16-03-2017 BA/B.Sc. (Hons.) V Semester Mathematics Course ...

REVISED BOS: 16-03-2017

B.A./B.Sc. (Hons.) V Semester

Mathematics

Course Title: Real Analysis I

Course Number: MMB-551

Credits: 04

Unit – I: Elements of Point Set Theory on R (14 LECTURES)

Sets, Intervals: Open and closed, Bounded and unbounded sets: Supremum and infimum,

Neighbourhood of a point, interior points, Open sets and results, Limits points of a set, Bolzano

– Weierstrass Theorem, Closed sets and related results, Countable and uncountable sets,

Compact sets and related results.

Unit – II: Limits and Continuity of Functions on R (14 LECTURES)

Limit of a function, Theorems on algebra of limits, Sequential approach, Cauchy’s criteria for

finite limits, Continuous functions, Discontinuous functions, Theorems on continuity,

Properties of continuous functions on closed intervals, Uniform continuous functions and

related results.

Unit – III: Differentiation of Functions on R (14 LECTURES)

Definitions of derivatives and related results, Increasing and decreasing functions, Darboux’s

theorem, Rolle’s theorem, mean value theorems of differential calculus and their applications,

Taylor’s theorem with various forms of remainder, Maculaurin’s theorem, Taylor’s infinite

series, Maculaurin’s infinite series expansion of some functions.

Unit – IV: Functions of Bounded Variations (14 LECTURES)

Functions of bounded variations and their properties, Variation function and related results,

Jordon theorem, Vector valued functions, Vector valued functions of bounded variation and

related results.

BOOKS RECOMMEND:

1. W. Rudin: Principles of Mathematical Analysis, Third Edition, McGraw Hill, New

York 1976.

2. S.C. Malik and Savita Arora: Mathematical Analysis, Second Edition, Wiley Eastern

Limited, New Age International (P) Limited, New Delhi, 1994.

OLD


Mathematics

Course Title: Real Analysis I

Course Number: MMB-501 Credits: 04

Unit – I: Elements of Point Set Theory on R (14 LECTURES)

Sets, Intervals: Open and closed, Bounded and unbounded sets: Supremum and infimum,

Neighbourhood of a point, interior points, Open sets and results, Limits points of a set, Bolzano

– Weierstrass Theorem, Closed sets and related results, Countable and uncountable sets, The

Heine –Borel covering Theorem, Compact sets and related results.

Unit – II: Limits and Continuity of Functions on R (14 LECTURES)

Limit of a function, Theorems on algebra of limits, Limit or a function, Sequential approach,

Cauchy’s criteria for finite limits, Continuous functions, Discontinuous functions, Theorems

on continuity, Properties of continuous functions on closed intervals, Uniform continuous

functions and related results, Functions of bounded variation and their properties, variation

function and related results, Jordon theorem.

Unit – III: Differentiation of Functions on R (14 LECTURES)

Definitions of derivatives and related results, Increasing and decreasing functions, Darboux’s

theorem, Rolle’s theorem, mean value theorems of differential calculus and their applications,

Taylor’s theorem with various forms of remainder, Maculaurin’s theorem, Taylor’s infinite

series, Maculaurin’s infinite series expansion of some functions.

Unit – IV: Real Sequences and Infinite Series (14 LECTURES)

Sequences: Bounded and convergent limit, Points of a sequence, Bolzano–Weierstrass

Theorem, Limit inferior and superior, Theorems for convergence sequences, non-convergent

(divergent) sequence, Cauchy’s general principle of convergence, Theorems on algebra of

sequences (statement only) and their application, monotonic sequences. Infinite series and their

convergence, Theorem non algebra of limits of series (without proof), Series of positive terms:

the comparison, Cauchy’s root and D’ Alembert ratio tests (without proof) and their

applications, Alternating series, Leibnitz test, absolute and conditional convergence, Series of

arbitrary terms, Abel’s and Dirichlet’s tests.

BOOKS RECOMMEND:



2. Robert G. Bartle and Donald R. Sherbert: Introduction to Real Analysis, Third Edition

Wiley India Pvt. Limited, New Delhi, 2011.


B.A./B.Sc. (Hons.) VI Semester

Mathematics

Course Title: Real Analysis II


Credits: 04

Unit – I: Real Sequence (14 LECTURES)

Concept of sequence, Limit points of a sequence, Limit inferior and superior, Convergent and

divergent sequences with related theorems; Cauchy’s general principle of convergence,

Algebra of sequence and statement of related theorems with applications, Monotonic

increasing and decreasing sequences, Several examples on this unit with solutions.

Unit–II: Real Sequence (14 LECTURES)

Introduction of infinite series, Sequence of partial sums and convergence of infinite series,

Necessary condition for the convergence of an infinite series with proof, Positive term series,

Comparison tests (first type and limit form), Cauchy root test, D’Alembert’s ratio test with

their applications, Alternating series, Leibnitz test, Absolute and conditional convergence,

Series of arbitrary terms, Abel’s and Dirichlet’s tests.

Unit – III: Riemann Integration (14 LECTURES)

Definition and existence of Riemann integral; Inequalities for Riemann integrals; Refinement

of partitions; Darboux’s theorem’; Theorems on conditions of integrability; Theorems on

integrability of the sum, difference, quotient and product of integrable functions (without

proof), Theorems on integrability of the modulus and square of integrable functions, The

Riemann integral as a limit of sums, Theorems on integrable functions, The fundamental

theorem: First and generalized first mean value; Theorems of integral calculus, Integartion by

parts, Change of variables, Second mean value theorem.

Unit –IV: Riemann Stieltjes Integration and Fourier Series (14 LECTURES)

Definition and existence of the integral, refinement of partitions, Condition of integrability and

related results, Integral as a limit of sums and related results; Fourier series; Bessel’s inequality;

Dirichlet’s criteria of convergence of Fourier series; Fourier series for even and odd functions;

Fourier series on [0,2], [-l,l] and [0,l], where 1 is a real number.

BOOKS RECOMMEND:

1. W. Rudin: Principles of Mathematical Analysis, Third Edition, McGraw Hill, New

York, 1976.



OLD


Mathematics

Course Title: Real Analysis II


Unit – I: Riemann Integration (16 LECTURES)

Definition and existence of Riemann integral, Inequalities for Riemann integrals, Refinement

of partitions, Darboux’s theorem, Theorems on conditions of integrability, Theorems on

integrability of the sum, difference, quotient and product of integrable functions (without

proof), Theorems on integrability of the modulus and square of integrable functions, The

Riemann integral as a limit of sums, Theorems on integrable functions, The fundamental

theorem, first mean value, Theorems of integral calculus, Integartion by parts, Change of

variables, Second mean value theorem.

Unit – II: Riemann Stieltjes Integration (12 LECTURES)

Definition and existence of the integral, some deduction, refinement of partitions, A condition

of integrability, Some Theorems, A definition integral as a limit sum,, some examples, Some

Important Theorems.

Unit – III: Uniform Convergence (16 LECTURES)

Pointwise convergence, Uniform convergence on an Interval, Cauchy’s Criterion for uniform

convergence, some examples, Tests for uniform convergence, A Test of uniform convergence

of sequences, Tests of uniform convergence of series, Properties of uniformly convergence

sequence and series, Uniform convergence and continuity, Uniform convergence and

integration, Uniform convergence and differentiation, Some associated examples, The

Weierstrass approximation theorem

Unit – IV: Fourier Series (12 LECTURES)

Fourier series, Some preliminaries theorem, Some definition and theorems, The main theorem,

Fourier series for even and odd functions, Half range series, Interval other than [-,], [0,2],

[-1,1], [a,b] and [0,1], where 1 is a real number.

BOOKS RECOMMEND:



2. Robert G. Bartle and Donald R. Sherbert: Introduction to Real Analysis, Third Edition

Wiley India Pvt. Limited, New Delhi, 2011.



Mathematics

Course Title: Complex Analysis


Credits: 04

Unit-I Revision of basic concepts; Triangle inequality; Roots of complex numbers; Region in

the complex plane; Functions of a complex variable; Theorems on Limits; Limits involving the

point at infinity; Continuity; Derivatives; Examples.

Unit-II Cauchy-Riemann (CR) equations; Sufficient conditions for the differentiability; Polar

form of CR equations; Analytic functions; Harmonic functions; Harmonic conjugate; Polar

form of Laplace equation; Complex valued functions; Complex line integrals/contour integrals;

Examples.

Unit-III Cauchy-Goursat theorem (without proof); Consequences of Cauchy-Goursat theorem;

Cauchy’s integral formula; Cauchy’s integral formula for higher order derivatives; Morera’s

theorem; Cauchy’s inequality; Liouville’s theorem; Fundamental theorem of algebra; Gauss’

mean value theorem; Maximum and minimum modulus theorems/principles; Examples.

Unit-IV Power series; Taylor’s series; Laurent’s series; Singular points; Classifications of

singularities; Zeros and poles of order m; Residues; Calculation of residues; Residue theorem;

Evaluation of integrals; Examples.

Recommended Book:

1. Ruel V. Churchill and James W. Brown: Complex Variables& Applications. New York

McGraw Hill, 4th Edition, 1984.

Reference Book:

1. Murray R. Spiegel: Schaum’s Outline Series: Theory and Problems of Complex

Variables.

OLD


Mathematics

Course Title: Complex Analysis


Unit- 1

Complex number system, Algebraic properties, Geometric interpretation, exponential forms,

powers and roots, Properties of moduli, Regions in complex plane, Limit, continuity and

derivatives.

Unit- 2

CR equations, sufficient conditions, polar conditions, Analytic functions, Harmonic functions,

Construction of analytic function, Line integral.

Unit- 3

Cauchy Goursat theorem, Cauchy integral formula, Derivatives of analytic function, Morera’s

and Liouville’s Th, Fundamental Theorem of Algebra, Maximum and Minimum modulus

principles, Taylor’s and Laurent sereis.

Unit- 4

Differentiations and examples, Zeros of analytic function, Residues and theorem of Residue,

Residue at poles, Evaluation of improper real integrals, Integrals involving series and coseries.

Books:

1. R.V. Churchill and J W Brown: Complex Variable & Applications. McGrow Hill,

International Book Company, London.

2. Huzoor H. Khan: Complex Analysis, Hamza Publishers, Delhi.


(OPTIONAL)


Mathematics

Course Title: Optimization Theory

Course Number: MMB656 Credits: 04

UNIT-1

Definitions and scope of O.R.(see Ch-1 in [1]), Linear programming problem (Sec:2.2 in [1]),

Formulation of linear programming problem (Sec:2.3, 2.4 in [1]), Graphical solution of L.P.P.

(Sec:3.2 in [1]), Some exceptional cases (Sec:3.3 in [1]), General L.P.P. and some definitions

(Sec:3.4 in [1]), Canonical and standard form of L.P.P. (Sec:3.5 in [1]).

UNIT-2

Hyperplane, Convex sets and their properties (Sec:0.13 in [1]), Some definitions (Sec:4.1 in [1]),

Fundamental theorem of linear programming (Theo 4.3 in [1]), Simplex method (Sec:4.3 in [1]),

Two-phase method, Big M method (Sec:4.4 in [1]), Duality in L.P.P., General Primal-Dual pair

(Sec:5.2, 5.4 in [1]), Weak duality theorem, Strong duality theorem (Sec:5.5 in[1]), Dual simplex

method (Sec:5.7 in [1]).

UNIT-3

Convex functions and their properties (Sec:7.2 in [2]), General NLPP (Sec: 27.3 in [1]),

Formulation of NLPP (Sec: 27.2 in [1]), Methods for solving NLPP: Graphical method (Sec:28.2

in [1]), Method of Lagrange's multipliers (Sec:27.4 in [1]), The Steepest Descent method

(unconstrained opt. prob.) (Sec:9.4 in [2]), Newton's method (unconstrained opt. prob.) (Sec:9.5 in

[2]).

UNIT-4

KKT necessary/sufficient optimality conditions, Solution of NLPP using KKT conditions

(Sec:27.5 in [1], Sec: 8.5 in [2]), Quadratic programming (Sec:28.4 in [1]), Wolfe's method for

quadratic programming (Sec: 28.5 in [1], Sec:7.7 in [2]), Convex programming problems (Sec:7.4

in [2]).

Text Book:

1. Kanti Swarup, P.K. Gupta, Man Mohan, Operations Research, Sultan Chand & Sons, 2009.

Books Recommended:

2. S. Chandra, Jayadeva, Aparna Mehra, Numerical Optimization with Applications, Narosa.

3. Hamdy A. Taha, Operations Research, An Introduction, 9th Edition, Pearson.

4. M.S. Bazarra, H.D. Sheral and C.M. Shetty, Nonlinear Programming theory and Algorithms,

John Wiley and Sons, Inc.

OLD


Mathematics

Course Title: Optimization Theory


Unit – I

Various definition of O.R., Scope of O.R., Formulation of problems linear programming in

matrix notation, Graphical solution of L.P.P. feasible solution, Basic solution, Basic feasible

solution. Optimal solution. Degenerate basic feasible solution. Convex set, Extreme point of a

convex set. convex cone and polarity.

Unit – II

Fundamental theorem of linear programming, convex and concave functions and their

properties, Differentiable convex function. Simplex method. Big M method. Two phase

method. Definition of dual problem,

Unit – III

Relationship between primal and optimal solutions. Economic interpolation of duality. Dual

simplex method. Primal dual computations. Weak duality theorem. Strong duality theorem.

Complementary slackness theorem. Formulation of transportation problem.

Unit – VI

Determination of the starting solution by North – West – Corner method. Least cost method,

Vogel’s approximation method. Iteration computations of the algorithms. Formulation of

assignment problem, the Hungarian method.

BOOKS RECOMMEND:

1. H.A. Taha: Operation Research-An Introduction, Macmillan Publishing Company

Inc., New York.

2. M.S. Bazarra, H.D. Sheral and C.M. Shetty, Nonlinear Programming theory and

Algorithms, John Wiley and Sons, Inc.

NEW COURSE BOS: 16-03-2017

B.A./B.Sc. (Hons.) IV Semester

Course Title: Elementary Mathematics


Unit 1: Relations and Functions

Definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Ordered pairs,

Cartesian product of sets. Number of elements in the Cartesian product of two finite sets. Cartesian

product of the reals with itself. Types of relations: Reflexive, symmetric, transitive and equivalence

relations. One to one and onto functions, composite functions, inverse of a function. Binary operations.

Function as a special kind of relation from one set to another. Pictorial representation of a function,

domain, co-domain and range of a function. Real valued function of the real variable, domain and range

of these functions, constant, identity, polynomial, rational, modulus, signum and greatest integer

functions with their graphs. Sum, difference, product and quotients of functions. Principle of

Mathematical Induction: Process of the proof by induction, motivating the application of the method by

looking at natural numbers as the least inductive subset of real numbers. The principle of mathematical

induction and simple applications.

Unit 2: Linear Inequalities and Permutation and combination

Linear inequalities, Algebraic solutions of linear inequalities in one variable and their representation on

the number line, Graphical solution of linear inequalities in two variables. Solution of system of linear

inequalities in two variables-graphically. Fundamental principle of counting. Factorial n. Permutations

and combinations derivation of formulae and their connections, simple applications.

Unit 3: Matrices and Determinants:

Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and

skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple

properties of addition, multiplication and scalar multiplication. Non commutativity of multiplication of

matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices

of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the

uniqueness of inverse, if it exists; (Here all matrices will have real entries). Determinant of a square

matrix (up to 3×3 matrices), properties of determinants, minors, cofactors and applications of

determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency,

inconsistency and number of solutions of system of linear equations by examples, solving system of

linear equations in two or three variables (having unique solution) using inverse of a matrix.

Unit 4: Mathematical Reasoning and Linear Programming

Mathematically acceptable statements, Connecting words/phrases-consolidating the understanding of

“if and only if (necessary and sufficient) condition”, “implies”, “and/or”, “implied by”, “and”, “or”,

“there exists” and their use through variety of examples related to real life and Mathematics. Validating

the statements involving the connecting words-difference between contradiction, converse and

contrapositive. Introduction, related terminology such as constraints, objective function, optimization,

different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems,

graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and

infeasible solutions, optimal feasible solutions (up to three non-trivial constrains).



Mathematics

Course Title: Group Theory


Credits: 04 Unit – I

Binary relation, Function, Binary Operation; Groups, its examples and basic properties, Order

of an element in a group, Subgroups, its examples and some basic properties; Centre of a

group, Normaliser of a set, Product of two subgroups; Cyclic groups, generators, its examples

and related results.

Unit – II

Cosets, Lagrange’s theorem and its related results; Index of subgroup of a group, Euler’s

theorem, Fermat’s theorem, Isomorphism and homomorphism of groups with examples and

related results; Inner automorphism; Normal subgroups and simple Groups , their examples

and related results.

Unit – III

Commutator subgroup and some basic properties; Quotient groups with examples; First,

second and third isomorphism theorems and their related results; Internal and External direct

product of groups and their related results; Characterization of a group as a direct product of

its two subgroups.

Unit – IV

Permutations, even and odd permutations, Order of a permutation, transposition, Cycle and its

length, disjoint cycles and their examples; Permutation groups, Altenating groups and their

related results; Signature of a permutation, Cayley’s theorem, Cauchy’s theorem for finite

abelian groups.

Text Book:

N.S. Gopalakrishnan: University Algebra.

Reference Book:

Contemporary Abstract Algebra by Joseph A. Gallian.



Mathematics

Course Title: Set Theory and Number Theory


Credits: 04

Unit–I Relations and their representations, Inverse relation, Composition of relations and their

properties, Equivalence relation and partition, Cross Partition, Fundamental theorem of

equivalence relation, Functions their restrictions and extensions, Invertible functions,

Characteristic functions and choice functions, Equipotent sets.

Unit–II Infinite sets, Denumerable sets, Countable sets, The continuum, Cardinals, Cardinal

arithmetic, Inequalities of cardinal numbers, Cantor’s theorem, Schroeder Bernstein theorem,

Continuum hypothesis, Partially ordered sets, Totally ordered sets, Similar sets and Well-

ordered sets, First and last elements, Maximal and minimal elements.

.

Unit–III The division algorithm and derived results, Least common multiple, Greatest

common divisor, Euclid’s algorithm, Prime numbers and related results, Fundamental theorem

of arithmetic, Relatively prime integers, Euler’s function.

Unit–IV Congruences, Euler’s Theorem, Fermat’s theorem, Order of an integer (mod m),

Linear congruences, Chinese remainder Theorem, Algebraic congruence mod p, Lagrange’s

theorem, Wilson theorem, Algebraic congruences with composite number.

Books Recommended:

1. Seymour Lipschutz: Set Theory and Related Topics. (Schuam’s Outline Series).

2. J. Hunter: Number Theory.

Books for Reference:

1. P.R. Halmos: Naive Set Theory.

2. David M. Burton: Elementary Number Theory, 6th Edition.

3. G.B. Mathews: theory of Number Part-I.



Mathematics

Course Title: Geometry of Curves and Surfaces


Credits: 04

Unit- I: Space Curves (10 Lectures)

Space curves. Examples. Plane curves. Parameterization of curves (Generalized and natural

parameters) Change of parameter regular curves and singularities. Contact of curves. Contact

of a curve and a plane. Frenet trihedron. Osculating plane. Serret-Fernet formulae. Involutes

and Evaluates, Fundamental Theorem for space curves.

Unit- II: Surface in R3 (10 Lectures)

Surface in R3. Implicit and explicit forms of the equation of a surface. Parametric curves on

surfaces. Tangent plane. First fundamental form. Angle between two curves on a surface. Area

of a surface. Invariance under co-ordinate transformation.

Unit- III: Extrinsic Geometry (10 Lectures)

Second fundamental form on a surface, Gauss map and Gaussian curvature. Gauss and

Weingrten formulae. Christoffle symbols. Some co-ordinate transformations. Goddazi

equation and Gauss theorem. Fundamental Theorem of surface Theory.

Unit- IV: Curves on a Surface (10 Lectures)

Curvature of a curve on a surface. Geodesic curvature and normal curvature. Geodesics.

Principal directions and lines of curvature. Rodrigue formula. Asymptotes Lines. Conjugate

Directions. Developable surfaces.

Books Recommend:

1. A. Goetz: Differential Geometry, Springer Verlag.

2. S.I. Husain: Lecture notes on Differential Geometry.



Mathematics (Optional)

Course Title: Tensor Analysis


Credits: 04

Unit-1: Dummy indices, free indices, summation convention, KRONECKER symbols,

permutation symbols, differentiation of a determinant, linear equations, Cramer’s Rule,

functional determinants, functional matrices, quadratic forms, real quadratic forms, pairs of

quadratic forms, quadratic differential forms, differential equations, Space of N-dimensions,

subspaces, directions at a point.

Unit-2: Transformations of coordinates, contravariant vectors, scalar invariants, covariant

vectors, scalar product of two vectors, tensors of the second order, tensors of any order,

symmetric and skew symmetric tensors, Addition and multiplication of tensors, contraction,

composition of tensors, quotient law, reciprocal symmetric tensors of the second order.

Unit-3: Riemannian space, Fundamental tensor, length of a curve, magnitude of a vector,

associate covariant and contravariant vectors, inclination of two vectors, orthogonal vectors,

coordinate hypersurfaces, coordinate curves, field of normals to a hypersurface, N-ply

orthogonal system of hypersurfaces, congruences of curves, orthogonal ennuples, principal

directions for a symmetric covariant tensor of the second order, Euclidean space of n-

dimensions.

Unit-4: The Christoffel symbols and their properties, second order derivatives of the metric

tensors, Covaraint derivative of a covariant vector, curl of a vector, covariant derivative of a

contravariant vector, derived vector in a given direction, covariant differentiation of tensors,

covariant differentiation of sums and products, divergence of a vector, Laplacian of a scalar

invariant, Intrinsic derivatives, Curvature tensor, Ricci tensor, scalar curvature, covariant

curvature tensor, Bianchi Identities.

Text Book:

1. Riemannian geometry and The Tensor Calculus by C. E. Weatherburn, CUP, 1938.

Reference Books:

1. Tensor Analysis: Theory and Applications by I. S. Sokolnikoff, Chapman and Hall,

1951.

2. Tensor Calculus by U. C. De, A. A. Shaikh and J. Sengupta, Narosa Publi. , New Delhi,

2nd Edition, 2008.

3. Tensors-Mathematics of Differential Geometry and relativity by Zafar Ahsan, PHI,

New Delhi, 2015.


B.A/B.Sc( Hons.) V Semester

Mathematics (Optional)

Course Title: Integral Equations

Course Number: MMB-557 Credit: 04

Unit –I

Integral equations, Volterra and Fredholm integral equations of first and second

kind. Singular integral equation. Relation between differential and integral equations

Solution of Volterra and Fredholm integral equations by the method of successive

substitutions and successive approximations.

Unit-2

Iterated and resolvent kernels. Neumann series reciprocal functions.Volterra

solution of Fredholm equations. Fredhlom first theorem .Fredholm second and third

theorem only statement. Fredholm associated equation.

Unit -3

Solution of integral equations using Fredholm’s determinant and minor.

Homogeneous integral equations, Fundamental function. Integral equations with

separable kernels. The Fredholm alternatives, Symmetric kernels.

Unit-4

Fundamental theorems on symmetric equations. Hilbert Schmidt Theorem. Solution

of symmetric integral equations.

BOOK RECOMMEND:

1. Shanti Swarup: Integral Equations Krishna Media (P)Ltd . Meerut, 1982

2. W. V. Lovitt:Linear integral Equation, Dover PublicationsInc New York,1950

3. K.F. Riley, M.P. Hobson and S.T.Bence: Mathematical Methods for Physics

and Engineering , Cambridge University Press, U.K. 1977.

4. M.D. Raisinghania Linear integaral equation Publication Kedar Nath Ram

Nath, Meerut.



Mathematics

Course Title: Ring Theory


Credits: 04

Unit – I

Rings, Zero divisors, Integral domains, Division rings, Fields, Subrings and Ideals, Congruence

modulo a subring relation in a ring, Simple ring, Algebra of ideals, Ideal generated by a subset,

Nilpotent ideals, Nil ideals, Quotient rings, Prime and Maximal ideals.

Unit – II

Homomorphism in rings, Natural homomorphism, Kernel of a homomorphism, Fundamental

theorem of homomorphism, First and second isomorphism theorems, Field of quotients,

Embedding of rings, Ring of endomorphisms of an abelian group.

Unit – III

Factorization in integral domains, Prime and irreducible elements, H.C.F. and L.C.M. of two

elements of a ring, Principal ideals domains, Euclidean domains, Unique factorisation domains,

Different relations between principal ideal domains, euclidean domains and unique

factorization domains.

Unit – IV

Polynomials rings, Algebraic and transcendental elements over a ring, Factorization in

polynomial ring R[x], Division algorithm in R[x], where R is a commutative rings with identity,

Properties of polynomial ring R[x] if R is a field or a U.F.D., Gauss lemma, Gauss Theorem

(statement only), Eisenstein irreducibility criteria and its applications, Division algorithm for

polynomial ring F[x], where F is a field, Reducibility test for polynomials of degree 2 and 3 in

F[x].

Book Recommended:

1. Surjeet Singh, Quazi Zameeruddin: Modern Algebra.

Books for Reference:

1. J.B. Fraleigh: A first Course in Abstract Algebra.

2. Joseph A Gallian: Contemporary Abstract Algebra.


(OPTIONAL)


Mathematics

Course Title: Programming in C and Matlab


Credits: 04 Unit – I (14 Lectures)

Introduction to Matlab, Standard Matlab windows (Command Window, Figure Window,

Editor Window, help window), The semicolon(;), The clc command, Using Matlab as

calculator, Display formats, Elementary math built in functions, Defining scalar variables,

Examples of Matlab applications, Creating arrays (one dimensional & two dimensional), The

zeroes, ones and eye commands, The transpose operators, Using a colon, Adding elements to

existing variables, Deleting elements, Built in functions for handling arrays.

Unit – II (14 Lectures)

Array multiplication, Inverse of a matrix, Solving three linear equations (array division),

element by element operations, Built in function for analysing arrays, Generation of random

numbers, Creating and saving a script files, output commands.

Unit – III (14 LECTURES)

The plot command, Plot of a function, Plotting multiple graph in the same plot, Using the hold

on, hold off commands, Function and Function files (basic), Conditional statements, Loops,

Nested loops and nested conditional statements, Calculating polynomials with Matlab

(Problems based on roots & derivatives of polynomials).

Unit – IV (14 LECTURES)

Line plots, Mesh and surface plots, plots with special graphics, The view commands, Solving

an equation in one variable, solving a nonlinear equation, finding a minimum or maximum of

a function, Numerical integration, Solving ordinary differential equations.

BOOKS RECOMMEND:

1. Amos Gilat: Matlab, An Introduction and its Applications, Wiley India Edition.


(Optional)

B.Sc. (Hons.) VI Semester

Mathematics

Course Title: Discrete Mathematics

Course No: MMB-657

Credits: 04

Unit 1:

Propositional Logic: Propositional Logic, Applications of Propositional Logic, Propositional

Equivalences, Predicates and Quantifiers, Nested Quantifiers, Rules of Inference, Introduction

to Proofs, Proof Methods and Strategies.

Unit 2:

Graphs and Graph Models: Graph Terminology and Special Types of Graphs, Representing

Graphs and Graph Isomorphism, Connectivity, Euler and Hamilton Paths, Shortest-Path

Problems, Planar Graphs.

Unit 3:

Trees: Introduction to Trees, Applications of Trees, Tree Traversal, Spanning Trees, Minimum

Spanning Trees. Boolean Algebra: Boolean Functions, Representing Boolean Functions, Logic

Gates, Minimization of Circuits

Unit 4:

Coding Theory: Introduction to coding theory, Error Correction, Group codes, Weight of codes

word, Distance between the code word, Parity-check and generator matrix, Linear codes and

cyclic codes.

Books:

1. Rosen, K. H. (2007), Discrete Mathematics and Its Applications, 7th edition, USA,

McGraw-Hill.

2. W. B. West, Introduction to Graph Theory, Prearson Education (Singapore) .

3. Ling, S. and Xing, C. P. (2004), Coding Theory: A First Course, Cambridge University

Press, Cambridge.


SYLLABUS

M.A./M./Sc. III SEMESTER

ADVANCED RING THEORY: MMM-3006

UNIT-1

Examples and fundamental properties of rings(Review), Direct and discrete direct sum of rings,

Ideals generated by subsets and their characterizations in terms of elements of the ring under

different conditions, Sums and direct sums of ideals, Ideal products and nilpotent ideals, Minimal

and maximal ideals.

UNIT-2

Complete matrix ring, Ideals in complete matrix ring, Residue class rings, Homomorphisms,

Subdirect sum of rings and its characterizations, Zorn’s Lemma, Subdirectly irreducible rings,

Boolean rings.

UNIT-3

Prime ideals and m-systems, Different equivalent formulation of prime ideals, Semi prime ideals

and n-systems, Equivalent formulation of semi prime ideals, Necessary and sufficient conditions

for an ideal to be a prime ideal, Prime radical of a ring.

UNIT-4

Prime rings and its characterization in terms of prime ideals, Primeness of complete matrix rings,

D.C.C. for ideals and the prime radical, Jacobson radical: Definition and simple properties,

Relationship between Jacobson radical and prime radical of a ring, Primitive rings, Jacobson

radical of primitive rings.

BOOK RECOMMENDED:

N.H. McCoy: The Theory of Rings

BOOKS FOR REFERENCE:

Anderson and Fuller: Rings and Categories of Modules

I.S. Luthar, I.B.S.Passi: Algebra Volume 2: Rings


SYLLABUS


RIEMANNIAN GEOMETRY AND SUBMANIFOLDS : MMM-3007

UNIT-1 Partition of unity, paracompactness, Riemannian matrix of a paracompact

manifold, First fundamental form on a Riemannian manifold, Riemannian

connexion, Riemannian curvature, Ricci and scalar curvature.

UNIT-2 Immersion, Imbedding, Distribution, Submanifold, Submanifold of Riemannian

manifold, Sypersurfaces, Gauss and Weingarten formulae, Equation of Gauss,

Coddazi and Ricci.

UNIT-3 Complex and almost manifolds, Nejenhuis tensor and integrability of a structure,

Almost Hermitian, Kaehler and nearly Kaehler manifolds, Almost contact and

Sasakian manifolds.

UNIT-4 Submanifolds of almost Hermitian manifolds, Invariant and Anti- Invariant

distributions of a Hermitian manifold, C.R.-submanifolds of Kaehler and nearly

Kaehler, Generic and slant submanifolds of Kaehler manifold.

Books Recommended:

1. Riemannian Geometry: R.S. Mishra

2. Geometry of Submanifolds: B.Y. Chen

3. Foundation of Geometry (Volume I): S. Kobayashi and K. Nomizu

4. Lecture Notes on Differentiable Manifolds: S.I. Husain


Optional Paper

Syllabus

M.A./M.Sc. III Semester

Variational Analysis and Optimization: MMM-3015

Unit 1: Convex Set, Hyperplanes, Convex function and its characterizations; Generalized convex

functions and their characterizations, Optimality criteria, Kuhn-Tucker optimality criteria.

Unit 2: Subgradients and subdifferentials; Monotone and generalized monotone maps, their

generalizations and their relations with convexity.

Unit 3: Variational inequalities and related problems, Existence and uniqueness results, Solution

methods.

Unit 4: Generalized variational inequalities and related topics; Basic existence and uniqueness

results.

Books:

1. Q.H. Ansari, C.S. Lalitha and M. Mehta, Generalized Convexity, Nonsmooth Variational

and Nonsmooth Optimization, CRC Press, Taylor and Francis Group, Boca Raton, London,

New York, 2014.


Optional Paper

M.A. / M.Sc. III-Semester

Wavelet Analysis

Paper Code: MMM-3016

Unit-I Gabor and Wavelet Transforms

Fourier and inverse Fourier transforms, Parseval identity, Convolution, Dirac delta function, Gabor

transform, Gaussian function, Centre and width of Gaussian function, Time-frequency window of

Gabor transform, Advantage of Gabor transform over Fourier transform, Continuous wavelet

transform, Time-frequency window of wavelets, Discrete wavelet transform, Haar wavelet and its

Fourier transform, Wavelets by convolution, Mexican hat wavelet, Morlet wavelet.

Unit-II Multiresolution Analysis and Wavelets

Parseval theorem for wavelet transform, Inversion formula for wavelets, Multiresolution Analysis,

Decomposition and reconstruction algorithms, Properties of scaling functions and orthonormal

wavelet bases, Construction of wavelets by generating functions, Conjugate Quadratic Filter

(CQF), The Shannon wavelet.

Unit-III Construction of Orthonormal Wavelets and its Applications

The Haar wavelet, Cardinal B-splines and spline wavelets, The Franklin wavelet, The Battle-

Lemarié wavelet, Daubechies wavelets, Application of wavelets in Image processing, Wavelet

packets, Best basis, Image compression and denoising.

Unite-IV Frames and its Applications

Concept of Frames in Hilbert space, Properties of frames and related theorems, Characterization

of frames, Frame multiresolution analysis, Gabor frames, Wavelet frames, Wavelet frames by

extension principles, Applications of tight frames in image deblurring.

Recommended Books:

1. C.K. Chui, An Introduction to Wavelets, Academic Press, New York, 1992.

2. I. Daubechies, Ten Lectures on Wavelets, CBS-NSF Regional Conferences in Applied

Mathematics, SIAM, Philadelphia, 1992.

3. O. Christensen, An Introduction to Frames and Riesz bases, Birkhäuser, Boston, 2003.

Reference Books:

1. Y. Meyer, Wavelets: Algorithms and Applications, SIAM, Philadelphia, 1993.

2. L. Debnath, Wavelet Transforms and their Applications, Birkhäuser, Boston, 2002.

3. M.W. Frazier, An Introduction to Wavelets through Linear Algebra, Springer, New

York, 1999.

4. M.K. Ahmad, Lecture Notes on Wavelet Analysis, Seminar Library, Department of

Maths, AMU, 2015.


OPTIONAL

SYLLABUS


LATTICE THEORY AND ALGEBRAIC STRUCTURES: MMM-3017

UNIT-1 Lattices (12 Lectures)

Partially order sets, Lattices, Modular Lattice, Schreier’s theorem, The chain conditions,

Decomposition theorem for lattices with ascending chain condition, Independence, Complemented

modular lattices, Boolean Algebras.

UNIT-2 Modules and Ideals (12 Lectures)

Generators, Unitary Modules, Chain conditions, Hilbert Basis Theorem, Noetherian Rings, Prime

and Primary ideals, Representation of an ideal as intersection of primary ideals, Uniqueness

Theorems, Integral dependence.

UNIT-3 Lie and Jordan Structures in Rings (12 Lectures)

Lie and Jordan ideals in ring R, Jordan simplicity of ring R, Lie structure of [R, R], Subring fixed

by automorphism, Simple rings with involutions, Involution of second kind, Skew elements and

related results.

UNIT-4 Homomorphisms and Derivations (12 Lectures)

Jordan Homomorphisms onto Prime rings, n–Jordan mappings, Derivations, Lie Derivations and

Jordan derivations, Some results of Martindale, Herstein theorem on Jordan derivation.

Books Recommended:

1. Lectures in Abstract Algebra by Nathan Jacobson

2. General Lattice Theory by George Gratzer

3. Topics in Rings Theory by I.N. Herstein

4. Rings with Involutions by I.N. Herstein


SYLLABUS

(Optional)

M.A./M./Sc. III- SEMESTER (CBCS System)

THEORY OF SEMIGROUPS : MMM-3018

UNIT-1

Basic definitions, Group with zero, Monogenic semigroups, Ordered sets, Semilattices and lattices,

Binary relations, Equivalences and related results.

UNIT-2

Congruences and related results, Free semigroups and monoids, Presentation of semigroups, Ideals

and Rees congruences, Lattices of equivalences and congruences.

UNIT-3

Green's Equivalences and related results, The structure of D-classes, Green's lemma and its

corollaries. Regular D-classes, Regular semi groups, The Sandwich set.

UNIT-4

Simple and 0-simple semigroups, Completely 0-simple semigroups.The Rees Theorem,

Completely simple semigroups, Isomorphism and normalization.

Book Recommended:

1. Fundamentals of semi group theory by John M Howie (Clarendon press. Oxford 1995).

Reference Books:

1. The Algebraic theory of semi groups, Vol. 1 and 2 by A H Clifford and G B Preston

(Mathematical surveys of the AMS- 1961 and 1967).

2. Techniques of Semi Group Theory by P M Higgins (Oxford University Press 1992).


SYLLABUS

M.A./M./Sc. IV SEMESTER

SEQUENCE SPACES: MMM-4014

UNIT-1

Classical sequence spaces, their topological properties, Maddox type sequence spaces, Linear

Metric spaces, Paranormed spaces, Frechet spaces, FK and BK-spaces, Schauder basis, AK-, AD-

, AB-, FH-, and BH-spaces.

UNIT-2

Dual of sequence spaces, Continuous duals, Kothe-Toeplitz, generalized Kothe-Toeplitz and

bounded Kothe-Toeplitz duals, Determination of duals of classical sequence spaces, their

relationships.

UNIT-3

Matrix transformations, matrix transformations between some classical sequence spaces,

Conservative and regular matrices, Schur matrix, Coregular and conull matrices, Matrix Classes

of some FK and BK spaces.

UNIT-4

Banach limit, Almost convergence, Relation between convergence and almost convergence,

Almost regular and almost conservative matrices, Duals of Maddox type sequence spaces and their

matrix transformations.

Books Recommended:

1. Element of Functional Analysis by I.J. Maddox, Cambridge University Press (1970) and

(1988).

2. Elements of Metric Spaces by Mursaleen, Anamaya Publ. Company, 2005.


SYLLABUS

M.A./M.Sc. IV SEMESTER

FIELD AND MODULE THEORY: MMM-4015 UNIT-1

Field extensions, Finite Field extensions, Finitely generated extensions of a field, Simple extension

of a field, Algebraic extension of a Field, Splitting (Decomposition) fields, Multiple roots, Normal

and separable extension of a Field.

UNIT-2

Automorphism and group of automorphisms of a Field, Galois group, Galois group of a separable

polynomial, Galois group of a polynomial of permutation of its roots, Finite Fields and Galois

Fields.

UNIT-3

Modules, Submodules and factor modules, Sum and intersection of sub-modules, Subsets and the

submodules they generate, Homomorphisms of modules, Isomorphisms theorems, Bimodules,

Inverse image of submodules, Annihilators, Torsion and torsion-free modules.

UNIT-4

Direct sums, Internal direct sums, Direct summands, Natural maps, Splitting maps, Projections

and injections, Idempotent endomorphisms, Essential and superfluous submodules, Semi-simple

modules, Socle and radical of modules, Linearly independent sets, Bases and free modules, Rank

of free modules, Divisible modules and their basic properties.

BOOKS RECOMMENDED:

1. I.T. Adamson: Introduction to Field Theory.

2. M.E. Keating: A first course in Module Theory.

BOOKS FOR REFERENCE

1. J.S. Milne: Fields and Galois Theory.

2. F.W. Anderson and K.R. Fuller: Rings and Categories of Modules.


SYLLABUS


STRUCTURES ON MANIFOLDS: MMM-4016

UNIT–I

Almost Complex and Hermitian manifolds. The fundamental two form and the Kaehler structure

on a manifold. Infinitesimal automorphism. Holomorphic vector fields. Characterizations for a

vector field to be an infinitesimal automorphism. Cayley algebra on R8 and almost Hermitian

structure on S6. The space of constant holomorphic sectional curvature.

UNIT –II

Almost contact structure on a smooth manifold. Contact manifolds. Torsion tensor of an almost

contact metric manifold. Sasakian manifold, Kenmotsu manifold. Trans-Sasakian manifold.

Sasakian and Kenmotsu space forms.

UNIT–III:

Invariant submanifolds in contact metric manifold. Semi-invariant submanifolds of a Sasakian

manifold. Umbilical submanifolds of almost contact metric manifolds. Semi-invariant products in

Sasakian manifolds. Some characterizations. Totally contact umbilical semi-invariant

submanifolds of Sasakian manifold. Pseudo-umbilical submanifold.

UNIT–IV

Topological groups. Subgroups and quotient spaces. Homomorphisms of topological groups.

Connected components of a topological group. Lie groups and Lie-algebras. Invariant differential

forms on Lie-groups. One parameter subgroup and exponential map. Example of Lie-groups.

Books Recommended:

1. Structures on manifolds, Kentaro Yano and Masahiro Kon, World Scientific Press.

2. Differentiable manifolds, Y.Matsushima, Marcel Dekker, inc.

Reference Books:

1. Foundations of Differential Geometry, vol II- S.Kobayashi and K.Nomizu, John Wiley and

sons.

2. Geometry of CR-submanifolds -Aurel Bejancu, D.Reidel publication co.

3. Contact manifolds in Riemannian geometry – D.E. Blair, Lecture notes in Math. 509,

Springer-Verlag.


Syllabus

M.A. /M. Sc. IV Semester

SPECIAL FUNCTIONS AND LIE THEORY: MMM-4017

UNIT-I Introduction; Gamma Function; Hypergeometric Functions: Definition and special cases,

convergence, analyticity, integral representation, differentiation, transformations and summation

theorems; Bessel Functions: Definition, connection with hypergeometric function, differential and

pure recurrence relations, generating function, integral representation; Neumann polynomials,

Neumann series and related results; Examples on above topics.

UNIT-II Legendre polynomials: (i) Generating function (ii) Special values (iii) Pure and differential

recurrence relations (iv) Differential equation (v) Series definition (vi) Rodrigues’ formula (vii)

Integral representation; Hermite polynomials: Results (i) to (vii) and expansion of xn in terms of

Hermite polynomials; Laguerre polynomials: Results (i) to (vii); Examples on above topics.

UNIT-III Simple sets of polynomials; Orthogonal polynomials: Equivalent condition for orthogonality; Zeros

of orthogonal polynomials; Expansion of polynomials; Three-term recurrence relation; Christoffel-

Darboux formula; Normalization and Bessel’s inequality; Orthogonality of Legendre, Hermite and

Laguerre polynomials; Ordinary and singular points of differential equations, Regular and irregular

singular points of hypergeometric, Bessel, Legendre, Hermite and Laguerre differential equations;

Examples on above topics.

UNIT-IV Lie groups; Tangent vector; Lie bracket; Lie algebra; General linear and special linear groups and

their Lie algebras; Exponential of matrix and its properties; Construction of partial differential

equation; Linear differential operators; Group of operators; Extended forms of the group generated

by the operators; Derivation of generating functions; Examples on above topics.

Books Recommended

1. E. D. Rainville: Special Functions, Reprint of 1960 First Edition. Chelsea Publishing Co., Bronx,

New York, 1971.

2. W. Jr. Miller: Lie Theory and Special Functions, Academic Press, New York and London, 1968.

3. E. B. McBride: Obtaining Generating Functions, Springer Verlag, Berlin Heidelberg, 1971.


SYLLABUS


NON COMMUTATIVE RINGS (MMM-4018)

Unit I

Basic terminology and examples, Free k-rings, Rings with generators and relations, Twisted

polynomial rings, Differential polynomial rings, Group rings, skew group rings.

Unit II

Radical of a ring, Prime radical of a ring, Jacobson radical of a ring, Neotherian rings, Artinian

rings, Simple rings, Semisimple rings, Simple Artinian rings, Semisimple artinian rings.

Unit III

Prime rings, Semiprime rings, Subdirectly irreducible rings, Primitive rings, Density Theorem,

Wedderburn Artin's Theorem.

Unit IV

Regular rings, Some commutativity Theorems, Wedderburn Theorem, Generalizations of

Wedderburn Theorem.

Text Book

Non commutative Rings, I.N. Herstein John Wiley and Sons, INC.

Reference Book

A First Course in Non commutative Rings Springer-Verlag.


SYLLABUS

M.A./M.Sc. IV SEMESTER

TOPOLOGICAL VECTOR SPACES: MMM-4019

UNIT-1

Review of topological space, metric space and normed linear space; Topological vector space,

Convex set, Cone; Balanced, absorbing and absolutely convex sets, Minkowski’s functional,

Boundedness and continuity, Subspace, product space and quotient space of a topological vector

space.

UNIT-2

Locally convex topological vector space, Locally bounded and locally compact topological vector

spaces, F-space, Frechet space, Separation property, Normable and Metrizable topological vector

space, Complete topological vector space.

UNIT-3

Linear transformations and linear functionals and their continuity, Finite dimensional topological

vector space, Linear varieties and hyperplanes, Banach-Alaoglu theorem, Extreme points and

extreme sets, Krein-Milman’s theorem.

UNIT-4

Seminorms and local convexity, Linear metric space, Paranormed space; Schauder basis, FK, FH

and BK spaces and their properties, Banach-Steinhaus theorem, AK property.

Books Recommended:

1. W. Rudin: Functional Analysis, Mc Graw Hill Education, 2nd Ed, 1991.

2. Elements of Metric Spaces by Mursaleen, Anamaya Publ. Company, 2005.


SYLLABUS

OPEN ELECTIVE: ELEMENTS OF ELEMENTARY CALCULUS

M.Sc. IV SEMESTER (MATHEMATICS)

PAPER CODE: MMM-4091

4 Credits

M.M.: 100

Sessional Marks: 10

Mid Term Marks: 30

Final Marks: 60

Unit 1: Sets, Function and Limit

Sets, and their properties, Functions and their properties, Some known functions, Domain, Range,

Graph of Functions, Limit and its basic properties.

Unit 2: Continuity and its Basic Properties

Derivative (as rate of change and slope of a tangent), Properties of derivatives, Derivatives of some

known functions, namely, polynomial, logarithmic functions, exponential functions, trigonometric

functions.

Unit 3: Application of Derivative

Rate of change, increasing and decreasing functions, maxima and minima of polynomials and

trigonometric functions (first and second derivative test motivated geometrically) simple problems

(that illustrates basic problems and understanding of the subject as well as real life situations. Mean

Theorem Functions.

Unit 4: Integration and its applications

Indefinite integral, standard formulae of indefinite integral, Definite integral as a limit of sum,

Basic properties and formulae of definite integral of simple functions (without proof) Applications

finding the area of under simple curves, especially lines, area of circle, parabolas, ellipse (in

standard form only).

Books Recommended:

1. Calculus, Finney and Thomas, Addison-Wesley Pub. Company.

2. Mathematics Vol. 1 and 2 Class 12, R.D. Sharma, Dhanpat Rai and Sons.

3. Problems in Calculus of one variable, I.A. Marun, Arihant Publication.

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