorem, MDS Codes

Unit-4 10 Hours

A Double-Error Correcting Decimal Code and an Introduction to BCH Codes, Cyclic

Codes, Hadamard Codes, Reed-Solomon Codes

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Elective-IV

Name of The Course Finite Element Method

Course Code MSCM6019

Prerequisite

Corequisite

Antirequisite

L T P C

4 0 0 4

Course Objectives: To make students to learn basic principles of finite element analysis

procedure, to learn the theory and characteristics of finite elements that represent

engineering structures.

Course Outcomes

CO1 Understand the fundamental theory of the FEA method.

CO2 Understand the use of the basic finite elements for different structural problems.

CO3 To Develop the ability to generate the governing FE equations for systems govern

by ordinary and partial differential equations.

CO4 To Demonstrate the ability to evaluate and interpret FE analysis results for design and

eva evaluation purposes.

CO5 To d Develop a basic understanding of the limitations of the FE method and understand

the possible error sources in its use.

Text Book (s)

1. Reddy J.N., “Introduction to the Finite Element Methods”, Tata McGraw-Hill. 2003

2. Bathe K.J., Finite Element Procedures”, Prentice-Hall. 2001

Reference Book (s)

3. Cook R.D., Malkus D.S. and Plesha M.E., “Concepts and Applications of Finite Element Analysis”, John Wiley.2002

4. Thomas J.R. Hughes “The Finite Element Method: Linear Static and Dynamic Finite Element Analysis”. 2000

5. George R. Buchanan “Finite Element Analysis”, 1994

Unit-1 10 Hours

Introduction to finite element methods, comparison with finite difference methods.

Methods of weighted residuals, collocations, least squares and Galerkin’s method, Variational formulation of boundary value problems equivalence of Galerkin and Ritz methods.

Unit-2 10 Hours

Applications to solving simple problems of ordinary differential equations, Linear, quadratic and higher order elements in one dimensional and assembly, solution of assembled system.

Unit-3 10 Hours

Simplex elements in two and three dimensions, quadratic triangular elements, rectangular elements, serendipity elements and isoperimetric elements and their assembly, discretization with curved boundaries.

Unit-4 10 Hours

Interpolation functions, numerical integration, and modelling considerations, Solution of two dimensional partial differential equations under different geometric conditions.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Computational Fluid Dynamics

Course Code MSCM6020

Prerequisite

Corequisite

Antirequisite

L T P C

3 0 0 3

Course Objectives: The course aims to shape the attitudes of learners regarding the field of

Computational Fluid Dynamics and its application.

Course Outcomes

CO1 Identify mathematical characteristics of partial differential equations

CO2 Explain the basic properties of computational methods – accuracy, stability,

consistency

CO3 Apply computational solution techniques for various types of partial differential

equations

CO4 Apply computational method to solve Euler and Navier-Stokes equations

Text Book (s)

1. C. A. J. Fletcher, “Computational Techniques for Fluid Dynamics”, Vol-I

and Vol-II, Springer, 1988.

2. J. C. Tanehill, D. A. Anderson, R. H. Pletcher, “Computational Fluid

Mechanics and Heat Transfer”, Taylor & Francis, 1997.

Reference Book (s)

3. P. Niyogi, S. K. Chakraborty and M. K. Laha, “Introduction to

Computational Fluid Dynamics”, Pearson Education, Delhi, 2005.

4. R. Peyret and T. D. Taylor, “Computational Methods for Fluid Flow”,

Springer, 1983.

5. J. F. Thompson, Z.U.A Warsi and C. W. Martin, “Numerical Grid

Generation, Foundations and Applications”, Prentice Hall, 1985.

6. J.D. Anderson, “Computational Fluid Dynamics”, Mc Graw Hill, 1995.

Unit-1 10 Hours

Classification of 2 order partial differential equations - parabolic, hyperbolic and elliptic types. Governing equations of fluid dynamics, Introduction to finite difference discretization. Explicit and Implicit schemes. Truncation error, consistency, convergence and stability analysis.

Unit-2 10 Hours

Thomas algorithm. ADI method for 2-D heat conduction problem. Splitting and approximate factorization for 2-D Laplace equation. Multigrid method. Upwind scheme, CFL stability condition. Lax-Wendroff and MacCormack schemes.

Unit-3 10 Hours

Finite Volume method: Preliminary concepts. Flux computation across quadrilateral cells.

Reduction of a BVP to algebraic equations. Illustrative example like, solution of Dirichlet

problem for 2-D Laplace equation. Conservation principles of fluid dynamics. Basic

equations of viscous and inviscid flow. Basic equations in conservative form. Associated

typical boundary conditions for Euler and Navier-Stokes equations. Grid generation using

elliptic partial differential equations.

Unit-4 10 Hours

Incompressible viscous flow field computation: Stream function vorticity formulation, Staggered grid, MAC method, SIMPLE algorithm.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Computational Fluid Dynamics Lab

Course Code MSCM6021

Prerequisite

Corequisite

Antirequisite

L T P C

0 0 2 1

Course Objectives: The course aims to shape the attitudes of learners regarding the field of

Computational Fluid Dynamics and its application.

Course Outcomes

CO1 Understand the classification of PDE

CO2 Understand the concept of solving Heat Conduction equation

CO3 Apply the concept of Finite volume method.

CO4 Understand the concept of viscous and inviscous flow.

CO5 Apply the concept of vorticity.

Text Book (s)

1. C. A. J. Fletcher, “Computational Techniques for Fluid Dynamics”, Vol-I

and Vol-II, Springer, 1988.

2. J. C. Tanehill, D. A. Anderson, R. H. Pletcher, “Computational Fluid

Mechanics and Heat Transfer”, Taylor & Francis, 1997.

3. P. Niyogi, S. K. Chakraborty and M. K. Laha,“Introduction to Computational

Fluid Dynamics”, Pearson Education, Delhi, 2005.

Reference Book (s)

4. R. Peyret and T. D. Taylor,“Computational Methods for Fluid Flow”, Springer,

1983.

5. J. F. Thompson, Z.U.A Warsi and C. W. Martin,“Numerical Grid Generation,

Foundations and Applications”, Prentice Hall, 1985.

6. J.D. Anderson,“Computational Fluid Dynamics”, Mc Graw Hill, 1995.

S. No. Experiment

1. Installation of the Scilab, Overview, Basic syntax, Mathematical Operators,

Predefined constants, Built in functions.

2. Determination of vector differential operators for the given tensors

3. Plotting of stream lines and plot lines.

4. Plots of solution curves/surfaces to both ODE and PDE.

5. Demonstration of plane-Couette flow.

6. Demonstration of the flow of a viscous and inviscous incompressible fluid

between two vertical plates placed at a finite distance

7. Demonstration of the radially symmetric incompressible steady flow between

two cylinders.

8. Finite Volume method: Preliminary concepts. Flux computation across

quadrilateral cells.

9. Demonstration of the pressure distribution on an idealized underwater vehicle

as it moves along near the ocean bottom

10. Demonstration of Laminar flow of an incompressible viscous fluid between

two parallel plates.

Continuous Assessment Pattern

Internal Assessment Lab (IA)

End Term Lab Test

(ETE)

Total Marks

50 50 100

Name of The Course Stochastic Processes

Course Code MSCM6022

Prerequisite

Corequisite

Antirequisite

L T P C

4 0 0 4

Course Objectives: The aim of this course is to make students understand the concept

of Random process and its applicability.

Course Outcomes

CO1 explain basic concepts of probability and stochastic process

CO2 apply various stochastic processes .

CO3 explain Discrete parameter Markov Chains and apply it to their application.

CO4 explain Continuous parameter MarkovChains and apply it to their application.

Text Book (s)

1. J. Medhi, Stochastic Processes, 3rd Edition, New Age International, 2009.

2. Liliana Blanco Castaneda, Viswanathan Arunachalam and S. Dharmaraja,

Introduction to Probability and Stochastic Processes with Applications, Wiley, 2012.

Reference Book (s)

3. S.M. Ross, Stochastic Processes, 2nd Edition, Wiley, 1996.

4. S Karlin and H M Taylor, A First Course in Stochastic Processes, 2nd edition,

Academic Press, 1975.

5. Kishor S. Trivedi, Probability, Statistics with Reliability, Queueing and

Computer Science Applications, 2nd edition, Wiley, 2001.

6. S. E. Shreve, Stochastic Calculus for Finance, Vol. I & Vol. II, Springer, 2004.

7. V. G. Kulkarni, Modelling and Analysis of Stochastic Systems, Chapman & Hall,

1995.

8. G. Sankaranarayanan, Branching Processes and Its Estimation Theory, Wiley,

1989.

Unit-1 10 Hours

Introduction to Stochastic Processes (SPs): Definition and examples of SPs, classification of random processes according to state space and parameter space, types of SPs, elementary problems. Stationary Processes: Weakly stationary and strongly stationary processes, moving average and auto regressive processes.

Unit-2 10 Hours

Discrete-time Markov Chains (DTMCs): Definition and examples of MCs, transition

probability matrix, Chapman-Kolmogorov equations; calculation of n-step transition probabilities, limiting probabilities, classification of states, ergodicity, stationary

distribution, transient MC; random walk and gambler’s ruin problem, applications.

Unit-3 10 Hours

Continuous-time Markov Chains (CTMCs): Kolmogorov- Feller differential equations, infinitesimal generator, Poisson process, birth-death process, stochastic Petri net, applications to queueing theory and communication networks. Martingales: Conditional expectations, definition and examples of martingales.

Unit-4 10 Hours

Brownian Motion: Wiener process as a limit of random walk; process derived from Brownian motion, stochastic differential equation, stochastic integral equation, Ito formula, Some important SDEs and their solutions, applications to finance.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Automata & Formal Languages

Course Code MSCM6023

Prerequisite

Corequisite

Antirequisite

L T P C

4 0 0 4

Course Objectives: Introduce students to the mathematical foundations of computation

including automata theory; the theory of formal languages and grammars; the notions of

algorithm, decidability, complexity, and computability.

Course Outcomes

CO1 Understand basic concepts of mathematical prilimnaries and finite automata and their

applications.

CO2 Understand the concepts as well as applications of regular expressions and regular

languages and their applications.

CO3 Understand the concepts context-free languages and pushdown automata.

CO4 Understant the basic concepts of Turing machines and their applications.

Text Book(s)

1. D. Kelly, Automata and Formal Languages: An Introduction, Prentice-Hall, 1995.

2. P. Linz, An Introduction to Formal Languages and Automata, 3rd Edition, Narosa, 2002.

References Book(s)

3. J. E. Hopcroft, R. Motwani, and J.D. Ullman, Introduction to Automata, Languages, and

Computation (2nd edition), Pearson Edition, 2001.

Unit-1 10 Hours

Alphabets and Languages: Alphabets, words, and languages. Operations on strings

and languages. Regular Languages and Automata: Regular languages and regular

expressions. Deterministic finite automata. DFAs and languages. Nondeterministic finite

automata.

Unit-2 10 Hours

Equivalence of NFA and DFA. ε-Transitions. Minimization and equivalence of finite automata. Finite automata with outputs. Moore and Mealy machines. Finite automata and regular expressions. Properties of regular languages. Pumping lemma.

Unit-3 10 Hours

Context-free Languages: Grammars. Regular grammars. Regular grammars and regular

languages. Context-free grammars. Derivation or parse tree and ambiguity. Simplifying

context-free grammars. The Chomsky normal form. Properties of context-free languages.

Pumping lemma for context-free languages. The CYK algorithm

Unit-4 10 Hours

Turing Machines: Basic definitions. Turing machines as language acceptors. Modifications to Turing machines. Universal Turing machines. Turing Machines and Languages: Languages accepted by Turing machines. Regular,

context-free, recursive, and recursively enumerable languages. Unrestricted grammars and

recursively enumerable languages. Context-sensitive languages and the Chomsky

hierarchy.

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100

Name of The Course Cryptography

Course Code MSCM6024

Prerequisite

Corequisite

Antirequisite

L T P C

4 0 0 4

Course Objectives: 1. To develop a mathematical foundation for the study of cryptography.

1. To Understand Number Theory and Algebra for design of cryptographic algorithms

2. To understand the role of cryptography in communication over an insecure channel.

3. Analyse and compare symmetric-key encryption public-key encryption schemes

based on different security models

Course Outcomes

CO1 Describe modern concepts related to cryptography and cryptanalysis.

CO2 Describe and implement the specifics of some of the prominent techniques for

public-key cryptosystems and digital signature schemes (e.g., Rabin, RSA,

ElGamal, DSA, Schnorr)

CO3 Explain the notions of public-key encryption and digital signatures, and sketch their

formal security definitions

CO4 Explain the notions of public-key encryption and digital signatures, and sketch their

formal security definitions

Text Book(s)

1. Douglas R. Stinson: Cryptography: Theory and Practice, Third Edition, CRC

Press.2006

2. Alfred Menezes, Paul C. van Oorschot: et. al., : Handbook of Applied

Cryptography, 5th ed. CRC Press, 2001

Reference Book(s)

3.Bruce Schnier: Applied Cryptography (2nd Edition):, John Wiley and Sons.

Unit-1 10 Hours

Introduction to Cryptography and Cryptanalysis. Features of Cryptography. Classical

methods and modern methods. Cryptographic Protocols and standards. Fiestel Ciphers,

Block Ciphers and Stream Cihpers. Symmetric key algorithms, Asymmetric Key Algorithms. Key Exchange algorithms and protocols. Digital Signatures. CA.

Review of number theory and finite field arithmetics. Random-Sequence and Random number generators.

Unit-2 10 Hours

Stream Ciphers: RC4, RC5. Symmetric Key Algorithms: DES, AES, Asymmetric

Key Algorithms: RSA, El-Gamal,

Unit-3 10 Hours

Key Exchange Algorithms, Public-key, Private Key, Signature Schemes, Introduction,

The ElGamal Signature Scheme, The Digital Signature Standard , One-time Signatures , Undeniable Signatures , Fail-stop Signatures

Unit-4 10 Hours

Hashing Functions: Signatures and Hash Functions ,Collision-free Hash Functions, The

Birthday Attack , The MD5, SHA1. Hash Function, Introduction to Stenography,

Timestamping , Zero-knowledge Proofs , Interactive Proof Systems , Perfect Zero-

knowledge Proofs , Bit Commitments , Computational Zero-knowledge Proofs , Zero-

knowledge Arguments , Elliptic Curve Cryptosystems. A Discrete Log Hash Function,

Extending Hash Functions, Hash Functions From Cryptosystems ,

Continuous Assessment Pattern

Internal Assessment

(IA)

Mid Term Test

(MTE)

End Term Test

(ETE)

Total Marks

20 30 50 100