BERHAMPUR UNIVERSITY

Syllabus
for

M.A./M.Sc. in Mathematics
(2-Year Programme)

P. G. Department of Mathematics
Berhampur University
Berhampur-760007 (Orissa)

2024-25
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 BERHAMPUR UNIVERSITY
Syllabus for M.A./M.Sc. in Mathematics
(Applicable for Students Taking Admission from the Session 2024-25)

Programme Outcome:

A two years regular course M.A./M.Sc. in Mathematics will develop a breadth of

understanding in Calculus, Complex analysis, Measure theory, Numerical analysis, Topology,
Differential equations, Functional analysis, Optimization techniques, Number theoretic
Cryptography, Graph theory and Statistics along with a depth of knowledge in algebra and
analysis. The course is designed to make the students competent to solve ordinary and partial
differential equations using Laplace transform and Fourier transform techniques, Eigen value
problems, systems of linear differential equations, problems concerning topological spaces,
continuous functions, product topologies, and quotient topologies, extension fields, roots of
polynomials, complex integrals, elliptic functions. The course also includes the initial value
problems by using single step methods, multi step methods, problems on interpolation,
numerical differentiation and integration, measurable sets, measurable functions, problems on
Green, Gauss and Stokes theorems, problems on probability distributions and generating
functions, problems on Hahn-Banach theorems, problems on primitive roots, quadratic
residues, and quadratic non-residues, cryptography, zero knowledge protocol and oblivious
transfer, the rho method, the continued fraction method. After completion of this course the
students will be capable in different competitive examinations like, TIFR, IISc, HRI, CSIR
(NET & JRF), GATE, Civil services and pursue research in any national and international
institutes of high repute. This course also makes the students cognizant on various features of
teaching, learning, and research. Students after completion of this course are expected to
operate the mathematical projects and magnify their skills in writing various research articles
and to publish the same in national and international reputed journals.

First Semester
Sl. Subject Subject Title Internal External Credits
No Code
1. MATH C101 PARTIAL DIFFERENTIAL EQUATIONS 30 70 4
AND ITS APPLICATIONS
2. MATH C102 TOPOLOGY 30 70 4
3. MATH C103 ALGEBRA-I 30 70 4
4. MATH C104 ELEMENTARY COMPLEX ANALYSIS 30 70 4
5. MATH C105 NUMERICAL ANALYSIS AND ITS 30 70 4
APPLICATIONS
6. MATH C106 INDIAN KNOWLEDGE SYSTEM IN 30 70 4
MATHEMATICS

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 Second Semester
7. MATH C201 ABSTRACT MEASURE 30 70 4
8. MATH C202 ADVANCED CALCULUS 30 70 4
9. MATH C203 ALGEBRA-II 30 70 4
10. MATH C204 ADVANCED COMPLEX ANALYSIS 30 70 4
11. MATH C205 GRAPH THEORY 30 70 4
12. MATH VAC206 AN INTRODUCTION TO MATLAB Grade Non-Credits
Third Semester
13. MATH C301 FUNCTIONAL ANALYSIS-I 30 70 4
14. MATH C302 NUMBER THEORETIC 30 70 4
CRYPTOGRAPHY - I
Elective - I A Student is allowed to opt any two papers
15. MATH E303 OPTIMIZATION TECHNIQUES-I 30 70 4
16. MATH E304 ORDINARY DIFFERENTIAL 30 70 4
EQUATIONS-I
17. MATH E305 MATRIX TRANSFORMATIONS IN 30 70 4
SEQUENCE SPACES-I
18. MATH E306 FLUID DYNAMICS-I 30 70 4
19. MATH E307 ABSTRACT MEASURE AND PROBABILITY-I 30 70 4
20. MATH E308 FUZZY SETS AND FUZZY LOGIC 30 70 4
21. MATH E309 MATHEMATICAL STATISTICS 30 70 4
22. MATH VAC310 AN INTRODUCTION TO LATEX Grade Non-Credits
CBCT Course Other Department students will opt this paper
23. MATH CT300 MATHEMATICAL METHODS 30 70 4
Fourth Semester
24. MATH C401 FUNCTIONAL ANALYSIS-II 30 70 4
25. MATH C402 NUMBER THEORETIC 30 70 4
CRYPTOGRAPHY-II
26. MATH D408 DISSERTATION 100 4
Elective - II A Student is allowed to opt any two papers
27. MATH E403 OPTIMIZATION TECHNIQUES-II 30 70 4
28. MATH E404 ORDINARY DIFFERENTIAL 30 70 4
EQUATIONS-II
29. MATH E405 MATRIX TRANSFORMATIONS IN 30 70 4
SEQUENCE SPACES-II
30. MATH E406 FLUID DYNAMICS-II 30 70 4
31. MATH E407 ABSTRACT MEASURE AND 30 70 4
PROBABILITY- II
32. MATH AC409 CULTURAL HERITAGE OF SOUTH ODISHA Grade Non-Credits
Total 2100 84

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Total Credit: 84
C- Core Course - 1500 (Mandatory with no choice)
E- Elective - 500 (Mandatory with choice departmentally)
CT- Credits Transformation - 100 (Students of Mathematics shall opt for CBCT courses
offered by other Departments)
VAC – Value Added Course (Non-Credits), AC - Add on Course (Non-Credits)
SWAYAM COURSE: All P.G. students are required to complete one SWAYAM course (Minimum
2 credits) on or before completion of 3rd Semester.
Dissertation – 100
Internal-30(Attendance-05+ Quizz-05+ Written Assesment-20)

DETAILED SYLLABUS
FIRST SEMESTER
Sub. Code: MATH C101 Partial Differential Equations and its Applications
Semester: I Credit: 4 Core Course
Pre-requisites: Basic knowledge in ordinary and partial differential equations
Course Outcome:

 To solve the Cauchy problems and wave equations with homogeneous and
Nonhomogeneous equations.
 To solve Eigen value Problems and Special Functions, Boundary Value Problems
of partial differential equations.
 To solve partial differential equations by applying Fourier Transforms and Laplace
Transforms.

Unit-I 10 hours

Basic Concepts and Classification of Second Order Linear Equations.

Unit-II 10 hours

The Cauchy Problem and Wave Equations, Method of Separation of Variables.

Unit-III 10 hours

Eigen value Problems and Special Functions, Boundary Value Problems.

Unit-IV 10 hours

Fourier Transforms and Laplace Transforms.

Text Book:
Tyn Myint, U. & Lokenath Debnath: Linear Partial Differential Equations for Scientists
and Engineers, Birkhauser Pub. (4th Edition). Chapters: 1(1.2-1.6), 4, 5(5.1-5.7), 7, 8, 9, 12
(12.1-12.6, 12.8-12.11).
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Reference Book:
Tyn Myint, U.: Partial Differential Equations of Mathematical Physics. (Elsevier Pub.)

Sub. Code: MATH C102 Topology

Semester: I Credit: 4 Core Course
Pre-requisites: Basic knowledge in Sets and Functions
Course Outcome:

 To learn about different Topological spaces, Open sets, Closed sets, Connected Sets
and Compact sets.
 To understand the Metric spaces, Regular and Normal Spaces.

Unit-I 10 hours

Open sets and Limit points, Closed sets and Closure, Bases and relative Topologies.

Unit-II 10 hours

Connected Sets and Components, Compact and Countable compact spaces, Continuous
functions, Homeomorphisms.

Unit-III 10 hours

T0 -and T1-spaces & sequence, T 2 Spaces, Regular and Normal Spaces, Completely regular
Spaces.

Unit-IV 10 hours

Urysohn's lemma, Urysohn's Metrization theorem, Finite products, Product invariant

properties, Metric products, Product topology.

Text Book:
W. J. Pervin: Foundations of General Topology, Academic Press. Chapters: 3(3.1, 3.2
and 3.4), 4(4.1 to 4.4), 5(5.1 to 5.3, 5.5 and 5.6), 8(8.1 to 8.4), 10(10.1 only).

Reference Books:
1. J. R. Munkers: Topology-A First Course, Prentice Hall, 1996.
2. K. D. Joshi: Introduction to General Topology, Willey Eastern Ltd., 1983.

Sub. Code: MATH C103 Algebra-I

Semester: I Credit: 4 Core Course
Pre-requisites: Basic concepts in group theory and ring theory
Course Outcome:

 To study p- Sylow’s Subgroups of a finite Group.

 To construct the maximal Ideals by using irreducible polynomials.

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  To learn about finite extension field, Algebraic element and transcendental
numbers.

Unit-I 10 hours

Automorphisms, Cayley's Theorem, Permutation Groups, Another Counting Principle.

Unit-II 10 hours

Sylow's Theorems, More Ideals and Quotient Rings, The Field of Quotients of an Integral
Domain, Euclidean Rings, A Particular Euclidean Ring.

Unit-III 10 hours

Polynomial Rings, Polynomial Rings over the Rational Field, Elementary Basic Concepts of
Vector Space, Linear Independence and Bases.

Unit-IV 10 hours

Extension Fields, The Transcendence of e, Roots of Polynomials, Construction with

Straightedge and Compass, More about Roots.

Text Book:
I. N. Herstein: Topics in Algebra, John Wiley and Sons (2nd Edition) 2002. Chapters:
2(2.8 to 2.l2), 3(3.5 to 3.10), 4(4.1, 4.2), 5(5.1 to 5.5).

Reference Books:
1. S. Singh end Q, Zameeraddin: Modern Algebra, Vikas Publishing House, 1590.
2. P. B .Bhattacharya. S. K. Jain and S. R. Nagpal: Basic Abstract Algebra,
Cambridge University Press, 1995.

Sub. Code: MATH C104 Elementary Complex Analysis

Semester: I Credit: 4 Core Course
Pre-requisites: Basic concepts in complex numbers and complex functions
Course Outcome:

 To find an analytic functions when its real or imaginary part is known.

 To establish a linear transformation through cross ratio.
 To compute the complex integrations.

Unit-I 10 hours

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Complex Numbers.

Unit-II 10 hours

Complex Functions.

Unit-III 10 hours

Conformality and Linear Transformations

Unit-IV 10 hours

Complex Integration: Fundamental theorems, Cauchy's Integral formula, Local properties

of analytic functions, Complex integration continued: General form of Cauchy's theorem.

Text Book:
Lars V. Ahlfors: Complex Analysis, McGraw-Hill International Editions (3rd Edition).
Chapters: 1, 2, 3 (2.1 to 2.4, 3.1 to 3.3), 4 (4.1 to 4.4).

Sub. Code: MATH C105 Numerical Analysis and its Applications

Semester: I Credit: 4 Core Course
Pre-requisites: Basic knowledge in interpolation and approximation, numerical
integration and differentiation.
Course Outcome:

 To obtain the interpolating polynomial by using different methods.

 To solve numerical integration by using various numerical methods.
 To solve the ordinary differential equations (IVP) by single and multi step methods.

Unit-I 10 hours

Interpolation & Approximation: Introduction, Lagrange and Newton interpolations, finite

difference operators, Interpolating Polynomials using finite differences, Hermite
Interpolation, Piecewise and spline interpolation.

Unit-II 10 hours

Interpolation and Approximation (contd.): Bivariate interpolations, Approximation, least

square approximation, uniform approximation, Rational approximation, choice of the
method.

Unit-III 10 hours

Differentiation and Integration: Introduction, Numerical differentiation, Optimum choice

of step length, extrapolation method, partial differentiation, Numerical Integration,

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Methods based on interpolation. Methods based on undetermined coefficients, Composite
Integration methods, Romberg Integration, Double integration.

Unit-IV 10 hours

Ordinary Differential Equations, Initial Value Problems: Introduction, Differential

Equations, Numerical methods, single step methods, stability analysis of single step
methods, Multi step methods.

Text Book:
M. K. Jain, S. R. K. Iyengar and R. K. Jain: Numerical Methods for Science and
Engineering Computations (4th Edition), New Age International Publishers, 2003. Chapters:
4, 5, 6(6.1 to 6.6).

Sub. Code: MATH C106 Indian Knowledge System In Mathematics

Semester: I Credit: 4 Core Course
Pre-requisites: Some ideas about the Indian Mathematicians and their Biography.
Course Outcome:

 To identify the Ancient Indian Mathematicians.

 To learn about the pioneering contributions of Indian Mathematicians.

Unit-I 10 hours
Mathematical Thought in Vedic India: The Vedas and mathematics, The Sulaba-sutras, The
Jyotisa-vedanga; The Genre of Medieval Mathematics: Chapters on mathematics in
siddhantas, The Bakhshali Manuscript, The Ganita-sara-sangraha

Unit-II 10 hours
The Development of “Canonical” Mathematics: Mathematicians and society, The
“standard” texts of Bhaskara (II), The School of Madhava in Kerala: Background, Lineage,
Infinite series and other mathematics.

Unit-III 10 hours
Congruences for p(n) and τ(n): Historical Background, Elementary Congruences for τ(n),
Ramanujan's Congruence p(5n+ 4) ≡ 0 (mod 5), Ramanujan's Congruence p(7n + 5) ≡ 0
(mod 7), The Parity of p(n).

Unit-IV 10 hours
The Rogers-Ramanujan Continued Fraction: Definition and Historical Background, The
Convergence, Divergence, and Values of R(q), The Rogers-Ramanujan Functions, Identities
for R(q), Modular Equations for R(q).

Text Books:

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 1. Kim Plofker: Mathematics in India, Princeton University Press, 2008, Chapters:
2(2.1, 2.2, 2.4), 5(5.1, 5.2, 5.3), 6(6.1, 6.2), 7(7.1, 7.2, 7.3).
2. Bruce C. Berndt: Number theory in the spirit of Ramanujan, Student Mathematical
Library, Volume 34, American Mathematical Society, Providence, Rhode Island,
Chapters: 2(2.1, 2.2, 2.3, 2.3, 2.4, 2.5), 7(7.1, 7.2, 7.3, 7.4, 7.5).
Reference Books:
1. Eric Temple Bell: Men of Mathematics, Simon and Schuster, Reissue Edition, 1986.
2. C.D. Olds: Continued Fraction, Random House and the I.W. Singer Company.

SECOND SEMESTER
Sub. Code: MATH C201 Abstract Measure
Semester: II Credit: 4 Core Course
Pre-requisites: Sets, Functions, Differentiation and Integration.
Course Outcome:

 To identify the measurable sets and measurable functions.

 To learn about Lebesgue Integrable functions.

Unit-I 10 hours

Introduction, Outer Measure, Measurable sets and Lebesgue Measure, A non-

Measurable set, Measurable functions, Littlehood's three Principles.

Unit-II 10 hours

The Lebesgue Integral.

Unit-III 10 hours

Differentiation and Integration.

Unit-IV 10 hours

The classical Banach Spaces.

Text Book:
H. L. Royden: Real Analysis (MacMillan Pub.) Chapters: 3, 4, 5, 6.

Sub. Code: MATH C202 Advanced Calculus

Semester: II Credit: 4 Core Course
Pre-requisites: Limit, Continuity and Differentiability of real valued functions.
Course Outcome:

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  To understand the derivatives for functions of several variables, Differentiations of
transformations and Inverse of transformations.
 To exhibit the set function, transformation of multiple Integrals.

Unit-I 10 hours

Derivatives for Functions on Rn - Differentiation of composite functions, Taylors Theorem.

Unit-II 10 hours

Transformations, Linear function and transformations, Differentials of transformations,

Inverse of transformations.

Unit-III 10 hours

Implicit function theorems, functional dependence, set function, transformation of

multiple Integrals.

Unit-IV 10 hours

Curves and Arc length, surfaces and surface area, Integrals over curves and surface,
Differential forms, Theorem of Green, Gauss and stokes, exact form and closed form.

Text Book:
R. C. Buck: Advanced Calculus (3rd Edition), McGraw Hill. Chapters: 3(3.3 to 3.3),
7(7.2 to 7.7), 8(8.2 to 8.6), 9(9.2, 9.4, 9.5).

Sub. Code: MATH C203 Algebra-II

Semester: II Credit: 4 Core Course
Pre-requisites: Basic knowledge in Linear transformation and inner product spaces
Course Outcome:

 To understand the basic knowledge of Golois Group and solvability by radicals.

 To gain the knowledge about the triangular, Nilpotent and Jordan Form of the linear
transformation.
 To know the Application of Hermitian, Unitary and normal Transformations.

Unit-I 10 hours

Dual Spaces, Inner Product Spaces, The Elements of Galois Theory, Solvability by Radicals.

Unit-II 10 hours

The Algebra of Linear Transformation, Characteristic Roots, Matrices.

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Unit-III 10 hours

Canonical Forms 1 Triangular Form, Nilpotent Transformations, Jordan Form.

Unit-IV 10 hours

Trace and Transpose Determinants, Hermitian, Unitary and normal Transformations.

Text Book:
I. N. Herstein: Topics in Algebra, John Wiley and Sons (2nd Edition), 2002.
Chapters: 4(4.3, 4.4), 5(5.6, 5.7), 6(6.1 to 6.6, 6.8 to 6.10).

Reference Books:
1. S. Singh end Q, Zameeraddin: Modern Algebra, Vikas Publishing House, 1590.
2. P. B.Bhattacharya, S. K. Jain and S. R. Nagpal: Basic Abstract Algebra,
Cambridge University Press, 1995.

Sub. Code: MATH C204 Advanced Complex Analysis

Semester: II Credit: 4 Core Course
Pre-requisites: Knowledge in Power series and special functions.
Course Outcome:

 To learn about various types of power series expansions and some special
functions.

Unit-I 10 hours

Complex Integration Calculus of Residues.

Unit-II 10 hours

Series and Product development: Power series expansion, partial fraction and
factorization.

Unit-III 10 hours

Series and product development continued: Entire function, Riemann Zeta Function.

Unit-IV 10 hours

Elliptic Functions: Simple periodic functions and Double periodic functions, Elliptic
Functions, Weierstrass Theory.

Text Book:

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 Lars V. Ahlfors: Complex Analysis, McGraw-Hill International Editions (3rd Edition).
Chapters: 4 (4.5), 5(5.1 to 5.4), 7(7.1 to 7.3).

Sub. Code: MATH C205 Graph Theory

Semester: II Credit: 4 Core Course
Pre-requisites: Basic knowledge in graphs
Course Outcome:

 To learn about various types of graphs and trees.

 To understand the colouring of the graphs.

Unit-I 10 hours

Introduction to Graphs.

Unit-II 10 hours

Trees and Connectivity.

Unit-III 10 hours

Euler Tours and Hamiltonian Cycles: Euler Tours, Hamiltonian graphs, Planar Graphs:
Plane and Planar Graphs, Euler’s Formula, Kuratowski’s Theorem.

Unit-IV 10 hours

Colouring.

Text Book:
John Clark and D. A. Holton: A First Look at Graph Theory, World Scientific and
Allied Publishers. Chapters: 1, 2, 3(3.1, 3.3), 5(5.1, 5.2 & 5.4), 6.
Reference Book:
N. Deo: Graph Theory and Applications to Engineering, Anil Computer Sciences,
Prentice Hall of India.

Sub. Code: MATH VAC206 An Introduction to MATLAB

Semester: II Credit: Nil Non-Credits Course
Pre-requisites: Basic knowledge of computer
Course Outcome:

 To analyze and design systems.

Unit-I 10 hours

Introduction: Matrices and arrays.

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Unit-II 10 hours

Basic functions and commands.

Unit-III 10 hours

Simulink: image processing, machine learning, parallel computing and more similar
concepts.

Unit-IV 10 hours

Modelling and Simulations.

Text Book:
MATLAB Programming, The MathWorks, Inc.(Pub.), Chapters: 1, 2, 3, 4, 5 and 6.

Sub. Code: MATH SC207 SWAYAM COURSE

Semester: II Credit: 4 Core Course
Students will have to opt any one of the Course from Mathematics Discipline from
SWAYAM PORTAL
Course Outcome:

 To learn about Mathematical Concepts and its Applications

 To acquire the knowledge of Mathematical Sciences

THIRD SEMESTER

Sub. Code: MATH C301 Functional Analysis-I

Semester: III Credit: 4 Core Course
Pre-requisites: Basic knowledge in linear space and different types of functions
Course Outcome:

 To learn about Normed spaces and Banach spaces

 To acquire the knowledge of Application of Uniform Boundedness Principle,
Closed Graph and Open Mapping Theorem.

Unit-I 10 hours

Normed spaces, Continuity of linear maps.

Unit-II 10 hours

Hahn-Banach Theorems, Banach spaces.

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Unit-III 10 hours

Uniform Boundedness Principle, Closed Graph and Open Mapping Theorems, Bounded
Inverse Theorem.

Unit-IV 10 hours

Spectrum of a Bounded operator, Duals and Transposes.

Text Book:
B. V. Limayee: Functional Analysis, New Age International Ltd. (2nd Edition).
Chapters: 5, 6, 7(Except Banach Limits), 8, 9(Except Divergence of Fourier Series of
continuous Functions and Matrix Transformations and Summability Methods), 10, 11, 12
(12.1 to12.6) and 13 (13.1 to 13.5).

Sub. Code: MATH C302 Number Theoretic Cryptography-I

Semester: III Credit: 4 Core Course
Pre-requisites: Basic knowledge in number theory
Course Outcome:

 To able time estimates for doing arithmetic, Divisibility and Euclidean algorithm.
 To able the factoring large number, to find the quadratic residues in Finite fields.
 To solve the some cryptosystems problems with enciphering matrices.
 To solve the cryptosystems problems by using RSA.

Unit-I 10 hours

Time estimates for doing arithmetic, Divisibility and Euclidean algorithm, Congruences,
Some applications to factoring.

Unit-II 10 hours

Finite fields, Quadratic residues and Reciprocity.

Unit-III 10 hours

Some simple Cryptosystems, Enciphering Matrices.

Unit-IV 10 hours

The idea of public key Cryptography, RSA.

Text Book:
Neal Koblitz: A Course In number theory and Cryptography, Springer Verlag, GTM
No. 114; (1987). Chapters: 1, 2, 3, 4(4.1 and 4.2).

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Reference Book:
J. Menezes, P. C. Van Oorchot and Scoff A. Vanstone: Hand Book of Applied
Cryptography, CRC Press (1997).

Sub. Code: MATH E303 Optimization Techniques-I

Semester: III Credit: 4 Core Course
Pre-requisites: Basic knowledge in operation research
Course Outcome:

 To solve the integer programming problems by applying different type of methods.

 To solve the game theory problems by using linear programming, graphical
methods and dominance principal.

Unit-I 10 hours

Integer Programming: Gomory's Algorithm for pure integer linear programs, Gomory's
mixed integer- continuous variable algorithm, Branch and Bound methods.

Unit-II 10 hours

Kuhn-Tucker optimality conditions: Some theorems, Kuhn-Tucker first order necessary

optimality conditions, Second order optimality condition, Lagranges method.

Unit-III 10 hours

Convex programming problem, Sufficiency of Kuhn-Tucker conditions, Legrangian saddle

point and duality, duality for convex programs.

Unit-IV 10 hours

Game Theory : Game theory problem, Two person zero sum Game, Finite matrix Game,
Graphical method for 2xn and mx2 matrix games, Some theorems, Dominance principal.

Text Book:
N. S. Kambo: Mathematical Programming, Affiliated EWP Ltd. New Delhi. Chapters:
6(6.4 to 6.6), 7(7.1 to 7.4), 8, 16.

Sub. Code: MATH E304 Ordinary Differential Equations-I

Semester: III Credit: 4 Elective Course
Pre-requisites: Derivative and Differential equations with solutions.

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Course Outcome:

 To solve the linear differential equations of higher order with variable coefficients
and constant coefficients.
 To learn the existence and uniqueness of solutions of first order ordinary differential
equations with initial conditions and systems of first order ordinary differential
equations with constant coefficients.

Unit-I 10 hours

Basic Concepts and Linear Equations of the First Order.

Unit-II 10 hours

Linear Differential Equations of Higher Order.

Unit-III 10 hours

Systems of Linear Differential Equations, Systems of First Order Equations, Existence and
Uniqueness Theorems, Fundamental Matrix Non Homogeneous Linear Systems, Systems
of Linear Differential Equations, Continued Linear Systems with Constant Coefficients,
Linear System with Periodic Coefficients.

Unit-IV 10 hours

Equations with Deriving Arguments, Existence and Uniqueness of Solutions.

Text Book:
S. G. Deo. V. Lakhimikantbam, V. Raghavendra: Text Book of Ordinary Differential
Equations (2nd Edition), Tata-Mc Graw-Hill Publishing Company Ltd. New Delhi. Chapters:
1, 2(except 2.10), 4, 5, 11.

Sub. Code: MATH E305 Matrix Transformations in Sequence Spaces-I

Semester: III Credit: 4 Elective Course
Pre-requisites: Knowledge in Infinite series, sequence of real numbers
Course Outcome:

 To learn about different types of limitation methods for matrix transformations.

 To understand various matrices such as Norlund and Riesz Musos, Scbur Matrices,
Cesaro and Holder Matrices, etc.

Unit-I 10 hours

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Limitation Methods: Limitation methods, Examples of Limitation Methods, Matrix
Limitation Methods, Norlund and Riesz Musos.

Unit-II 10 hours

Limitation Methods: Scbur Matrices: Consistency of Matrix Methods.

Unit-III 10 hours

Some particular Limitation Matrices: Norlund Mean, Cesaro and Holder Matrices.

Unit-IV 10 hours

Hausdorff Methods, Abels method, Tauberin Theorem, Banach Limits, Strongly Regular
Matrices, Counting function.

Text Book:
G. N. Peterson: Regular Matrix Transformation, McGraw-Hill Publishing Company.
Chapters: 1, 2, 3(3.1 to 3.3).

Sub. Code: MATH E306 Fluid Dynamics-I

Semester: III Credit: 4 Elective Course
Pre-requisites: Ordinary and Partial differential equations with solutions
Course Outcome:

 To study different types of fluids and various governing equations of it.

 To solve equations of the flow of viscous compressible and incompressible fluids.

Unit-I 10 hours

Kinematics of Fluids, Methods describing Fluid motion. Legarangian and Eulerian

Methods. Translation Rotation and Rate of Deformation. Streamlines, Pathlines and
Streaklines. The Material derivative and Acceleration Vorticity in Polar and Orthogonal
Curvilinear Coordinates.

Unit-II 10 hours

Fundamental equations of the flow of viscous compressible fluids, Equations of continuity,

motion and energy is Cartesian coordinate systems.

Unit-III 10 hours

The equation of state. Fundamental equations of continuity, motion and energy in

Cylindrical and Spherical coordinates.

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Unit-IV 10 hours

2-D and 3-D in viscid incompressible flow. Basic equations and concepts of flow.
Circulation theorems, Velocity potential, Rotational and Irrotational flows. Integration of
the equations of motion. Bernoulli's Equation, The momentum theorem and the moment
of momentum theorem. Laplace's equations in different coordinate systems. Stream
function in 2-D motion.

Text Book:
S. W. Yuan: Foundations of Fluid Mechanics, Prentice-Hall of India. Chapters: 3, 5
(5.1 to 5.6), 7 (7.1 to 7.9).

Sub. Code: MATH E307 Abstract Measure and Probability-I

Semester: III Credit: 4 Elective Course
Pre-requisites: Basic concept in Probability and set theory
Course Outcome:

 To introduce the Measures on Boolean semi-Algebra and -algebra.

 To understand the several Distributions such as Binomial Distribution, Poisson
Distribution and Normal Distribution and several Approximations to such
Distribution.

Unit-I 10 hours

Sets and Events, Probability on Foslesn Algebra, Probability Diminutions and Elementary
Random Variables, Repeated Trials and Statistical Independence, Poisson Approximation
to the Binomial Distribution, Normal Approximation to Binomial Distribution.

Unit-II 10 hours

Multivariate Normal Approximation to Multinomial Distribution, some applications of the

normal approximation. Independent simple Random variables and central limit theorem,
Conditional probability, Law of large numbers An application of the law of large numbers
to a problem is Analysis.

Unit-III 10 hours

-algebra and Borel spaces, Monotone classes, Measures on Boolean semi-Algebra and
Algebra Extension of Measure to -Algebra, Uniqueness of extensions of measures.

Unit-IV 10 hours

Extension and completion of measures, measures on matrix spaces, probability contents,

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the lebesgue measure on the Real line, Elementary properties of Borel Maps, Borel Maps
into Matrix Spaces, Borel Maps on measure Spaces.

Text Book:
K. R. Parthsarathy: Introduction to probability and measure, MacMillan Company.
Chapters: 1, 2, 3 (22, 23, 24).

Sub. Code: MATH E308 Fuzzy Sets and Fuzzy Logic

Semester: III Credit: 4 Core Course
Pre-requisites: Sets, Functions and Relations
Course Outcome:

 To introduce Fuzzy sets versus crisp sets, types of Fuzzy set.

 To learn about Fuzzy Arithmetic, Fuzzy numbers, Fuzzy Relation.

Unit-I 10 hours

From Classical (CRISP) sets to Fuzzy sets: Fuzzy sets: Basic types, Basic concept. Fuzzy
sets versus crisp sets: Additional properties of α-cuts, Representations of fuzzy sets,
extension principle of fuzzy sets.

Unit-II 10 hours

Operations on Fuzzy sets: Types of operations, Fuzzy complements, Fuzzy intersections:

t-norms, Fuzzy unions: t-conorms, Combinations of Operations, Aggregation operations.

Unit-III 10 hours

Fuzzy Arithmetic: Fuzzy numbers, linguistic variables, Arithmetic operations on Intervals

and Fuzzy numbers, Lattice of Fuzzy numbers, Fuzzy equations.

Unit-IV 10 hours

Fuzzy Relation: Crisp versus Fuzzy relations, Projections and cylindric extensions, Binary
Fuzzy relations, Binary relations on a single set, Fuzzy equivalence relations, compatibility
relations and ordering relations, Fuzzy morphisms, Sup-i compositions of Fuzzy relations,
Inf-𝑤𝑖 compositions of Fuzzy relations.

Text Book:
George J. Klir & Bo Yuan: Fuzzy sets and Fuzzy Logic: Theory and Applications,
Prentice Hall PTR under Saddle River, New Jersey 07458.

Reference Books:

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 1. S. K. Pundir and R. Pundir: Fuzzy sets and their applications, A Pragati Editions,
8th Editions.
2. A. K. Bhargava: Fuzzy set theory fuzzy logic and their applications, S. Chand &
Co, New Delhi.

Sub. Code: MATH E309 Mathematical Statistics

Semester: III Credit: 4 Core Course
Pre-requisites: Basic knowledge in probability theory
Course Outcome:

 To solve the probability problems of discrete and continuous random variables.

 To solve the probability problems of probability distribution and generating
functions.

Unit-I 10 hours

Elements of Theory of Probability : Classical definition of probability, Theorems on

probability of union of events, Conditional probability : Theorem of compound
probability, Independence of events, The Bayes Theorem, Statistical and empirical
definition of probability, Geometric probability, Axiomatic definition of probability,
Conditional probability (Axiomatic definition of probability).

Unit-II 10 hours

Probability distribution on R: Random variables, probability distribution of a random

variables, discrete and continuous random variables, independent random variables,
lebesgue-stieltjes integrals, Integration of a random variables.

Unit-III 10 hours

Some characteristic of probability distribution: Expectation, Moments, some inequalities

concerning moments, Different measures of central tendency, measures of dispersion,
Measures of skewness and kurtosis, some probability inequalities.

Unit-IV 10 hours

Generating functions: probability generating function, Moment generating function,

Factorial generating function, Cummulant generating function, characteristic function,
Exercises, Some discrete distribution on R: The discrete uniform distribution, the
Bernoulli distribution, the binomial distribution, The hypergeometric distribution, The
Poisson distribution, The geometric distribution, The negative binomial distribution, The
power series distribution.

Text Book:

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 Parimal Mukhopadhyay: Mathematical Statistics, Books and Allied (P) Ltd. Kolkata.
Chapters: 1, 2, 3, 4 and 5.

Reference Books:
1. Robert V. Hogg and Allen T. Craig: Introduction to mathematical statistics,
Pearson Education Asia, Indian Branch :482 F.I.E Pratapgaanj, Delhi 110092
2. John E. Freund and Ronald E. Walpole: Mathematical statistics, Prentice Hall
India Pvt. Ltd. New Delhi-110001.

Sub. Code: MATH VAC310 An Introduction to LATEX

Semester: III Credit: Nil Non-Credits Course
Pre-requisites: Knowledge about computer programming.
Course Outcome:

 To be capable to write a research article in LaTeX.

Unit-I 10 hours

Basics: Introduction to LaTeX, Text, Symbols and Commands, Document layout and
organization, displayed text.

Unit-II 10 hours

Mathematical formulas, Graphics inclusion and color.

Unit-III 10 hours

Floating tables and figures, User customizations.

Unit-IV 10 hours

Beyond the Basics: Document management, Postscript and PDF, Bbliographic data bases
and BiBTeX, Presentation material.

Text Book:
Helmut Kopka & Patrick W. Daly: A Guide to LATEX and Electronic Publishing
(Fourth Edition), Addison-Wesley Longman Ltd. Chapters: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15.

Sub. Code: MATH CT300 Mathematical Methods

Semester: III Credit: 4 CBCT Course
Pre-requisites: knowledge of sets, functions, limit, differentiation, Interpolation
Course Outcome:

 To solve functions using limit, differentiation.

 To obtain the interpolating polynomial by using different methods.
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  To solve numerical integration by using various numerical methods.

Unit-I 10 hours

Transcendental and polynomial equations: Introduction, Bisection method, Iteration

methods based on first degree equation, Rate of convergence of Secant method, Regula-
Falsi method, Newton-Raphson method; System of Linear Algebraic equations:
Introduction, Direct methods, Cramer Rule, Gauss elimination method, Gauss-Jordan
elimination method.

Unit-II 10 hours

Interpolation & Approximation: Introduction, Lagrange and Newton interpolations, finite

difference operators, Interpolating Polynomials using finite differences, Hermite
Interpolation, Piecewise and spline interpolation.

Unit-III 10 hours

Limit and Continuity of real valued functions.

Unit-IV 10 hours

The Derivatives, Maxima and Minima.

Text Books:
1. M. K. Jain, S. R. K. Iyengar and R. K. Jain: Numerical Methods for Science and
Engineering Computations (4th Edition), New Age International Publishers, 2003.
Chapters: 2(2.1 to 2.3, 2.5), 3(3.1, 3.2), 4(4.1 to 4.6).
2. Shanti Narayan and M. D. Raisinghania: Elements of Real Analysis, S. Chand &
Company Pvt. Ltd., New Delhi. Chapter: 8(8.1 to 8.21), 9(9.1 to 9.6), 11(11.1 to 11.4).

FOURTH SEMESTER

Sub. Code: MATH C401 Functional Analysis-II

Semester: IV Credit: 4 Core Course
Pre-requisites: Basics concepts in convergence of sequence and inner product
spaces.
Course Outcome:

 To learn the Weak and Weak *convergence Reflexivity.

 To Normal, Unitary and Self-Adjoint Operators.

Unit-I 10 hours

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Weak and Weak *convergence Reflexivity.

Unit-II 10 hours

Inner product spaces, Orthonormal sets.

Unit-III 10 hours

Approximation and Optimization Projection and Riesz Representation Theorems.

Unit-IV 10 hours

Bounded Operators and Adjoints, Normal, Unitary and Self-Adjoint Operators.

Text Book:
B. V. Limayee: Functional Analysis, New Age International Ltd. (2nd Edition).
Chapters: 15, 16 (16.1 to 16.3), 21, 22, 23, 24, 25 and 26 (26.1 to 26.5).

Sub. Code: MATH C402 Number Theoretic Cryptography-II

Semester: IV Credit: 4 Core Course
Pre-requisites: Basic ideas of RSA, Factorization in finite fields, primes.
Course Outcome:

 To solve the Discrete log problems by using Silver-Pihlog-Samir method and

Knapsack problems.
 To find the factor of large number.

Unit-I 10 hours

Discrete log, Knapsack.

Unit-II 10 hours

Zero knowledge protocols and oblivions transfer, pseudo primes.

Unit-III 10 hours

The rho method, Fermat factorization and factor bases.

Unit-IV 10 hours

The continued fraction method, The quadratic sieve method.

Indian Knowledge System: Contributions of Indian Mathematician Srinivas Ramanujan
on continued fraction.

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Text Book:
Neal Koblitz: A Course on number theoretic Cryptography, Springer Verlag, GTM
No. 114 (1987). Chapters: 4(4.3, 4.4, 4.5), 8.

Reference Book:
J. Menezes. P. C. Van Oorschot and Scott A. Vanstone: Hand Book of Applied
Cryptography, CRC Press (1997).

Sub. Code: MATH D408 Dissertation

Semester: IV Credit: 4 Core Course
Pre-requisites: All semester theory.
Course Outcome:

 To acquire knowledge for writing research proposal for pursuing higher studies in
mathematics.

Course Details:
Chapter Contents Hours
1 Literature Review 15
2 Learning objectives 15
3 Dissertation work 50
4 Report writing in proper format 20
Total 100

NB: 1. The students will be informed regarding their supervisors. Each student has to work for
at least 100 hours for writing his/her dissertation under the guidance.
2. The research work will be submitted in the form of a dissertation before one week of last
theory examination/as instructed by HOD. The student has to present his/her work in power
point before the External examiner and internal examiners for evaluation.

Sub. Code: MATH E403 Optimization Techniques-II

Semester: IV Credit: 4 Core Course
Pre-requisites: Basic knowledge in operation research
Course Outcome:

 To solve the quadratic programs by using Wolfe's algorithm, Beales Algorithm,

Fletchers method.
 To solve the non linear programs by using Frank-Wolfe's method, Reduced
gradient method and Kelley’s cutting method.
 To solve the Geometric programming.

Unit-I 10 hours

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Quadratic program, Wolfe's algorithm, Beales Algorithm, Fletchers method.

Unit-II 10 hours

Dual quadratic program, Complementarity problem.

Unit-III 10 hours

Nonlinear programming methods: Frank-Wolfe method, Reduced Gradient method,

Kelley's cutting plane method.

Unit-IV 10 hours

Geometric programming: Proto type primal and dual Geometric Programs, Reduction to
proto type Geometric program, Dynamic Programming: Principle of optimality, Reliability
of system in series, Height of projectile, Cargo-Loading problem, Inventory problem.

Text Book:
N. S. Kambo: Mathematical Programming , Affiliated EWP Pvt Ltd, New Delhi. Chapters:
10(10.1 to 10.5, 10.8), 11(11.1 to 11.3), 12 (12.1 to 12.2), 15 (15.1 to 15.5).

Sub. Code: MATH E404 Ordinary Differential Equations -II

Semester: IV Credit: 4 Elective Course
Pre-requisites: Basic knowledge in ordinary differential equations and its solutions
Course Outcome:

 To analyze the stability of Nonlinear Systems of first order ordinary differential

equations.
 To explain the oscillatory solutions of Nonlinear Differential Equations.

Unit-I 10 hours

Analysis and Methods of Nonlinear Differential Equations.

Unit-II 10 hours

Boundary Value Problems.

Unit-III 10 hours

Oscillations of Second Order Equations.

Unit-IV 10 hours

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Stability of Linear and Nonlinear, Systems: Elementary Critical Points, System of
Equations with constant coefficients, linear Equations with constant coefficients, Stability
of Linear and Nonlinear Systems (continued) Lyapunov stability, stability of Quasi-linear
systems, Second Order Linear Differential Equations.

Text Book:
S. G. Deo. V. Lakhsmikantham, V. Raghavendra: Text Book of Ordinary Differential
Equations (2nd Edition), Tata Mc Graw Hill Publishing Company Ltd. New Delhi. Chapters: 6,
7, 8, 9.

Sub. Code: MATH E405 Matrix Transformations in Sequence Spaces -II

Semester: IV Credit: 4 Elective Course
Pre-requisites: Convergent and divergent of sequence and series.
Course Outcome:

 To demonstrate the universal Tauberian Theorem, some special types of matrices.

 To understand the summability theory.

Unit-I 10 hours

Strongly Regular Matrices: Some Matrices of a special Type, A universal Tauberian

Theorem.

Unit-II 10 hours

Bounded sequence, Uniformly limitable sequence, Intersection of Bounded Convergence

Fluids.

Unit-III 10 hours

Set of Matrices, Bounds on Limits of sequences, Matrix Norms, Pairs of consistent

matrices.

Unit-IV 10 hours

Matrix and linear transformations Algebras of matrices, Summability, Tauberian

theorems.

Text Books:
1. O. M. Peterson: Regular Matrix Transformations, Chapters: 3 (8.4 and 3.5), 4.
2. I. J. Maddox: Elements of Functional Analysis, Cambridge University Press,
Chapter: 7.

Sub. Code: MATH E406 Fluid Dynamics-II

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Semester: IV Credit: 4 Elective Course
Pre-requisites: Basic ideas in nonlinear ODE and PDE
Course Outcome:

 To understand nonlinear Navier-Stokes equations of motion and its solutions.

 To learn about the various types of flow of fluid through different mediums.

Unit-I 10 hours

Laminar Sow of viscous incompressible fluids, Similarity' of flows, The Reynolds number,
Flow between parallel flat plates, Couette flow, plane Poiseuille flow, Steady flow in pipes,
The Hagen-Poiseuille flow, Flow between two coaxial cylinders*.

Unit-II 10 hours

Flow between two Coaxial rotating cylinders. Steady flow around a sphere Theory of very
slow motion. Unsteady motion of a flat plate.

Unit-III 10 hours

The laminary boundary layer. Properties of Navier-Stokes equations. The boundary layer,
equations in 2-D flow. The boundary layer along a flat plate. Boundary layer on a surface
with pressure gradient, Momentum integral theorems for the boundary layer.

Unit-IV 10 hours

Von Karman-Pohlhausen method. Boundary layer for axially symmetrica' flow. Separation
of boundary layer flow. Boundary layer control. Separation prevention by boundary layer
suction, The origin of turbulence. Reynolds modification of the Navier-Stokes equations
for trubulent flow. Reynolds equations and Reynolds stresses, PrandtPs mixing length
theory. The universal velocity profile near a wall. Turbulent flow in pipes, Turbulent
boundary layer over a smooth flat plate.

Text Book:
S. W. Yuan: Foundations of Fluid Mechanics, Prentice-Hall of India. Chapters: 8 (8.1
to 8.3, 8.7 to 8.8), 9, 10.

Sub. Code: MATH E407 Abstract Measure and Probability -II

Semester: IV Credit: 4 Elective Course
Pre-requisites: Vector spaces, Integration and differentiation,
Course Outcome:

 To know about Riemann and Lebesgue Integrals of different functions and

probability measure on Rn.

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  To understand the convolution theory on Lp spaces.

Unit-I 10 hours

Integration of non-negative Functions, Integration of Borel Functions, Riemann and

Lebesgue Integrals.

Unit-II 10 hours

Riesz Representation theorem, some Integral Inequality.

Unit-III 10 hours

Transition Measures and Fubinis theorem, convolution of probability measure on Rn

Lebesgue measure on Rn

Unit-IV 10 hours

Convolution Algebra L1(Rn) approximation on Lp spaces with respect to Lebesgue

Measure on Rx , Elementary properties of Banach spaces, projections in Hilbert space,
orthogonal squences.

Text Book:
K. R. Parthsarathy: Introduction to probability and measure, MacMillan Company.
Chapters: 4 (except 4.30, 4.31), 5, 6 (6.40 to 6. 42).

Sub. Code: MATH AC409 Cultural Heritage of South Odisha

Semester: IV Credit: Nil Non-Credits Course
Pre-requisites: Know about Kabi Samrat Upendra Bhanja along with the Arts,
Culture and Folk Tradition of Ganjam.
Course Outcome:

 To acquire a valuable understanding of the literary and cultural heritage of South

Odisha.
 To promote the literature and culture of Odisha on a global scale.

Unit-I 10 hours

Literary works of Kabi Samrat Upendra Bhanja

Unit-II 10 hours

Other Litterateurs of South Odisha.

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Unit-III 10 hours

Cultural Heritage of South Odisha.

Unit-IV 10 hours

Folk and Tribal Traditions of South Odisha.

Text Book:

Assessment and Expectations from Class: Mentor-Mentees class, attendance, discipline,

punctuality, doubt clearing class.

Model Questions Paper:

MA/M.Sc.-Math-

YEAR

Time : 3 hours Full Marks: 70

Answer from both the Sections as per direction.
The figures in the right-hand margin indicate marks
(Paper: )
SECTION –A
1. Answer all questions from the following : 1×10
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
SECTION -B
Answer all questions : 15×4
2.
(a)
OR
(b)
3.
(a)
OR
(b)

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4.
(a)
OR
(b)
5.
(a)
OR
(b)

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