syllabus of m. sc mathematics
TRANSCRIPT
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