Faculty of Mathematical and Statistical Sciences INSH

Faculty

of

Mathematical

and

Statistical Sciences

INSH

PROGRAMME

M.Sc. (Mathematics)

Programme Learning Objective Mathematics Postgraduate curriculum should such that it offers our student

To train qualified, adaptable, motivated, and responsible mathematicians

who will contribute to the scientific and technological development.

To develop the mathematical skills needed in modeling and solving

practical problems.

To prepare students to face the present challenges in different field of

Mathematics.

To develop an outlook and high level study skills that will be hugely

valuable whatever career path follow after post graduation.

To impart an intensive and in-depth learning to the students in the field

of Mathematics.

To develop students’ skills in dissertation writing.

The programme also provides students with the knowledge and

opportunity to progress towards a Doctorate programme.

Programme Learning Outcomes After doing Mathematics Postgraduate Course, our student will be able to

PLO1: Identify fundamental concepts of higher mathematics as applied to applied mathematics.

PLO2: Apply analytic, numerical and computational skill to analyze and solve higher Mathematical

problems.

PLO3: Apply knowledge of higher Mathematics with integrative approach in diverse fields.

PLO4: Acquire appropriate skill in higher Mathematics to handle research oriented scientific problems.

PLO5: Gain specific knowledge and understanding will be determined by your particular choice of

modules, according to your particular needs and interests.

PLO6: Communicate scientific information effectively in written and oral formats.

PLO7: The basic and advanced knowledge attained by student will also make him/her enable to inspire

them to pursue higher studies.

PLO8: Apply their responsibilities in social and environmental context.

M. Sc.: Mathematics I Year: I Semester

S.

No.

Subject

Code Subject L T P CIE ESE Total C

THEORY

1. MMA1001 Algebra-I 3 2 - 40 60 100 5

2. MMA1002 Real Analysis 3 2 - 40 60 100 5

3. MMA1003 Differential Geometry of

Manifolds 3 2 - 40 60 100 5

4. MMA1004 Differential Equations 3 2 - 40 60 100 5

5. MMA1005 Mechanics 3 2 - 40 60 100 5

PRACTICAL/TRAINING/PROJECT

6. MMA1501 MATHEMATICA Lab - - 2 80 20 100 1

7. MMA1502 Seminar - - 2 80 20 100 1

Total 15 10 4 360 340 700 27

L - Lecture

T -Tutorial

P -Practical

CIE -Continuous Internal Evaluation

ESE -End Semester Exam

C -Credit

STUDY & EVALUATION SCHEME

(Effective from the session 2017-2018)

STUDY & EVALUATION SCHEME (Session 2019-2020)

M. Sc.: Mathematics I Year: II Semester

S.

No.

Subject

Code Subject L T P CIE ESE Total

[[[[[[

C

THEORY

1. MMA2001 Algebra-II 3 2 - 40 60 100 5

2. MMA2002 Complex Analysis 3 2 - 40 60 100 5

3. MMA2003 Topology 3 2 - 40 60 100 5

4. MMA2004 Calculus of Variations and

Integral Equations 3 2 - 40 60 100 5

5. MMA2005 Numerical Methods 3 2 - 40 60 100 5


6. MMA2501 MATLAB - - 2 80 20 100 1

7. MMA2502 Seminar - - 2 80 20 100 1

Total 15 10 4 360 340 700 27




M. Sc.: Mathematics II Year: III Semester

S.

No.

Subject


C

THEORY

1. MMA3001 Measure and Integration Theory 3 2 - 40 60 100 5

2. MMA3002 Probability and Statistics 3 2 - 40 60 100 5

3. MMA3003 Fluid Dynamics 3 2 - 40 60 100 5

4. --- Elective-I 3 1 - 40 60 100 4

5. --- Elective-II 3 1 - 40 60 100 4


6. MMA3501 LATEX - - 2 80 20 100 1

7. MMA3502 Seminar - - 2 80 20 100 1

Total 15 8 4 360 340 700 25




University Mandatory Non-Credit Course

1. XHUX601 Human Values and Ethics 2 - - 100 - 100 0

M. Sc.: Mathematics II Year: IV Semester

S.

No.

Subject


C

THEORY

1. MMA4001 Functional Analysis 3 2 - 40 60 100 5

2. MMA4002 Operations Research 3 1 - 40 60 100 4

3. --- Elective-III 3 1 - 40 60 100 4


4. MMA4501 Project - - 2 80 20 100 4

5. MMA4502 Comprehensive Viva-Voce - - - 100 - 100 2

Total 9 4 2 120 180 500 19


M. Sc.: Mathematics

List of Electives

S. No. Subject Code Subject

Elective-I (Semester-III)

1. MMA3101 Discrete Mathematical Structure

2. MMA3102 Mathematical Biology

3. MMA3103 Stochastic Processes

Elective-II (Semester-III)

1. MMA3201 Number Theory and Cryptography

2. MMA3202 Dynamical Systems

3. MMA3203 Fuzzy Sets and Applications

Elective-III (Semester-IV)

1. MMA4101 Mathematical Modeling

2. MMA4102 Mathematics of Finance and Insurance

3. MMA4103 Optimization Techniques

4. MMA4104 Fractional Calculus and Nonlinear Dynamics


Algebra-I (MMA1001)

L T P C

3 2 0 5

(40 Hours)

Course Objective (CO):

CO1: To introduce some fundamental concepts of Algebra.

CO2: To provide some understanding of the concepts like group and ring.

CO3: To introduce some understanding of the field and module.

CO4: To explore the application based problems of the related topics and use this knowledge in more

advanced and complex situations of pure and applied mathematics.

UNIT – I: GROUP THEORY (10 Hours)

Normal subgroups, Quotient groups, Homomorphism and Isomorphism theorems of groups, Cayley’s

theorem, class equations, Direct product of groups (External and Internal), Cauchy’s Theorem for finite

abelian groups, p-groups , Sylow subgroups, Sylow’s Theorem, applications of Sylow subgroups.

UNIT-II: SYLOW’S THEOREM AND SOLVABILITY (07 Hours)

Normal and subnormal series, composition series, Jordan holder theorem, Solvable groups, simplicity of

)5( nAn , Nilpotent groups.

UNIT-III: ADVANCED RINGS THEORY (08 Hours)

Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal

domain, Euclidean domain, Polynomial rings and irreducibility criteria, Eisenstein’s criterion of

irreducibility.

UNIT-IV: FINITE FIELDS (08 Hours)

Extension fields, Finite, algebraic and transcendental extensions, Simple and algebraic field extensions,

Splitting fields and normal extensions, algebraically closed fields.

UNIT-V: MODULE THEORY (07 Hours)

Modules, Submodules, Quotient modules and cyclic modules, Homomorphism and Isomorphism

theorems, Simple modules, free modules, Rank of module.

TEXT BOOKS:

T1. Ramji Lal, Algebra, Vols. I & II, Shail Publications, Allahabad, 2002.

T2. P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra.

T3. Vivek Sahai and Vikas Bist, Algebra, Narosa Publishing House 1999.

T4. Khanna & Bhamri, “A course in Abstract Algebra”, Vikas Publishing House.





REFERENCES BOOKS:

R1. I.N.Herstein.. Topics in Algebra, John Wiley & Sons.

R2. M. Artin.. Algebra, Prentice Hall of India.

R3. D. S. Malik, J. N. Mordeson, and M. K. Sen, Fundamentals of Abstract Algebra, McGraw-Hill

International Edition, 1997.

R4. N. Jacobson, Basic Algebra, Vol. I, Hindustan Publishing Co., New Delhi, 1984.

R5. Joseph & Gallian, “Contemparay Abstract Algebra”, Narosa Publishing House.

Course Learning Outcome (CLO):

After completing this course, our Student will be able to

CLO1: recall, apply and analyze the basic properties of group theory.

CLO2: Define, illustrate and interpret about Sylow’s theorem and Solvability.

CLO3: understand, analyze and apply advanced Ring theory.

CLO4: remember, comprehend, apply and analyze finite fields and module theory.

Matching of PLOs and CLOs:

PLO1 PLO2 PLO3 PLO4 PLO5 PLO6 PLO7 PLO8

CLO1 H H H M H M L M

CLO2 H H L M H M L M

CLO3 H M M M H M L L

CLO4 H L H M H M L M

Real Analysis (MMA1002)

L T P C

3 2 0 5

(40 Hours)


CO1: To introduce some fundamental ideas about sequence, series and sequence, series of the

functions.

CO2: To provide some understanding of the basic concept and theories of metric spaces.

CO3: To aim that understanding the concept of Riemann-Stieltjes integral and their properties.

CO4: To furnish students with mathematical tools that have been found essential in dealing with

variety of problems as they arise in the physical world. UNIT-I: SEQUENCES AND SERIES (08 Hours) Sequences 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, Bolzano Weierstrass

theorem, Heine-Borel theorem, Continuity, uniform continuity, differentiability, mean value theorem.

UNIT-II: SEQUENCES AND SERIES OF FUNCTIONS (08 Hours) Pointwise convergence, uniform convergence on interval, Cauchy’s criterian for uniform convergence,

test for uniform convergence, test for uniform convergence of series, Weierstrass M-test, Abel’s and

Dirichlet’s test, properties of uniform convergence.

UNIT- III: METRIC SPACE-I (08 Hours) Definition and examples of metric spaces, Neighborhoods, closure and interior, boundary points, Limit

points, Open and closed sets, Subspaces, Convergent and Cauchy sequences.

UNIT- IV: METRIC SPACE-II (08 Hours) Continuous functions, Uniform continuity, Complete metric space and its properties, Cantor’s

intersection Theorem, Compact metric space and its properties, Finite intersection property, Balzano-

Weierstrass property, Connectedness.

UNIT-V: RIEMANN-STIELTJES INTEGRAL (08 Hours) Riemann sums and Riemann integral, Improper Integrals. Monotonic functions, types of discontinuity,

functions of bounded variation, Definition and existence of Riemann-Stieltjes integral, Conditions for R-

S integrability, Properties of the R-S integral, Integration of vector valued functions.

TEXT BOOKS: T1. T.M. Apostol, Mathematical Analysis, Narosa Publishing House , New Delhi, 1985.

T2. S.C. Malik., Mathematical Analysis, Wiley Eastern Ltd., New Delhi..

T3. Walter Rudin, Principles of Mathematical Analysis- McGraw Hill International Editions,

Mathematics series, Third Edition, 1964.

REFERENCE BOOKS:

R1. Gabriel Klambauer, Mathematical Analysis, Marcel Dekkar, Inc. New York, 1975.

R2. R.R. Goldberg, Real Analysis, Oxford & I.B.H. Publishing Co., New Delhi, 1970.

R3. D. Soma Sundaram and B. Choudhary, A first Course in Mathematical Analysis , Narosa

Publishing House, New Delhi, 1997.

R4. P.K. Jain and S.K. Kaushik, An Introduction to Real Analysis, S.Chand & Co. New Delhi,

2000.

R5. Murray R. Spiegel, Theory and Problems of Advanced Calculus, Schaum’s outline series,

Schaum Publishing Co. New York.

Royden – Real Analysis, PHI, 1989.



CLO1: Recall, comprehend, apply and analyze the basics of sequence and series, functions of single

& several variables.

CLO2: Recall, understand, use and analyze the basic concept of metric spaces and their properties.

CLO3: Understand, apply, analyze and communicate the concept of Riemann-Stieltjes integral and

their properties

CLO4: Remember, comprehend, apply and analyze some fundamental mathematical tools of Real

Analysis to dealing with a variety of problems in various branches.



CLO1 H H H H H H H M

CLO2 H H H L H H M L

CLO3 H

H

H L M H M L

CLO4 H H H H H H M L

Differential Geometry of Manifolds (MMA1003)

L T P C

3 2 0 5

(40 Hours)


CO1: To introduce basic ideas about differential manifolds and its importance.

CO2: To provide some understanding different type of differential manifolds.

CO3: To aim at understanding and discussing differential manifolds, topological groups and bundles.

CO4: To explore the connection between differential manifolds and fields of applications

of the subject.

UNIT- I: DIFFERENTIAL MANIFOLDS-I (08 Hours)

Definition and examples of differentiable manifolds, Tangent Spaces, Vector fields, Jacobian map, One

parameter group of transformations, Lie derivatives, Immersions and Imbeddings, Distributions,

Hypersurface of .

UNT- II: DIFFERENTIAL MANIFOLDS-II (08 Hours)

Standard connection on , covariant derivative, Sphere map, Weingarten map, Gauss equation, Gauss

curvature equation and Coddazi-Mainardi equations.

UNIT-III: DIFFERENTIAL MANIFOLDS-III (08 Hours)

Invariant view point, Cortan view point, coordinates view point, Difference Tensor of twoconnections,

Torsion and curvature tensors.

UNIT-IV: TOPOLOGICAL GROUPS (08 Hours)

Topological groups, Lie groups and Lie algebras, examples of Lie groups Products of Two Lie groups,

one parameter subgroups and exponential maps, Homomorphism and Isomorphism, Lie transformations

groups, General linear groups.

UNIT- V: BUNDLES (08 Hours)

Principal fibre bundle, linear frame bundle, associated fibre bundle, vector bundle, Induced bundle,

bundle homomorphism

TEXT BOOKS:

T1. B. B. Sinha, An Introduction of Modern Differential Geometry, Kalyani Publishers, New Delhi,

1982.

T2. R.S. Mishra, Structures on a differentiable manifold and their applications, ChandramaPrakashan,

Allahabad, 1984.

REFERENCE BOOKS:

R1. R.S. Mishra, A Course in Tensors with applications to Riemannian geometry, Pothishala, Pvt.

Ltd, 1965.

R2. NJ Hicks: Notes on Differential Geometry, D. Van Nostrand, 1965.

R3. K. Yano and M. Kon, Structures on manifold, Worlds Scientific Publishing, Co., Pvt, Ltd, 1984.



CLO1 comprehend, apply and analyze the tangent Spaces, vector fields, Lie derivatives,

sphere map, weingarten map applied in differential manifolds.

CLO2 recall, understand, use and analyze prescribed differential manifolds.

CLO3 understand, analyze and apply topological groups.

CLO4 comprehend, apply and analyze bundles.



CLO1 H H H H H M L L

CLO2 H H H H H M M L

CLO3 H M M H M H M L

CLO4 H M M H M M L L

DIFFERENTIAL EQUATIONS (MMA1004)

L T P C

3 2 0 5

(40 Hours)


CO1: To introduce some ideas and basic concept of existence and uniqueness of differential

equations.

CO2: To provide understanding of the basic concepts of local/global existence and uniqueness of

system of differential equations and finding their solutions.

CO3: To aim at understanding and finding the solution of ordinary and partial differential

equations through various methods.

CO4: To explore the connection between basics as well the advance tools of the subject to

demonstrate the link between theory and its real world applications

UNIT- I: EXISTENCE THEOREMS FOR FIRST ORDER EQUATIONS (08Hours)

Lipschitz conditions, Picard’s Theorem for local existence and uniqueness of solution of an initial value

problem of first order which is solved for the derivative, Initial and Boundary Value Problems,

Envelopes of one parameter family of curves, singular solutions.

UNIT- II: SYSTEMS OF DIFFERENTIAL EQUATIONS (08 Hours)

Local existence and uniqueness theorems for systems of first order equations, Gromwell’s inequality,

Global existence and uniqueness theorems of solutions over whole of the give interval and over whole of

R, Existence theory for equations of higher order, Conditions for transformability of a system of first

order equations into an equation of higher order.

UNIT- III: LINEAR SYSTEMS OF FIRST ORDER EQUATIONS (08 Hours)

Wronskians, General solutions for homogeneous and non-homogeneous linear systems, Abel’s formula,

Method of variation of parameters for particular solutions, Linear systems with constant coefficients,

Matrix methods, Different cases involving diagonalizable and non-diagonalizable coefficient matrices,

Real solutions of systems with complex eigenvalues.

UNIT- IV: PARTIAL DIFFERENTIAL EQUATIONS-I (08 Hours)

Formation of P.D.E.’s First order P.D.E.’s Classification of first order P.D.E.’s Complete, general and

singular integrals, Lagrange’s or quasi – linear equations, Integral surfaces through a given curve,

Orthogonal surfaces to a given system of surfaces, Characteristic curves.

UNIT- V: PARTIAL DIFFERENTIAL EQUATIONS-II (08 Hours) Method of separation of variables: Laplace, Diffusion and Wave equations in Cartesian, cylindrical and

spherical polar coordinates, Boundary value problems for transverse vibrations in a string of finite length

and heat diffusion in a finite rod.

TEXT BOOKS:

T1. I.N. Sneddon, Elements of Partial Differential Equations, McGraw-Hill, 1957.

T2. R.P. Kanwal, Linear Integral Equations, Birkhauser, Inc., Boston, MA, 1997.

T3. B. Rai, D.P. Choudhury and H.I. Freedman, A Course in Oridinary Differential Equations,

Narosa Publishing House, New Delhi, 2002.

REFERENCE BOOKS:

R1. T. Amaranath,An Elementary Course in Partial Differential Equations, Narosa Publishing

House, New Delhi, 2005.

Course Learning Outcome (CLO): After completing this course, our Student will be able to

CLO1 Recall, comprehend, apply and analyze the existence and uniqueness of differential equations.

CLO2 Recall, understand, and analyze of local/global existence and uniqueness of system of

differential equations and finding their solutions.

CLO3 Understand, apply and analyze solution of ordinary and partial differential equations through

various methods.

CLO4 Remember, comprehend and analyze the differential equations and their application in daily

life.



CLO1 H H H M H M H M


CLO3 H M M M H M H M

CLO4 H M H M H M H M

Mechanics (MMA1005)

L T P C

3 2 0 5

(40 Hours)

Course Objectives (CO):

CO1: To introduce some fundamental concepts of the motion of rotating frame, Eulerian, Lagrangian

and Hamiltonian approach to rigid body motion and also canonical transformations.

CO2: To develop the ability to provide a treatment of basic knowledge in mechanics used in deriving a

range of important results and problems related to rigid bodies.

CO3: To understand and appreciate the beauty of the classical mechanics.

CO4: To provide and develop the classical mechanics approach to solve a mechanical problem.

UNIT –I: MOTION OF ROTATING FRAMES (09 Hours)

Rotation of vector in two dimensions, Radial and Transverse directions, tangential and Normal

directions, motion of a particle in rotating plane and space, rotations of vector in three dimensions,

motion of a rigid body in rotating frame, effect of earth rotation.

UNIT – II: EULERIAN APPROACH TO RIGID BODY MOTION (07 Hours)

Linear velocity, Kinetic energy, angular momentum, Euler’s dynamical equations of motion, Euler’s

geometrical equations of motion, principle axes of rigid body.

UNIT – III: LAGRANGIAN APPROACH TO RIGID BODY MOTION (08 Hours)

Linear and rotational motions, generalized coordinates, geometrical equations, generalized momentum

variables, generalized force components, Lagrange equations of motion under finite forces, Lagrange

function, conservation of energy, cyclic coordinates, Lagrange equations for constrained motion,

Lagrange equations under impulses.

UNIT – IV: HAMILTONIAN APPROACH TO RIGID BODY MOTION (08 Hours)

Equations of motion, energy equation, some known dynamic problems, Hamilton principle of least

action, Lagrange and Hamilton equations of motion, Hamilton-Jacobi equation of motion and Hamilton-

Jacobi theorem.

UNIT – V: CANONICAL TRANSFORMATIONS AND POISSON BRACKET (08 Hours)

Phase space, Canonical transformations, conditions of canonicality and cyclic relations, Canonical

transformations form a group, Liouville’s theorem, Poisson bracket, Poisson first theorem, Poisson-

Jacobi identity, Poisson second theorem, invariance of Poisson bracket, Lagrange bracket.

TEXT BOOKS:

T1. Naveen Kumar, Generalized Motion of Rigid Body, Narosa Publishing House, New Delhi.

T2. K. C. Gupta, Classical Mechanics of Particles and Rigid Bodies, Willey Eastern Ltd, 1988.

T3. J.L. Synge & B.A. Griffith - Principles of Mechanics, Tata McGraw-Hill, 1959.

REFERENCE BOOKS:

R1. Narayan Chandra Rana and Pramod Sharad Chandra Joag, Classical Mechanics, TMH, 1991.

R2. H. Goldstein, Classical Mechanics, Narosa Publishing House, New Delhi.


After successful completion of this course, the student will be able to

CLO1: Recall, apply and analyze the concept of system of particle in finding moment inertia, directions of

principle axes and consequently Euler’s dynamical equations for studying rigid body motions.

CLO2: Define, illustrate and interpret about the equation of motion for mechanical systems using the Lagrangian

and Hamiltonian formulations of classical mechanics.

CLO3: Solve, apply, understand and utilize application based problems of the related topics.

CLO4: Obtain canonical equations using different combinations of generating functions and subsequently

developing Hamilton Jacobi method to solve equations of motion.

Matching of PLOs and CLOs: PLO1 PLO2 PLO3 PLO4 PLO5 PLO6 PLO7 PLO8

CLO1 H H H H H M L M


CLO3 H M M H H H L M

CLO4 H L M H H M L M

Mathematica Lab (MMA1501)

L T P C 0 0 2 1 Course Objective (CO):

CLO1: To introduce fundamental downloading of MATHEMATICA software.

CLO2: To introduce problem solving and programming techniques using MATHEMATICA

mathematical computing techniques.

CLO3: To provide understanding of writing various analytical, mathematical and numerical solution

program.

CLO4: To aim at understanding of various tools for simulation and its application.

1. Introduction to basic concepts of Mathematicia.

2. Write a Mathematica program to find out the inverse of the matrix.

3. Write a Mathematica program to find out the eigen value and eigen vectors of the matrix.

4. Write a Mathematica program to find out the stability analysis of mathematical model.

5. Write a Mathematica program to find out the root of the Algebraic and Transcendental

equations using Bisection, Newton-Raphson Methods.

6. Write a Mathematica program to find the derivative and the integral of a function.

7. Write a Mathematica program to find out the solution of ordinary differential equation.

8. Write a Mathematica program to find out the solution of partial differential equation.

9. Write a Mathematica program to plotting two dimensional graphics.

10. Write a Mathematica program to plotting three dimensional graphics.

TEXT BOOKS:

T1. Stephen Wolfram, The Mathematica Book , 5th

Edition, Wolfram Media, 2003.

REFERENCE BOOKS:

R1. Peter Mulquiney, Philip W. Kuchel, Modelling Metabolism with Mathematica, 1st Edition, CRC

Press.



CLO1: Write correct, efficient, and well-documented programs using MATHEMATICA.

CLO2: Gain basic knowledge about the MATHEMATICA computation techniques and will be

prepared for computing higher complex problems in the field of scientific and engineering

research.

CLO3: Acquire basic knowledge about efficient tools such as

https://www.wolfram.com/mathematica/resources/.

CLO4: Explore the connection between basics as well the advance tools of the MATHEMATICA to

demonstrate the link between theory and its real world applications.



CLO1 H H M H H H H M

CLO2 H H M M H H M M

CLO3 M M M M H M H H

CLO4 H H H HM M M H M

https://www.wolfram.com/mathematica/resources/

SEMINAR (MmA1502)

L T P C 0 0 2 1


CO1: To facilitate transfer of knowledge acquired by a student to a field of own choice for applications

to solve a problem.

CO2: To provide some understanding of the concepts and teach them the process of generating options

and making choices.

CO3: To introduce and inculcate a logical approach to decision making and problem solving.

CO4: To enhance presentation skills and explore the connection between history, science and

philosophy.

Methodology: The student will have to collect and study relevant material under mentorship of a faculty

member working in similar area; identify a suitable problem and propose methodology towards its

solution.

The topics selection covering the latest and relevant topics related to the emerging areas. Ideally, some

recent reputed journal papers abstraction and presentation shall be encouraged for presentation. The

evaluation shall be continuous and through components evaluation viz. content, coverage, depth,

presentation, response to the queries, and seminar report.

One topic out of three may be assigned every week to the students and each of them will be asked to

prepare 15 minutes presentation on a topic of their choice in that topic. Students will have to present it.

WEEK.

No.

NAME OF THE ACTIVITY

1. Consultation about theoretical and application aspects and to investigate topics of

Mathematics.

2. Discussion to the students about topics of current trends and need.

3. Selection and submission of topics by students.

4. Topic finalization and guidelines for the preparation of seminar report and presentation.

5. Exhibit schedule for Presentation and discussion about contents of topics selected by

students.

6-12 Submission of seminar reports and presentations (30 minutes) by students.

13-16. Submission of seminar reports and presentations by repeated presentations.



CLO1 Explain, comprehend, apply and analyze the knowledge of the topic of own interest.

CLO2 Examine, evaluate, develop and make use of the knowledge in the process of

generating options and making choices.

CLO3 Extend, identify and elaborate the knowledge in various diverse fields of mathematics.

CLO4 Conclude, deduct, interpret and build the mathematics in physical problems by

exploring the connection between history, science and philosophy.



CLO1 H H M L H M L M

CLO2 H H L L H M L L

CLO3 M M M L H M L H

CLO4 H L M L M M L M

Algebra-II (MMA2001)

L T P C

3 2 0 5

(40 Hours)


CO1: To introduce some fundamental concepts about Vector spaces and their elementary properties as

well as Linear transformations and their matrix representations.

CO2: To develop the ability to write the mathematical proofs.

CO3: To understand and appreciate the beauty of the abstract nature of mathematics.

CO4: To provide and develop a solid foundation of theoretical mathematics.

UNIT-I: VECTOR SPACES (08 Hours)

Vector Spaces over fields, Sub Spaces, Basis and Dimensions, Linear Transformations, Rank Nullity

theorem, representation of transformations by matrices, Eigen values and Eigen Vectors, Characterstics

Polynomials, minimal polynomials, Cayley Hamilton’s theorem, Diagonalization,.

UNIT-II: INNER PRODUCT SPACES (08 Hours)

Inner Product Space, Cauchy Schwarz inequality, Gram-Schmidt Orthonormalization. Orthogonal

Projections, Linear functional, Dual Space.

UNIT-III:LINEAR OPERATORS: (08 Hours)

Positive definite matrix, Hermitian, Unitary and normal linear operators, Spectral theorem for normal

operators, Quadratic forms, Bilinear forms, symmetric and skew symmetric bilinear forms.

UNIT-IV: CANONICAL FORMS: (08 Hours)

Invariant subspaces, Reduction to triangular forms, Nilpotent transformations, Index of nilpotency,

Invariants of a nilpotent transformation. The primary decomposition theorem, Jordan blocks and Jordan

canonical forms, Rational canonical form.

UNIT-V: MODULE THEORY (08 Hours)

Cyclic modules, simple modules, semi simple modules, Schuler’s lemma, free modules, Noetherian and

artinian modules.

TEXT BOOKS:

T1. K. Hoffman and R. Kunze.. Linear Algebra , Prentice Hall of India, 1991

T2. S. Mclane and G. Birkhoff: Algebra, AMS Chelsea Publishing, 1999.

T3. Bisht and Sahai: Linear Algebra, Narosa Publishing House, second edition, 2013

REFERENCE BOOKS:

R1. Surjit Singh. Linear Algebra, Vikas Publishing.

R2. David W. Lewis Matrix Theory, Allied Publisher

R3. Malik, Mordeson & Sen – Fundamentals of Abstract Algebra (Tata MaGraw-Hill)

R4. Sen, Ghosh & Mukhopadhyay – Topics in Abstract Algebra (University Press).

R5. S. Kumareson – Linear Algebra.

R6. Rao & Bhimsankaran – Linear algebra.


After successful completion of this course, the student will be able to

CLO1: Recall, apply and analyze the basic properties of the vector spaces and linear transformations.

CLO2: Define, illustrate and interpret about inner product spaces, linear operators, Quadratic forms,

Bilinear forms, invariant subspaces and Module theory.


CLO4: Conclude, deduct and make use of the knowledge in more advance and complex situations of

higher algebra.



CLO1 H H H L H M L M


CLO3 H M M L H H L H

CLO4 H L H L H M L M

Complex Analysis (MMA2002)

L T P C

3 2 0 5

(40 Hours)


CO1: To introduce some fundamental ideas about complex analysis

CO2: To provide some understanding of the basic concept and theories of complex analysis.

CO3: To aim that understanding the concept of conformal mapping.

CO4: To explore the connection between real and complex integration.

UNIT-I: COMPLEX INTEGRATION (08 Hours)

Properties of line integrals, Goursat Theorem, Cauchy theorem and its consequence of simply

connectivity, index of a closed curve, Cauchy’s integral theorem and formula for derivatives of analytic

functions, Morera’s theorem, Liouville’s theorem.

UNIT-II: CALCULUS OF RESIDUE (08 Hours)

Power series representation of a function, Taylor’s Theorem, Zeros of Analytic functions, Laurent’s

series, singularities, classification of singularities, Residue theorem, evaluation of real integrals of the

type

and

.

UNIT-III: MODULUS THEOREMS (08Hours)

Maximum modulus theorem, Minimum modulus theorem, Hadamard’s three circle theorem, Schwarz’s

Lemma, Rouche’s theorem, Meromorphic functions, Argument principle, Rouche’s theorem.

UNIT-IV: CONFORMAL MAPPINGS and MÖBIUS TRANSFORMATIONS (08Hours)

Principle of Conformal Mapping, Basic Properties of M⍤bius Maps, Fixed Point and M⍤bius Maps, The

Cross ratio and its Invariance Property, Determination of Mobius transformations mapping real line onto

itself, upper half plane onto open disc and an open disc onto an open disc.

UNIT-V: MEROMORPHIC AND ENTIRE FUNCTIONS (08 Hours)

Infinite Sums and Meromorphic Functions, Infinite Product of Complex Numbers, Infinite Product of

Analytic Functions, Factorization of Entire Function, The Gamma Function, The Zeta Function.

TEXT BOOKS:

T1. J. B. Conway, Complex Analysis, Narosa Publishing House, New Delhi.

T2. S. Ponnusamy, Foundations of Complex analysis (Second Edition), Narosa Publishing House,

1997.

T3. Walter Rudin, Real and Complex Analysis, McGraw Hill Book Com., 1966.

T4. H. S. Kasana, Complex Variables Theory and Application, PHI, 2019

REFERENCE BOOKS:

R1. Ruel V. Churchill, Complex variables and Applications.

R2. L. V. Ahlfors, Complex Analysis (Third edition), McGraw Hill Book Co., 1979.



CLO1: Recall, comprehend, apply and analyze the fundamental properties of Complex Analysis.

CLO2: Recall, understand, use and analyze and communicate is to impart student’s basic

concept of theories of complex analysis.

CLO3: Understand, apply, analyze and communicate to impart student’s , conformal mapping

and mobius transformation.

CLO4: Remember, comprehend, apply and analyze the connection between real and complex

integration.

. Matching of PLOs and CLOs: PLO1 PLO2 PLO3 PLO4 PLO5 PLO6 PLO7 PLO8

CLO1 H H H M H M H L


CLO3 H M M L H H M M

CLO4 H M H L H M L M

Topology (MMA2003)

L T P C

3 2 0 5

(40 Hours)


CO1: To introduce basic ideas about Topology and its importance.

CO2: To provide some understanding different type of topological spaces.

CO3: To aim at understanding and discussing separation axioms, compact set and connected spaces.

CO4: To explore the connection between topological spaces and fields of applications

of the subject.

UNIT-I: SET THEORY (08 Hours)

Countable and uncountable sets, Infinite sets and the Axiom of Choice, Cardinal numbers, Schroeder-

Bernstein theorem, Cantor’s theorem and the continuum hypothesis, Zorn’s lemma, Well-ordering

theorem.

UNIT-II: TOPOLOGICAL SPACES (08 Hours)

Definition and examples of topological spaces, Closed sets, Closure, Dense Subsets, Neighborhoods,

Interior, exterior and boundary, Accumulation points and derived sets, Bases and sub-bases, Sub-spaces

and relative topology.

UNIT- III: SEPARATION AXIOMS (08 Hours)

Continuous functions and homomorphism, First and Second Countable spaces, Lindelof’s theorems,

Separable spaces, Second Countability and Separability, Separation axioms, 0T , 1T , 2T , 3T , 4T

their Characterizations and basic properties, Urysohn’s lemma.

UNIT-IV: COMPACT SET (08 Hours)

Compactness, Continuous functions and compact sets, Basic properties of compactness, Compactness

and finite intersection property, Sequentially and countably compact sets, Compactness and product

spaces.

UNIT-V: CONNECTED SPACES (08 Hours)

Connected spaces, Connectedness on the real line, Path connected spaces, connected components,

Locally connected spaces, Tychonoff product topology in terms of standard sub-base and its

characterizations, Projection maps.

TEXT BOOKS:

T1. James R. Munkres, Topology, A First Course, Prentice Hall of India Pvt. Ltd., New Delhi, 2000.

T2. K.D. Joshi, Introduction to general Topology, Wiley, Eastern Ltd. 1983.

T3. Walter Rudin, Principles of Mathematical Analysis- McGraw Hill International Editions,


T4. G F Simmons: Introduction to Topology & Modern Analysis (Mc Graw Hill).

REFERENCE BOOKS:

R1. J. Dugundji, Topology, Allyn and Bacon, 1966 (Reprinted in India by Prentice Hall of India Pvt.

Ltd.).

R2. George F. Simmons, Inroduction of Topology and Modern Analysis, McGraw-Hill Book

Company, 1963.

R3. J.L. Kelley, General Topology, Van Nostrand Reinhold Co., New York, 1995.

R4. N. Bourbaki, General Topology, Part I (Transl.), Addison Wesley, Reading, 1966.

W.J. Pervin, Foundations of General Topology, Academic Press Inc. New York, 1964.



CLO1 comprehend, apply and analyze the set theory applied in Topological spaces.

CLO2 recall, understand, use and analyze prescribed Topological spaces .

CLO3 understand, analyze and apply Compact set.

CLO4 comprehend, apply and analyze Connected spaces.


CLO1 H H H H H M L L

CLO2 H H H H H M M L

CLO3 H M M H H H M L

CLO4 H M M H M M L L

CALCULUS OF VARIATIONS & INTEGRAL EQUATIONS

(MMA2004) L T P C

3 2 0 5

(40 Hours)


CO1: To introduce some fundamental ideas and basic concept of fundamental theories of calculus

of variation (CV) and integral equations.

CO2: To provide understanding of the basic concepts of CV and integral equations with various

methods.

CO3: To aim at understanding of CV and integral equations with various search techniques.


Demonstrate the link between theory and its real world applications

UNIT-I: CALCULUS OF VARIATIONS-I (08 Hours)

Functionals, Geodesics, isoperimetric problems, Euler-Lagrange equation, necessary and sufficient

condition for extrema, Functional dependence on functions of several independent variables.

UNIT-II: CALCULUS OF VARIATIONS-II (07 Hours)

Variational problems with fixed boundaries, variational problems with subsidiary conditions,

approximate solution of boundary value problems by Rayleigh- Ritz method.

UNIT-III: INTEGRAL EQUATIONS-I (07 Hours)

Definition and classification, Conversion of initial and boundary value problems to an integral equation,

Eigen values and Eigen functions, Convolution integral, Inner product Integrals, Equations with

separable kernels, reduction to a system of algebraic equations.

UNIT-IV: INTEGRAL EQUATIONS-II (10 Hours)

Fredholm theorem, Fredholm alternative theorem, approximate method, method of successive

approximations- Iterative schemes, Solution of homogeneous and general Fredholm integral equations of

second kind with separable kernels, Solution of Fredholm and Volterra integral equations of second kind

by methods of successive substitutions, successive approximations and by Resolvent kernel, Conditions

of uniform convergence and uniqueness of series solution.

UNIT-V: INTEGRAL EQUATIONS-III (08 Hours)

Classical Fredholm theory- Method of solution of Fredholm equations, Fredholm first theorem with

proof, Fredholm’s second and third theorem (statement only).

TEXT BOOKS:

T1. A.S. Gupta.. Calculus of Variation, Prentice Hall of India Pvt. Ltd.

T2. R.P. Kanwal 1971, Linear Integral Equations Academic Press, New York.

T3. Shair Ahmad & M.R.M. Rao, 1999, Theory of ordinary differential equations, Affiliated East-

West Press Pvt. Ltd., New Delhi.

T4. F. G. Tricomi, Integral equations, Interscience, New York.

REFERENCES BOOKS:

R1. P. Hartman, 1964, Ordinary Differential Equations, John Wiley.

R2. I.M.Gelfand and S. V. Francis, Calculus of Variation, Prentice Hall, New Jersey.

R3. L. G. Chambers, Integral Equations, International Text Book Company Ltd., London.



CLO1 Recall, comprehend, apply and analyze the of fundamental theories of calculus of variation

(CV) and integral equations.

CLO2 Recall, understand, use and analyze CV and integral equations with various solving methods.

CLO3 Understand, apply and analyze CV and integral equations with various solving techniques.

CLO4 Remember, comprehend and analyze CV and integral equations and various techniques.






Numerical Methods (MMA2005)

L T P C

3 2 0 5

(40 Hours)


CO1: To introduce some fundamental concepts of interpolation.

CO2: To provide some understanding of the concepts of numerical differentiation and integration.

CO3: To introduce some understanding of the numerical solution of system of linear equation and

ordinary differential equation.

CO4: To explore this knowledge in solving many problems of higher mathematics involving

mathematics using numerical techniques.

UNIT-I: INTERPOLATION (08 Hours)

Finite differences, Newton’s forward and backward interpolation, Lagrange’s and Newton’s divided

difference formula, Spline interpolation, Hermite interpolation, Errors in polynomial interpolation, .

CURVE FITTING, B-SPLINE AND APPROXIMATION

Linear and non-linear curve fitting by least squares method, Cubic spline fitting, Representation of B-

splines, Approximations of functions, Chebychev Polynomials.

UNIT-II: NUMERICAL SOLUTION OF SYSTEMS OF LINEAR EQUATIONS (08 Hours)

LU decomposition method, Error analysis, Solution of tridiagonal systems, Methods for ill-conditioned

linear system, Eigen value and Eigen vectors, Jacobi, Householder transformations for eigen value

problems, relaxation method.

UNIT-III: NUMERICAL DIFFERENTIATION & INTEGRATION (08 Hours)

Numerical differentiation, Numerical integration, Trapezoidal rule, Simpson’s one third and three eight

rules, Gauss Legendre quadrature, method of undetermined parameters.

UNIT-IV: NUMERICAL SOLUTION OF ODE (08 Hours)

Numerical solution of first order differential equations, Euler’s method, Picard’s method, Runge-Kutta

method.

UNIT-V: NUMERICAL SOLUTION OF PDE (08 Hours)

Classification of partial differential equations of second order, Numerical Solution of elliptic, parabolic

and hyperbolic partial differential equations specially Laplace, Poisson, heat conduction and wave

equations by finite differences and cubic spline methods.

TEXT BOOKS:

T1. V. Rajaraman, Computer Oriented Numerical Methods, PHI.

T2. S.S. Shastri, “Introductory Methods of Numerical Analysis”, PHI.

REFERENCE BOOKS:

R1. F.Acton, Numerical Methods that Work, Harper and Row.

R2. S.D.Conte and C.D.Boor, Elementary Numerical Analysis, McGraw Hill.

R3. C. F. Gerald and P.O. Wheatley, Applied Numerical Analysis, Addision Wesley



CLO1 recall, apply and analyze the basic properties of interpolation, curve fitting, B-spline

and approximation.

CLO2 Define, illustrate and interpret about the numerical solution of system of linear equation.

CLO3 understand, analyze and apply the numerical differentiation and integration.

CLO4 remember, comprehend, apply and analyze the numerical solution of ODE and PDE.



CLO1 H H H L H L H H

CLO2 H L H L H L H M

CLO3 H M M L H L H L

CLO4 H L M L H L H M

MATLAB (MMA2501)

L T P C

0 0 2 1

Course Objective (CO)

CO1: To introduce fundamental downloading of MATLAB software.

CO2: To introduce problem solving and programming techniques using MATLAB mathematical

computing techniques.

CO3: To provide understanding of writing various mathematical and numerical solution program.

CO4: To aim at understanding of various tools for simulation and its industrial application.

1. Introduction to basic concepts of MATLAB

2. Write a MATLAB program to find out the inverse of the matrix

3. Write a MATLAB program to find out the eigen value and eigen vectors of the matrix

4. Write a MATLAB program to find out the Laplace and Inverse Laplace Transform of the

function

5. Write a MATLAB program to find out the root of the Algebraic and Transcendental equations

using Bisection, Newton-Raphson Methods.

6. Write a MATLAB program to find out the analytical and numerical solution of ordinary

differential equation

7. Write a MATLAB program to find out the analytical and numerical solution of system of


8. Write a MATLAB program to find out the analytical and numerical solution of partial


9. Write a MATLAB program to plotting two dimensional graphics

10. Write a MATLAB program to plotting three dimensional graphics



CLO1 Write correct, efficient, and well-documented programs using MATLAB.

CLO2 Gain basic knowledge about the MATLAB computation techniques for higher complex

problems in the field of scientific and industrial research.

CLO3 Acquire basic knowledge about efficient tools such as Simulink and GUI for advanced

simulation.

CLO4 Explore the connection between basics as well the advance tools of the MATLAB to








SEMINAR (MmA2502)

L T P C 0 0 2 1



to solve a problem.


and making choices.



philosophy.



solution.







WEEK.

No.



Mathematics.





students.

















MEASURE and integration theory (MMA3001)

L T P C

3 2 0 5

(40 Hours)

COURSE OBJECTIVES (CO):

CO1: To introduce students higher mathematics related topics by boosting their confidence so

that they don’t feel any fear about the subject.

CO2: Making the students capable so that they can handle the mathematics related problem in

other subject also able to discriminate the related problem in various competitive exams.

CO3: Creation an interest about the measurable theory related research so that the students will

be able to apply such tools in real phenomena, by clearing the basic concept of related

topics.

CO4: To explore the connection between history, Mathematics and current fields of applications

of the subject.

UNIT-I: MEASURE THEORY (08 Hours)

Ring and algebra of sets, algebras generated by a class of sets, Borel algebra and Borel sets,

outer Measure, Lebesgue outer measure, Measurable Sets and their properties, Regularity, non-

measurable sets, Borel and Lebesgue measurability.

UNIT- II: MEASURABLE FUNCTIONS (08 Hours)

Measurable functions, operation on collection of measurable functions, pointwise limit of a sequence of

measurable functions, Functions of bounded variation, Lebesgue differentiation theorem.

UNIT-III: LEBESGUE INTEGRATION (08 Hours)

Riemann and Lebesgue integration, Lebesgue integral of non-negative measurable functions, Integrable

functions and Lebesgue integral of integrable functions, Linearity, Comparison of Lebesgue integrability

of Riemann integrable functions.

UNIT-IV: CONVERGENCE THEOREMS (08 Hours)

Monotone convergence theorem, Fatou‘s lemma, Dominated convergence theorem, Applications of

convergence theorems.

UNIT-V: MEASURE SPACES (08 Hours)

The pL -Spaces, convex functions, Jensen’s inequality, Holder and Minkowski inequalities, completeness

of pL , Convergence in Measure, almost uniform convergence.

TEXT BOOKS:

T1. T.M. Apostol, Mathematical Analysis, Narosa Publishing House , New Delhi, 1985.

T2. Inder K. Rana, An introduction of Measure and Integration (2nd

ed.) , Narosa Publishing House,

New Delhi, 2004.

T3. P. R. Halmos, Measure Theory, Graduate Text in Mathematics, Springer-Verlag 1979.

REFERENCE BOOKS:

R1. Gabriel Klambauer, Mathematical Analysis, Marcel Dekkar, Inc. New York, 1975.

R2. R.R. Goldberg, Real Analysis, Oxford & I.B.H. Publishing Co., New Delhi, 1970.

R3. D. Soma Sundaram and B. Choudhary, A first Course in Mathematical Analysis , Narosa

Publishing House, New Delhi, 1997.

R4. P.K. Jain and S.K. Kaushik, An Introduction to Real Analysis, S.Chand & Co. New Delhi,

2000.

R5. Walter Rudin, Principles of Mathematical Analysis- McGraw Hill International Editions,


R6. Royden – Real Analysis, PHI, 1989.

COURSE LEARNING OUTCOME (CLO):


CLO1: Upon successful completion of the course, students will be able to analyze the

mathematics related problem.

CLO2: Students will fill confident to give the analytical solution of syllabus related topics

questions with effective manner.

CLO3:

Students will be aware about the basics as well the advance tools and formulae of the

subject to promote creativity and demonstrate the link between theory and its real

world applications.

CLO4: subject to promote creativity and demonstrate the link between theory and its real

world applications.






Probability & Statistics (MMA3002)

L T P C

3 2 0 5

(40 Hours)


CO1: To introduce and evaluate the Probabilities of various events using a sample space, also find

conditional probability with the help of total probability and Bayes’ theorem.

CO2: To provide some understanding and evaluation of statistical independence of events and random

variables, also expectation of random variable.

CO3: To aim at understanding and debating what is meant by density and distribution functions,

multiple random variables, joint distribution and correlation and regression.

CO4: To explore several well-known distributions, including Binomial, Geometrical, Negative

Binomial, Beta, Gamma, Normal and testing of significance with t, Chi-square and ANOVA.

UNIT-I: PROBABILITY THEORY (08 Hours)

Axiomatic definition, Properties, Independence of events, Random variables, Conditional probability,

Multiplication law of probability, Multiplication theorem, Use of Binomial theorem, Use of multinomial

expansions, Bayes rule, Geometrical Probability.

UNIT-II: STATISTICS-I (08 Hours)

Measures of central Tendency, Measure of dispersion, Skewness, Moments and Kurtosis, Method of least

squares (Curve Fitting), Distribution function, Probability mass and density functions, Expectation,

Moment generating function, Chebyshev’s inequality, Covariance, Correlation and Regression, Multiple

regression.

UNIT-III: DISTRIBUTION FUNCTIONS (08 Hours)

Special Probability distributions: Bernoulli, Binomial, Geometric, Negative Binomial, Hypergeometric,

Poisson, Gamma, Normal distributions, Normal and Poisson approximations to binomial

UNIT-IV: SAMPLE MOMENTS AND THEIR FUNCTIONS (08 Hours)

Population, Sample, Parameters, Sample size problem, Elementary theory of testing of Hypothesis,

Concept of Statistical Hypothesis, Types of hypothesis, Procedure of testing the hypothesis, Types of

Errors, Level of Significance, Degree of freedom. Chi-Square Test, Properties and Constants of Chi-

Square Distribution, Contingency tables, Student’s t-Distribution, Properties & Applications of t-

Distribution, F-Test, Properties & Applications of F-Test.

UNIT-V: STATISTICS-II (08 Hours)

Analysis of Variance: Completely randomized design and randomized block design, The design of

Experiments, Quality Control, Theory of Estimators.

TEXT BOOKS:

T1. J. S. Milton and J. C. Arnold: Introduction to Probability and Statistics, 4th Edn, Tata McGraw-

Hill, 2003.

T2. Hogg., R.V and Craig, A.T.: Introduction to Mathematical Statistics, MacMillan, 2002.

T3. Goon, A.M. : Fundamental of Statistics, Vol. 1, 7th Edition, 1998

T4. An Introduction to probability and Mathematical Statistics, Rohatgi V, Wiley Eastern Ltd. New

Delhi

T5. M. Ray, Sharma and Chaudhary, Mathematical Statistics, Ram Prasad & sons publications, 2004.

REFERENCE BOOKS:

R1. R. Murray, Probability and Statistics.

R2. FrederichMosteller, Probability and Statistics.

R3. S. C. Gupta and V. K .Kapur, Probability and Statistics.



CLO1: Recall, comprehends, apply and analyze the probability definitions in concerned problems.

CLO2: Recall, understand, use and analyze problems of density and distributions.

CLO3: Understand, analyze and apply correlation and regression for finding various coefficients.

CLO4: Remember, comprehend, apply and analyze statistical methods for solving sampling problems.






Fluid Dynamics (MMA3003)

L T P C

3 2 0 5

(40 Hours)


CO1: To introduce some ideas and basic concept of various types of Fluids.

CO2: To provide understanding of the basic concepts of Fluids in different medium and coordinates

system and finding their solutions.

CO3: To aim at understanding and finding the solution of Fluids in different medium through various

methods.



UNIT-I: KINEMATICS OF FLUIDS IN MOTION (08 Hours)

Real fluids and Ideal fluids, Velocity of a fluid at a point, Stream lines , path lines , steady and unsteady

flows, Velocity potential, The vorticity vector, Local and particle rates of changes, Equations of

continuity , Acceleration of a fluid, Conditions at a rigid boundary.

UNIT-II: EQUATIONS OF MOTION OF A FLUID (08 Hours)

Pressure at a point in a fluid at rest, Pressure at a point in a moving fluid, Conditions at a boundary of

two inviscid immiscible fluids- Euler’s equation of motion, Bernoulli’s equation and discussion of the

case of steady motion under conservative body forces.

UNIT-III: SOURCE AND SINK (08 Hours)

Sources, sinks and doublets, Images in a rigid infinite plane - Axis symmetric flows, stokes stream

function

UNIT-IV: SOME TWO DIMENSIONAL FLOWS (08 Hours)

Meaning of two dimensional flow, Use of Cylindrical polar coordinate, The complex potential for two

dimensional, irrotational incompressible flow, Complex velocity potentials for standard two dimensional

flows, Two dimensional Image systems.

UNIT-V: VISCOUS FLOWS (08 Hours)

Stress components in a real fluid, Relations between Cartesian components of stress, Translational

motion of fluid elements, The rate of strain quadric and principal stresses, Some further properties of the

rate of strain quadric, Stress analysis in fluid motion, Relation between stress and rate of strain, The

coefficient of viscosity and Laminar flow.

TEXT BOOKS:

T1. F. Chorlton, Text Book of Fluid Dynamics ,CBS Publications. Delhi ,1985.

T2. R.W.Fox and A.T.McDonald. Introduction to Fluid Mechanics, Wiley, 1985.

T3. E.Krause, Fluid Mechanics with Problems and Solutions, Springer, 2005.

T4. Frank Chorlton, Text Book of Fluid Dynamics, C.B.S. Publishers, Delhi, 1985.

REFERENCE BOOKS:

R1. B.S.Massey, J.W.Smith and A.J.W.Smith, Mechanics of Fluids, Taylor and Francis, New York,

2005.

R2. P.Orlandi, Fluid Flow Phenomena, Kluwer, New York, 2002.

R3. T.Petrila, Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics,

Springer, berlin, 2004.

Course Learning Outcome (CLO): After completing this course, our Student will be able to

CLO1: Recall, comprehend, apply and analyze the various types of Fluids.

CLO2: Recall, understand, and analyze of Fluids in different medium and coordinates system and

finding their solutions.

CLO3: Understand, apply and analyze solution of Fluids in different medium through various

methods.

CLO4: Remember, comprehend and analyze the Fluids in different medium and their application in

daily life.






DISCRETE Mathematical Structures (MMA3101)

L T P C

3 1 0 4

(40 Hours)


CO1: To introduce general techniques of problem solving and explores the creation of propositional

logic, lattices, combinatory and graph theory.

CO2: To study methods for finding explicit formulae for the terms of sequences that satisfy certain

types of recurrence relations.

CO3: To aim that understanding the concept of mathematical arguments and graph theory.

CO4: To explore the creation of mathematical logic, combinatory and graph theory.

UNIT- I: COMBINATORICS (08 Hours)

Property of binary relations, equivalence, compatibility, partial ordering relations, partial order sets,

Hasse diagram, recursive relation and functions. Recurrence relation (nth

order recurrence relation with

constant coefficient, Homogenous and non- homogenous), generating function, solution of recurrence

relation using G.F.

UNIT- II: LATTICES AND BOOLEAN ALGEBRA (08 Hours)

Definition properties of Lattices- Bounded, Complemented, complete lattice.Introduction, Axioms and

theorems of Boolean algebra, Simplification of Boolean function, Karnaugh maps, Logic gates, Digital

circuits and Boolean.

UNIT-III: PROPOSITIONAL LOGIC (08 Hours)

Proposition logic, basic logic, logical connectives, truth tables, tautologies, contradiction, normal forms

(conjunctive and disjunctive), modus ponens and modus tollens, validity, predicate logic, universal and

existential quantification, Notion of proof:Proof by implication, converse, inverse, contra positive,

negation, and contradiction, direct proof, proof by using truth table, proof by counter example.

UNIT-IV: GRAPH THEORY (08 Hours)

Graphs, subgraph, some basic properties, various operations on graphs, Hamiltonian paths and circuits,

Euler graphs, the traveling sales man problem, connected and disconnected graphs & component, Cuts

sets and cut verities, Planner graphs, Kuratowski graphs, detection of planarity, thickness and crossings

coloring , chromatic number, chromatic polynomials.

UNIT-V: TREE (08 Hours)

Tree and fundamental circuits, distance diameters, radius & pendent vertices, spanning trees, finding all

spanning trees of graphs and a weighted graph, algorithm of primes, Kruskal and DijKstra Algorithms.

TEXT BOOKS:

T1. Kenneth H. Rosen, “Discrete Mathematics & its Application”, Mc. Graw Hill, 2002.

T2. Seymour Lipschutz, M.Lipson, “Discrete Mathematics”, Tata Mc. Graw Hill, 2005.

T3. Deo, N, “Graph theory with application to Engineering and computer Science”, PHI.

REFERENCE BOOKS:

R1. Tremblay &Manohar, “Discrete Mathematical structure with Applications to computer

science”TMH.

R2. J.P.Tremblay& R. Manohar, “Discrete Mathematical structure with Applications to computer

science” Mc. Graw Hill, 2005.

R3. V.Balakrishnan, Schaum’s “Outline of graph theory”, TMH.

R4. Uditagrawal, “Discrete Mathematics” DhanpatRai publication.

R5. Babu Ram, “Discrete Mathmatics”, Pearson



CLO1: Recall, comprehend, apply and analyze the mathematical arguments using logical connectives

.

CLO2: Recall, understand, use and analyze the basic concept of lattices and Boolean algebra

CLO3: The students will be able to solve application based problems of the related topics.

CLO4: Inculcate a logical approach to decision making and problem solving in the field of graph theory.



CLO2 H H H M H M L M

CLO3 H M M L H H L M

CLO4 H L H L M M L M

MATHEMATICAL BIOLOGY (MMA3102)

L T P C

3 1 0 4

(40 Hours)


CO1: To introduce some fundamental ideas about the mathematical modelling.

CO2: To provide some understanding of the concepts to formulate and analyse mathematical models.

CO3: To understand and appreciate the beauty of the real world problems.

CO4: To provide and develop the study of a real world problem mathematically and to make

predictions about behaviour of the system.

UNIT-I: INTRODUCTION (08 Hours)

Some fundamental concepts, mathematical modeling, formulation of a mathematical model, solution of a

mathematical model, interpretation of solution, types of models, limitation of models, areas of modeling,

mathematical modeling in biology or biomathematics, Single species models, stability and classification

of equilibrium points, relationship between eigenvalues and critical points, Exponential and logistic

growth models of single species.

UNIT-II: TWO SPECIES MODELS (08 Hours)

Predator-prey systems, two species models, two dimensional models without and with carrying capacity,

competition models, mutualism models, Introduction of time delay in prey predator models.

UNIT-III: MATHEMATICAL MODELS IN EPIDEMIOLOGY (08 Hours)

Introduction, Basic concepts, Formulation, solution and interpretation of SI, SIS, SIR models, epidemic

models with time delay, Basic reproduction ratio, models with demographic effects, model with fixed

period of temporary immunity.

UNIT-IV: MODELLING SPREAD OF EPIDEMICS & ITS CONTROL (08 Hours)

Modelling various methods of transmission of diseases: horizontal, vertical, vector etc, Epidemic models

with vaccination, epidemic models with drug therapy, models with quarantine, isolation

UNIT-V: MATHEMATICAL PRINCIPLES OF VIROLOGY (08 Hours)

Basic models of virus dynamics, mathematical modelling of antiviral drug therapy and dynamics of

immune response, Simple mathematical models on tumour growth and effect of immune response in its

control.

TEXT BOOKS:

T1. Mathematical Models in Biology and Medicine, J.N. Kapoor

T2. Mathematical Models in Population Biology and Epidemiology, Fred Brauer, Carlos Castillo

Chavez, 2001, Springer.

T3. Virus dynamics: Mathematical Principles of Immunology and Virology, Martin A. Nowak,

Robert M. May, 2000, Oxford University press.

REFERENCE BOOKS:

R1. Mathematical Biology, J.D. Murray, 1989, Springer-Verlag, Berlin Heidelberg- New York.

R2. Population Dynamics of Infectious Diseases: Theory and Applications, R.M. Anderson, 1982,

Chapman and Hall, London-New-York.



CLO1: recall, comprehend, apply and analyze the introduction and two species of model.

CLO2: recall, understand, use and analyze the mathematical models in Epidemiology.

CLO3: Define, illustrate and interpret about the modelling spread of epidemic and its control.

CLO4: Solve, apply, understand and utilize the mathematical principles of virology.






CLO4 H M H L H M L H

Stochastic Processes (MMA3103)

L T P C

3 1 0 4

(40 Hours)


CO1: To introduce and evaluate the Probabilities of various events using a probability space, also find

conditional probability with the help of total probability and Bayes’ theorem.

CO2: To provide some understanding and evaluation of statistical independence of events and random

variables for univariate and multivariate cases.

CO3: To aim at understanding and finding joint densities & Random sequences in probability with

probability bonds.

CO4: To explore the concept of Random processes, also find mean-covariance functions and

transmission of random process & power spectral density.

UNIT-I: RANDOM PROCESS (08 Hours)

Definition and classification of general stochastic processes, Random Process, Moments of a Stochastic

Process, Stationary and Ergodic theorem, Spectral density and correlation function.

UNIT-II: MARKOV PROCESS (08 Hours)

Classification of Stochastic processes, Chapman and Kolmogorov equation, Markov Chains:

definition,transition probability matrices, classification of states, limiting properties. Markov Chains

withDiscrete State Space

UNIT-III: POISSON PROCESS (08 Hours)

Poisson process, birth and death processes. Renewal Process: renewalequation, mean renewal time,

stopping time. Markov Process with Continuous State Space

UNIT-IV: BROWNIAN MOTION (08 Hours)

Introduction to Brownian motion, Congestion Process: Queuing Process, M/M/1 Queue

UNIT-V: STOCHASTIC ANALYSIS (08 Hours)

Limits, Mean square continuity, stochastic differentiation, stochastic integration

TEXT BOOKS:

T1. Chung, K.L. – Elementary Probability Theory and Stochastic Processes.

T2. Srinivasan, S.K. and Mehata, K.M. – Stochastic Processes.

T3. Hoel, P.G.,Port, S.C. and Stone, C.J. – Introduction to Stochastic Processes.

REFERENCE BOOKS:

R1. M. Scott, Applied Stochastic Processesin science and engineering.

R2. Oliver Knill, Probability Theory and Stochastic Processeswith Applications, Overseas Press

(INDIA).


After completing this course, our Students will be able to

CLO1: Recall, comprehend, apply and analyze the axiomatic formulation of modern Probability Theory

and think of random variables as an intrinsic need for the analysis of random phenomena.

CLO2: Understand, use and Characterize probability models and function of random variables based on

single & multiples random variables.

CLO3: Understand, analyze, evaluate and apply moments & characteristic functions and understand the

concept of inequalities and probabilistic limits.

CLO4: Understand the concept of random processes and determines covariance and spectral density of

stationary random processes







NUMBER THEORY AND CRYPTOGRAPHY (MMA3201)

L T P C

3 1 0 4

(40 Hours)

Course Objectives:

CO1: To introduce basic ideas about Cryptography and its importance in day to day life.

CO2: To provide some understanding of various mathematical principles used for crypto analysis.

CO3: To aim at understanding how much it is useful for military, diplomatic and commercial

domains.

CO4: To explore the connection between the traditional methods and the methods using

technology.

UNIT-I: ELEMENTARY NUMBER THEORY (08 Hours)

Time Estimates for doing arithmetic - Divisibility and Euclidean algorithm – Congruence Applications to

factoring.

UNIT-II: NUMBER THEORY (08 Hours)

Chinese Remainder Theorem, Structure of U(Z/nZ), Quadratic Reciprocity, Quadratic Gauss Sum,Finite

Fields, Gauss and Jacobi Sums, Cubic and Biquadratic Reciprocity, Equations over FiniteFields, Zeta

Function, Algebraic Number Fields and the Ring of Integers, Units and Primes, Factorisation, Quadratic

and Cyclotomic Fields, Dirichlet L Function, Diophantine Equations, Elliptic Curves.

UNIT-III: CRYPTOGRAPHY (07 Hours)

Some simple crypto systems - Enciphering matrices, Finite fields - Quadratic residues and Reciprocity

UNIT-IV: PUBLIC KEY CRYPTOGRAPHY (10 Hours)

The idea of public key cryptography - RSA Cryptosystem, Diffie-Hellma and the Digital Structure

Algorithm, Secret Sharing, Coin Flipping, Passwords, Signatures, and Ciphers, Practical Cryptosystems

and Useful Impractical Ones. Complexity of Computations, Big-O Notation

UNIT-V: PRIMALITY AND FACTORING (07 Hours)

Pseudoprimes - The rho method - Fermat factorization and factor bases - The Continued fraction method

-The quadratic sieve method.

TEXT BOOKS:

T1. I. Niven and Zuckermann H.S. : An Introduction to the theory of numbers, Wiley Eastern Ltd.

1972

T2. . Ireland & Rosen, A Classical Introduction to Modern Number Theory, Springer.

T3. Tom Apostol, Introduction to Analytic Number theory, Narosa Publications, New Delhi.

T4. Delfs, H., Knebl, H., Introduction to Cryptography, Springer, 2003.

REFERENCE BOOKS:

R1. Serre, J.-P., A Course in Arithmetic, Springer.

R2. Cassels, J.W.S., Frolich, A., Algebraic Number Theory, Cambridge

R3. Koblitz, N., Algebraic Aspects of Cryptography, Springer.



CLO1: recall, comprehend, apply and analyze idea of divisibility and Congruence

CLO2: recall, understand, use and analyze the Number theory applied in Cryptography

CLO3: understand, analyze and apply crypto-systems

CLO4: remember, comprehend and apply Cryptanalylis





CLO3 H M M M H H L M

CLO4 H L M H M M L M

Dynamical Systems (MMA3202)

L T P C

3 1 0 4


CO1: To introduce some fundamental ideas about the autonomous and non-autonomous dynamical

system.

CO2: To provide some understanding of the basic concepts and analyze dynamical system.

CO3: To understand the bifurcation theory, nonlinear oscillators and topological study of nonlinear

differential equation.

CO4: To provide and develop the study of dynamical system and to make predictions about behavior of

the system

UNIT-I: STABILITY ANALYSIS (12 Hours)

Phase space and flows, Limit sets and trajectories, Stability and Liapunov functions, Strong linear

stability, Orbital stability, Concept of limit cycles and tours, Critical point analysis via examples.

Autonomous and non-autonomous systems. Stability of nonlinear systems, Lyapunov methods

UNIT-II: BIFURCATION THEORY (08 Hours)

Bifurcation, types of bifurcation, Local bifurcation, Saddle-node bifurcation, Period doubling and Hopf

Bifurcation.

UNIT-III: NONLINEAR OSCILLATORS (10 Hours)

Conservative system, Hamiltonian System, Various types of Oscillators in nonlinear System, Solution of

nonlinear differential equations, Quasiperiodic motion.

UNIT-IV: POINCARE MAP (10 Hours)

Topological study of nonlinear differential equation, Poincare map, Poincare Benedixson theory and

Poincare index.

TEXT BOOKS:

T1. Robert L. Davaney, An Introduction to Chaotic dynamical system, Addison-Wesley

Publishing Co.Inc., 1989.

T2. D. K. Arrowosmith, Introduction to Dynamical systems, Cambridge University Press,

1990.

T3. V.I. Arnold, Ordinary Differential Equations, rentice Hall of India, New Delhi, 1998.

T4. L. Perko, Differential Equations and Dynamical Systems, Springer Verlag, NY, 1991.

REFERENCE BOOKS:

R1. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, TAM

Vol.2, Springer-Verlag, NY, 1990..

R2. K. T. Aligood et. al.Chaos Springer-Verlag.

R3. D. K. Arrowsmith et. al. Dynamical Systems, Chapman and Hall, London.



CLO1: Recall, comprehend, apply and analyze the introduction of stability analysis of linear

and nonlinear dynamical system.

CLO2: Recall, understand, use and analyze the bifurcation theory.

CLO3: Define, illustrate and interpret about the various type of oscillator in nonlinear system.

CLO4: Solve, apply, understand and utilize the topological study the nonlinear differential

equation and Poincare map.




CLO3 H M H L H M M M

CLO4 H M H L H M M L

Fuzzy Sets and Applications (MMA3203)

L T P C

3 1 0 4

(40 Hours)

COURSE OBJECTIVES (CO):

CO1: To introduce students higher mathematics related topics by boosting their confidence so

that they don’t feel any fear about the subject.

CO2: Making the students capable so that they can handle the fuzzy logic related problem in

other subject also able to discriminate the related problem in various competitive exams.

CO3: Creation an interest about the fuzzy set theory related research so that the students will

be able to apply such tools in real phenomena, by clearing the basic concept of related

topics.

CO4: To explore the connection between history and applications in engineering sciences of the

subject.

UNIT-I : FUZZY SETS (10 Hours)

Concepts of Crispness and fuzziness, fuzzy set, types of fuzzy set, -cuts, Convex fuzzy sets,

operations on fuzzy sets, type-2 fuzzy sets, fuzzy numbers and its operations, LR- representation of

fuzzy sets and extended operation on them, t- norms and t-conorms, increasing and decreasing

generators, interval equations, fuzzy equations

UNIT-II: FUZZY RELATION AND FUZZY GRAPHS (10 Hours)

Fuzzy relation on fuzzy sets, composition of fuzzy relations, Properties of Min-Max compositions, fuzzy

graphs, Fuzzy function and their extrema, integration of fuzzy function, fuzzy differentiation.

UNIT-III: FUZZY CONTROL (10 Hours)

Fuzzy control, Process of fuzzy control, fuzzy decisions, fuzzy linear programming problem, fuzzy

tranportation problem, fuzzy dynamic problem.

UNIT-IV: FUZZY LOGIC (10 Hours)

Fuzzy logic, truth table, Fuzzy theory and weather classifications, Water demand forecasting, Soil water

movement and applications in environmental science, Medical Diagnosis, Financial markets, Uncertainty

in Business management.

TEXT BOOK:

T1. George J. Klir, Bo Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice

Hall PTR, 1995.

T2. Hao Ying, Fuzzy Control and Modeling: Analytical Foundations and Applications, IEEE

Press, 2000.

REFERENCES:

R1. Zimmermann, H.J., Fuzzy Set Theory and its applications, Allied Publishers Limited.

R2. Ross, T.J., Fuzzy Logic with Engineering Applications, 2nd edition, John Wiley & Sons Ltd.

R3. Lai, Y and Hwang, C., Fuzzy Mathematical Programming, Springer-Verlag.

COURSE LEARNING OUTCOME (CLO):


CLO1: Upon successful completion of the course, students will be able to analyze the fuzzy

related problem.

CLO2: Students will fill confident to give the analytical solution of syllabus related fuzzy logic

topics questions with effective manner.

CLO3: Students will be aware about the basics as well the advance tools and formulae of the

subject to promote creativity and demonstrate the link between theory and its real

world applications. CLO4: subject to promote creativity and demonstrate the link between theory and its

application in engineering field.






LATEX (MMA3501)

L T P C

0 0 2 1


CO1: To introduce fundamental downloading and installation of Latex.

CO2: To introduce basic symbols of Latex.

CO3: To provide understanding of writing various mathematical equations with the help of Latex.

CO4:

To aim at understanding of various tools for design mathematical 3D diagrams with the help of.

Latex.

1. Introduction to basic concepts of Latex.

2. Create a document containing different fonts, page style, page numbering and bibliography

on Latex.

3. Create a title page of thesis on Latex

4. Create a front page of research paper on Latex.

5. Create a document containing table of contents, index and glossary on Latex.

6. Create a document containing simple mathematical symbols on Latex.

7. Create a mathematical document containing algebraic and transcendental functions on

Latex.

8. Create a mathematical document containing differentiation and integrations on Latex.

9.

Create a mathematical document containing matrices, and system of linear equations on

Latex.

10. Create a mathematical project or thesis on latex.



CLO1: Write correct, efficient, and well-documented programs using LATEX.

CLO2: Gain basic knowledge about the LATEX, drafting techniques and will be prepared for writing

higher complex problems in the field of scientific and allied research.

CLO3: Demonstrate system of linear equations and matrices with the help of LATEX.

CLO4: Explore the connection between basics as well the advance tools of the LATEX to discuss three

dimensional analyses of mathematical equations.







SEMINAR (MmA3502)

L T P C 0 0 2 1



to solve a problem.


and making choices.



philosophy.



solution.







WEEK.

No.



Mathematics.





students.

















HUMAN VALUES AND ETHICS

XHUX-601

Course Objectives:

CO1: To make students recall and apply the basic human values. CO2: To recommend them training in human relationship for appreciating family and society. CO3: To organize and prioritize them in a manner in which they can be adaptable to the nature. CO4: To develop the skills to identify the characteristics of people and the Eco-friendly production

system.

UNIT-I (06 Hours)

Requirement of Value Education, Fundamental Guidelines, Content, and Methodology for Value

Education

Appreciating the need, Fundamental Guidelines, Content and Methodology for Value Education, Self-

Investigation: Methodology and Content. Fundamental Human Aspirations: Uninterrupted Happiness

and Prosperity. A Critical Evaluation of the present situation: Understanding Happiness and Prosperity

rightly, Harmonious Living at all levels.

UNIT-II (04 Hours)

Understanding Harmony in the Individual - Harmony within Self

An individual as Co-existence of Sentient Self (‘I’) and Physical Body.

Learning by distinguishing the needs of Self (‘I’) and ‘Body’ - Happiness and Prosperity

Problems related to identity crisis, depression, lack of motivation, traumatic childhood.

Harmony of Self (‘I’) with the physical Body: Self-control and Health; Understanding Physical Needs.

Understanding suicide and psychosomatic problems.

UNIT– III (06 Hours)

Human Relationship: Appreciating Harmony in Family and Society

Fundamentals for Harmony in Family as basic unit of human interaction Understanding values in human

to human relationship. Family problems, domestic violence, relationship issues among youth.

Understanding society as an extension of Family;Social crimes; causes, factors leading to such.

UNIT- IV (06 Hours)

Existence as Co-existence; Understanding Harmony in the Nature and Existence Harmony in Nature and its critical appraisal in the present scenario.

UNIT– V (06 Hours)

Positive Psychology & Professional Ethics

Competence in Professional Ethics:

a) Capacity to utilize the professional competence for expanding Universal Human Order

b) Skill to identify the scope and characteristics of people-friendly and eco-friendly production systems

c) Ability to identify and develop appropriate technologies and management Patterns for above

production systems

L T P C

2 0 0 2

Text Book:

T1. Gaur, R.R.,Sangal, R., and Bagaria, G.P. (2009) “A Foundation Course in Human Values and

Professional Ethics” Excel Books Private Limited. New Delhi.

Reference Books:

R1.Illich, I.(1974) “Energy & Equity” The Trinity Press, Worcester and Harper Collins,USA.

R2.Schumacher, E.F. (1973) “Small is Beautiful: A Study of Economics as if People Mattered” Blond &

Briggs, Britain.

R3.Sussan, G.(1976) “How the Other Half Dies”, Penguin Press, Reprinted 1991.

R4.Meadows, D.H.,Meadows, D.L. Randers, J., Bchrens, W.W.III (1972) “Limits lo Growth Club of

Rome’s report”, Universe Books.

R5.Nagraj, A. (1998) “Jeevan Vidyaek Parichay” Divya Path Sansthan, Amarkantak.

R6.SeebauerE.G. and Berry, R.L. (2000) “Fundamentals of Ethics for Scientists &Engineers” Oxford

University Press.

Course Learning Outcomes (CLO)

On completion of this course, the students will be able to: CLO Description Bloom’s

Taxonomy Level

CLO1 Requirement of Value Education, Fundamental Guidelines,

Content, and Methodology for Value Education-

Explain the need, Fundamental Guidelines,Content and

Methodology for Value Education,Self-Inspect:Methodology

and Content. Fundamental Human Aspirations:Uninterrupted

Happiness and Prosperity. A Critical Evaluation of the

present situation:Understanding Happiness and Prosperity

rightly, Harmonious Living at all levels.

2, 4, 6

Understanding

Analyzing,

Evaluating

CLO2 Outlining Harmony in the Individual -Harmony within Self -

An individual as Co-existence of Sentient Self (‘I’) and

Physical Body.

Learning by distinguishing the needs of Self (‘I’) and ‘Body’

- Happiness and Prosperity

Problems related to identity crisis, depression, lack of

motivation, traumatic childhood.

Harmony of Self (‘I’) with the physical Body: Self-control and

Health;Interpret Physical Needs. Analyzing suicide and

psychosomatic problems.

2,5, 4,

Understanding,

Evaluating

Analyzing

CLO3 Human Relationship:Demonstrating Harmony in Family and

Society

Fundamentals for Harmony in Family as basic unit of human

interaction Identifying values in human to human

relationship. Family problems, domestic violence, relationship

issues among youth. Understanding society as an extension of

Family;Social crimes;causes, and determining factors

2,3, 5,

Understanding,

Apply

Evaluating

leading to such problems..

CLO4

Existence as Co-existence; Illustrate Harmony in the Nature

and Existence

Harmony in Nature and its critical appraisal in the present

scenario.

2,5,

Understanding

Evaluating,

CLO 5 Positive Psychology & Professional Ethics

Competence in Professional Ethics:

a) Developing skill to utilize the professional competence for

expanding Universal Human Order

b) Skill to identify the scope and characteristics of people-

friendly and Eco-friendly production systems.

c) Ability to identify and prioritize appropriate technologies

and management Patterns for above production systems

6,3,5

Creating,

Applying,

Evaluating

Mapping of CLOs with PLOs & PSOs

Course

Learning

Outcomes

CLOs

Programme Learning Outcomes (PLOs) Programme Specific Outcomes

(PSOs)

PL

O1

PL

O2

PL

O3

PL

O4

PL

O5

PL

O6

PL

O7

PL

O8

PL

O9

PL

O1

0

PL

O11

PL

O1

2

PS

O1

PS

O2

PS

O3

PS

O4

PS

O5

PS

O6

CLO1 H H M M M H H L H H M L H M L L L L

CLO2 H M M M H H H L H H L H M M H M M L

CLO3 M H H M M H H M M H H M M H H H H H

CLO4 M H H H M H H M M M M M H H H H H H

Functional Analysis (MMA4001)

L T P C

3 2 0 5

(40 Hours)


CO1: To introduce some fundamental concepts about Normed linear spaces, Banach spaces, Hilbert

spaces, dual spaces, functionals and operators on Hilbert spaces.

CO2: To develop the ability to write the mathematical proofs.

CO3: To understand and appreciate the beauty of the abstract nature of mathematics.

CO4: To provide and develop a solid foundation in various fields of mathematics and its applications.

UNIT-I: NORMED SPACES AND BANACH SPACES (08 Hours)

Normed linear spaces, subspaces, Banach spaces, some concrete examples of Banach spaces, Quotient

space of normed linear spaces and its completeness, equivalent norm, Riesz Lemma, basic properties of

finite dimensional normed linear spaces and compactness.

UNIT-II: BOUNDED LINEAR FUNCTIONALS (08 Hours)

Weak convergence and bounded linear transformations, normed linear spaces of bounded linear

transformations, Bounded linear operators, spaces of bounded linear operators, equivalent norms, open

mapping and closed graph theorems and their consequences, uniform boundedness principle.

UNIT-III: DUAL SPACES (08 Hours)

Examples and basic properties, Forms of dual spaces, Hahn-Banach theorem and its consequences,

embedding and reflexivity, adjoint of bounded linear operators, weak convergence.

UNIT-IV: HILBERT SPACES (08 Hours)

Inner product spaces, Definition and examples of Hilbert space and simple properties, orthogonal sets

and complements. Separable Hilbert spaces, Bessel's inequality, Parseval’s identity, Structure of Hilbert

spaces, the conjugate space, Riesz representation theorem, orthogonality of vectors, orthogonal

complements and projection theorem, orthonormal sets, complete orthonormal sets.

UNIT-V: FUNCTIONALS AND OPERATORS ON HILBERT SPACES (08 Hours)

Adjoint of an operator on a Hilbert space, Reflexivity of Hilbert spaces,bounded linear functionals,

Riesz-Frechet theorem, Hilbert-adjoint operators, self-adjoint operators, normal operators and unitary

operators, the spectral theorem on a finite dimensional Hilbert space.

TEXT BOOKS:

T1. E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, New York,

1978.

T2. A.H. Siddiqui, Functional Analysis with applications, Tata McGraw-Hill Publishing Company

Ltd. New Delhi.

T3. Walter Rudin, Functional Analysis, Tata McGraw-Hill Publishing Company Ltd., New Delhi,

1973.

REFERENCE BOOKS:

R1. P.K. Jain, O.P. Ahuja and Khalil Ahmad, Functional Analysis, New Age international (P) Ltd. &

Wiley Eastern Ltd., 1997.

R2. G. Bachman and L. Narici, Functional Analysis, Academic Press, 1966.

R3. R.E. Edwards, Functional Analysis, Holt Rinehart and Winston, New York, 1965.

R4. B. Choudhary and Sudarsan Nanda, Functional Analysis with Applications, Wiley Eastern Ltd.

1989.

R5. K.K. Jha, Functional Analysis, Students’ Friends, 1986.

R6. B.K. Lahiri, Elements of Functional Analysis, The world Press Pvt. Ltd., Calcutta, 1994.

Course Learning Outcome (CLO): After successful completion of this course, the student will be able to

CLO1: Recall, apply and analyze the basic properties of Normed linear spaces, Banach spaces, Hilbert

spaces, dual spaces.

CLO2: Define, illustrate and interpret about metric spaces, complete metric spaces, linear operators,

inner product spaces, Hilbert spaces.


CLO4: Conclude, deduct and make use of the knowledge of functional analysis to solve mathematical

problems.






OPERATIONS RESEARCH (MMA4002)

L T P C

3 1 0 4

(40 Hours)


CO1: To introduce some fundamental concepts of Operations Research.

CO2: To provide some understanding of the concept broad education in the techniques and modeling

concepts used to analyze and design complex systems.

CO3: To introduce some understanding application based problems of the related topics.

CO4: To explore the knowledge in more advanced and complex situations of higher mathematics.

UNIT-I: LINEAR PROGRAMMING (08 Hours)

Introduction, Mathematical formulation of the problem, Graphical Solution methods, Mathematical

solution of linear programming problem, Slack and Surplus variables. Matrix formulation of general

linear programming Problem,

UNIT-II: SIMPLEX METHOD (08 Hours)

The Simplex Method: Artificial variables, two phases Simplex Method, infeasible and unbounded LPP's,

alternate optima, Dual problem and duality theorems, dual simplex method and its application in post

optimality analysis, Revised Simplex method.

UNIT- III: ASSIGNMENT AND TRANSPORTATION MODELS (08 Hours)

Construction and solution of these Models, Hungarian method of solving assignment problem,

unbalanced assignment problem, matrix form of transportation problem, Initial basic feasible solution,

Balanced and unbalanced transportation problems, u-v method for solving transportation problems

Selecting the entering variables, Selecting the leaving variables, Degeneracy in transportation Problem.

UNIT-IV: THEORY AND QUEUING MODELS GAME (08 Hours)

Two person Zero sum games, Pure and mixed strategy, minimax and maximin principle, Rule of

Dominance, Different methods of solving mixed strategy games. Elementary queuing models, Steady-

state solutions of Markovian queuing models: M/M/1, M/M/1 with limited waiting space, M/M/C,

M/M/C with limited waiting space, M/G/1.

UNIT-V: REPLACEMENT AND INVENTORY CONTROL MODELS (08 Hours)

Replacement and Reliability models: Replacement of items that deteriorate, Replacement of items that

fail completely. Inventory Models: EOQ models with and without shortages, EOQ models with

constraints.

TEXT BOOKS:

T1. Operation Research, Theory and Application by J.K. Sharma, Macmillan India

T2. Quantitative techniques in Management by N. D. Vohra, TMH

T3. Linear Programming by N.P. Loomba

T4. Operations Research by P.K. Gupta and D.S. Hira, S Chand and Sons.

REFERENCES BOOKS:

R1. Operation Research: An Introduction by H.A. Taha

R2. S. S. Rao, Optimization Techniques, Wiely Eastern

R3. Operations Research, Kanti Swarup, S Chand and Sons



CLO1 Recall, apply and analyze the basic properties of linear programming.

CLO2 Define, illustrate and interpret about Simplex method.

CLO3 understand, analyze and apply assignment and transportation models.

CLO4 remember, comprehend, apply and analyze the theory and queuing models game,

replacement and inventory control models.


CLO1 H H H L H M L H

CLO2 H M H L H M L M

CLO3 H H M L H M L L


MATHEMATICAL MODELING (MMA4101)

L T P C

3 1 0 4

(40 Hours)


CO1: To introduce some fundamental ideas about the mathematical modelling.

CO2: To provide some understanding of the concepts to formulate and analyse mathematical models.

CO3: To understand and appreciate the objective will be to learn how to take a phenomena arising in

physics, chemistry, biology, even the social sciences, then study it using Mathematics.

CO4: To provide and develop various mathematical models. This will help them to apply the

mathematical techniques to real life situations.

UNIT-I: Mathematical Modelling- Need, Techniques, Classification and Simple Illustrations

(08 Hours)

Simple situations requiring mathematical modeling, technique of mathematical modeling, classification

of mathematical models, some characteristics of mathematical models, mathematical modeling through

Geometry, Algebra, Trigonometry, Calculus, Limitations of mathematical modeling.

UNIT-II: Mathematical modeling through ordinary differential equations of first order

(08 Hours)

Mathematical modeling through Differential equations, linear growth and decay models, Non-linear

growth and decay models, compartment models. Mathematical modeling through system of ordinary

differential equation of first order, Mathematical modelling in population dynamics and mathematical

modeling of epidemics.

UNIT-III: Mathematical modeling through ordinary differential equations of second order

(08 Hours)

Mathematical modeling of Planetary motion, mathematical modeling of circular motion and motion of

satellites, mathematical modeling through linear differential equations of second order.

UNIT-IV: Mathematical modeling through difference equations (08 Hours)

The need for mathematical modeling through difference equations: some simple models, basic theory of

linear difference equations with constant coefficients, mathematical modeling through difference

equations in Economic and Finance, mathematical modeling through difference equations in population

dynamics and genetics, mathematical modeling through difference equation in probability theory.

UNIT-V: Mathematical modeling through partial differential equations (8 Hours)

Situations giving rise to partial differential equations models, mass balance equations: First method of

getting PDE models, Momentum balance equations: the second method of getting PDE models,

Variational principles: third method of obtaining PDE models

TEXT BOOKS:

T1. J.N. Kapur, Mathematical Modeling, New Age International (P) Limited Publishers, 2009.

T2. J. N. Kapoor, Mathematical Models in Biology and Medicine, Affliated East West Press, 1985.

REFERENCE BOOKS:

R1. Fred Brauer, Carlos Castillo-Chávez: Mathematical Models in Population Biology and

Epidemiology, Springer, 2011.

R2. Mathematical Biology : J.D. Murrey, Springer.

R3. Population Biology : Alan Hastings Concepts and Models , Springer.

R4. J. B. Shukla, T. G. Hallam, V. Capasso, Mathematical Modelling of Environmental and

Ecological Systems, Elsevier Science Publishers, Amsterdam, 1987.



CLO1: recall, comprehend, apply and analyze the mathematical modelling- needs, techniques,

classification and simple illustrations

CLO2: Solve, apply, understand and utilize the mathematical modelling through first and second

order differential equation.

CLO3: Define, illustrate and interpret about the mathematical modelling through difference

equation.

CLO4: Recall, apply and analyze the mathematical modelling through partial differential

equation.



CLO2 H L H L H M L H


CLO4 H M M L H M L H

Mathematics of Finance and Insurance (MMA4102)

L T P C

3 1 0 4

(40 Hours)


CO1: To introduce some fundamental concepts of essential mathematical skill, teach them the process

of generating options.

CO2: To provide some understanding of the concepts of making choices and to inculcate a logical

approach to decision making and problem solving.

CO3: To introduce some understanding of the learn time value of money, life Insurance, life annuities.

CO4: To explore this knowledge in solving many problems of higher mathematics involving in market

analysis.

UNIT-I: TIME VALUE OF MONEY (08 Hours)

Elements of Theory of Interest, Cash Flow Valuation, Annuities, Amortization and Sinking Funds.

UNIT-II: MATHEMATICS FOR LIFE INSURANCE (08 Hours)

Brief Review of Probability Theory ,Survival Distributions, Life Tables, Valuing Contingent Payments,

Life Insurance, Life annuities, Net Premiums, Insurance Models including Expenses.

UNIT- III: FINANCIAL INSTRUMENTS AND VALUATION (08 Hours)

A Brief Introduction to Financial Markets, Basics of Securities, Stocks, Bonds and Financial Derivatives,

Viz Forwards, Features, Options and Swaps.

UNIT- IV: STOCHASTIC CALCULUS (08 Hours)

An Introduction to Stochastic Calculus, Stochastic Process, Geometric Brownian motion, Stochastic

Integration and Ito’s Lemma.

UNIT- V: OPTION PRICING (08 Hours)

Option Pricing Models-Binomial Model and Black Scholes Option Pricing Model for European Options,

Black Scholes Formula and Computation of Greeks.

TEXT BOOKS

T1. John C. Hull: Options, Futures and other Derivatives, Prentice-Hall of India Pvt. Ltd.

T2. Sheldon M. Ross: An introduction to Mathematical Finance, Cambridge University Press.

REFERENCE BOOKS:

R1. Salih N. Neftci: An Introduction to Mathematics of Financial Derivatives, Academic Press,

Inc.

R2. Robert J. Elliot and P. E. Kopp: Mathematics of Financial Markets, Springer Verlag, New York

Inc.



CLO1: recall, apply and analyze the time value of money.

CLO2: Define, illustrate and interpret about the mathematics for life insurance.

CLO3: understand, analyze and apply the financial instruments and valuation.

CLO4: remember, comprehend, apply and analyze the stochastic calculus and option pricing.



CLO1 H H M L H L L H

CLO2 H M M L H L M M

CLO3 H H M L H L H H

CLO4 H L M L H L M M

OPTIMIZATION Techniques

(MMA4103)

L T P C

3 1 0 4

(40 Hours)


CO1: To introduce some fundamental ideas about optimization techniques and operation research.

CO2: To provide understanding of the basic concepts of LPP and various solving techniques.

CO3: To aim at understanding of NLPP, dynamic programming and various search techniques.



UNIT- I: FORMULATION OF PROBLEM (08 Hours)

Mathematical Formulation of optimization problem, Convex sets, Definiteness of Q and its Application,

Graphical method.

UNIT- II: LINEAR PROGRAMMING PROBLEMS (08 Hours)

Simplex method, artificial variable method, Sensitivity Analysis, Integer Programming problem.

UNIT- III: NON-LINEAR PROGRAMMING PROBLEMS (08 Hours)

Convex Functions, Kuhn-Tucker theory, Quadratic Problems, Wolfe’s and complementary pivot

algorithms Separable Programming problem, Geometric Programming Problem.

UNIT- IV: DYNAMIC PROGRAMMING (08 Hours)

Bellman’s principle of optimality and method of recursive optimization (Problem involving one

contraint).

UNIT- V: SEARCH TECHNIQUES (08 Hours)

Unimodal function, Direct search and gradient method, Fibonacci method, Golden section method,

Steepest decent method, Newton-Raphson method, Hooke’s and Jeeve’s method, Conjugate gradient

method, Fletcher Reeves Method.

TEXT BOOKS:

T1.“Operations Research – An Introduction” H. A. Taha

T2. “Operations Research” S. D. Sharma, Ram NathKedarNath.

T3. “Introduction of optimization techniques” J. C. Pant, 5th edition, Jain Brothers.

REFERENCES BOOKS:

R1.Operations Research – Theory and Application, J. K. Sharma, Macmillian Pub.

R2.Mathematical Programming ,S. M. Sinha, Elsevier India Pvt. Ltd.

R3.Introduction to Operations Research, Hillier and Lieberman, McGraw Hill.

R4.Linear Programming , G. Hadly, Narosa Publishing House



CLO1: Recall, comprehend, apply and analyze the LPP its formulation with solving techniques.

CLO2: Recall, understand, use and analyze sensitivity analysis and integers programming.

CLO3: Understand, apply and analyze NLPP with various solving techniques

CLO4: Remember, comprehend and analyze Dynamic programming and various search techniques.


CLO1 H H H M H M M M

CLO2 H H H M H M M M


CLO4 H M H M H M M M

Fractional Calculus & Nonlinear Dynamics (MMA4104)

L T P C

3 1 0 4

(40 Hours)


CO1: To introduce some fundamental ideas about fractional calculus and some special functions.

CO2: To provide understanding of the basic concepts of fractional order differential equations.

CO3: To aim at understanding of solving and applications of fractional differential equations, chaos and

its synchronization.

CO4: To explore the connection between basics as well the advance knowledge of the subject to


UNIT-I: FRACTIONAL CALCULUS (8 Hours)

Introduction of fractional calculus, special functions of the fractional calculus, Gamma function, Mittag-

Leffler function of kind I & II.

UNIT-II: TYPES OF FRACTIONAL ORDER DIFFERENTIAL EQUATION AND ITS TRANSFORM

(12 Hours)

Definition and properties of Riemann-Liouville and Caputo fractional derivatives and integrals,

Laplace and Fourier transform of fractional derivative.

UNIT-III: METHOD AND APPLICATION FRACTIONAL ORDER DIFFERENTIAL EQUATION

(10 Hours)

Methods for solution of fractional order differential equation, Application of generalized fractional

calculus in science and engineering fields.

UNIT-IV: CHAOS AND ITS SYNCHRONIZATION (10 Hours)

A brief introduction of chaos and synchronization, Strange attractors, Stability of nonlinear systems,

Lyapunov methods, Study of some well known chaotic systemviz.Lotka-Volterra’s system, Lorenz’s

system, financial system, Control of chaotic systems, Methodology of chaos synchronization.

TEXT BOOKS:

T1. Podlubny, I. Fractional Differential Equations. Mathematics in Science and Engineering, vol.

198. Academic Press, 1999.

T2. Liao, S. J., Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman &

Hall/CRC Press, Boca Raton, 2003.

T3. Ivo Petras, fractional order nonlinear systems: modeling, analysis and simulation, Higher

education press, Beijing, 2011.

REFERENCE BOOK: R1. Shantanu Das, Functional Fractional Calculus for System Identification and Cont Springer-

Verlag Berlin Heidelberg 2008.



CLO1: Recall, comprehend, apply and analyze the introduction of fractional calculus.

CLO2: Recall, understand, use and analyze the introduction of types of fractional order

differential equation and its transform.

CLO3: Define, illustrate and interpret about the method and application fractional order

differential equation.

CLO4: Solve, apply, understand and utilize the chaos and its synchronization.






CLO4 H M H L H M L H

Project (MmA4501)

L T P C 0 0 2 1



to solve a problem.


and making choices.



philosophy.



solution.







WEEK.

No.



Mathematics.





students.

6-12 Submission of project reports and presentations (30 minutes) by students.

13-16. Submission of project reports and presentations by repeated presentations.















Comprehensive Viva Voce (MMA4502)

L T P C 0 0 0 2


CO1: To evaluate the awareness of the entire syllabus right from first year to second year.

CO2: To judge the overall ability of the student to find linkage among cross functional areas.

CO3: To evaluate the knowledge gained in various subjects

CO4: To evaluate the students’ preparedness to take up the future challenges of pursuing higher

studies or research or any other career in diverse field.

Assessment Criteria and Schedule

Students in the final semester just before the End Semester Examination have to appear in

Comprehensive Viva-Voce examination as per Study Evaluation Scheme which is to test his/her overall

comprehension of various subjects and its linkages.

. The examination will be of Interview type in oral form for 100 Marks and is of Summative type.

a. Appearing in Viva-Voce examination is mandatory.

b. The focus of this activity is to probe students understanding of concepts, theories, procedures,

problem solving skills, interpersonal competencies, intrapersonal qualities and integrated practices.

c. The format for evaluation may be different for different courses and will be based on above

parameter with varying weightage.

d. The Comprehensive Viva-Voce Examination Committee is purely internal committee which will

consist of three faculty members of cross functional areas of same Faculty and/or one odd member from

Inter Faculty on requirement if any.

e. The Committee will be constituted by Dean in consultation with the Course Coordinator of the

respective class which must have approval of Director of the Institute.



CLO1 Explain, comprehend, apply and analyze the knowledge of mathematics.

CLO2 Examine, evaluate, develop and make use of the knowledge of mathematics in

application in day to day life.

CLO3 Extend, identify and elaborate the knowledge in various diverse fields of mathematics

and effectively communicate it.

CLO4 Conclude, deduct, interpret and build the understanding of mathematics by exploring

the connection between history, science and philosophy.



CLO1 H M M L H H M H

CLO2 H L M L H H M H

CLO3 H H L L M H M H

CLO4 H L M L H H M M

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