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UNIVERSITY OF MUMBAI

Syllabus

for

T.Y.B.A./B.Sc. (CBCS)

Program: B.A/B.Sc.

Course: Mathematics

with eect from the academic year 2018-2019

1

UNIVERSITY OF MUMBAI

Syllabus

for

T.Y.B.A./B.Sc. (CBCS)

Program: B.A/B.Sc.

Course: Mathematics

with eect from the academic year 2018-2019

1

T.Y.B.A./T.Y.B.Sc. (CBCS)

Semester V

Discrete Mathematics

Course Code Unit Topics Credits L/W

USMT501,UAMT501Unit I Preliminary Counting

Unit II Advanced Counting 2.5 3

Unit III Permutations

Algebra V

USMT502, UAMT502Unit I Quotient Vector spaces and Orthogonal

Transformations

Unit II Eigenvalues, Eigenvectors 2.5 3

Unit III Diagonalisation

Topology of Metric Spaces

USMT503, UAMT503Unit I Metric Spaces

Unit II Sequences, Closed subsets, Limit Points 2.5 3

Unit III Continuity

Numerical Analysis-I (Elective A)

USMT5A4,UAMT5A4Unit I Error Analysis

Unit II Transcendental & Polynomial 2.5 3

Equations

Unit III Linear System of Equations

Number Theory and its Applications-I (Elective B)

USMT5B4,UAMT5B4Unit I Congruences and Factorisations

Unit II Diphantine Equations and their solutions 2.5 3

Unit III primitive Roots and Cryptography

Graph Theory (Elective C)

USMT5C4,UAMT5C4Unit I Basics of Graphs

Unit II Trees 2.5 3

Unit III Eulerian and Hamiltonian Graphs

Basics Concepts of probability and Random Variables (Elective D)

USMT5D4,UAMT5D4Unit I Basics Concepts of probability and

Random Variables

Unit II Properties of Distribution Function and

Joint Density Function 2.5 3

Unit III Weak Law of Large Numbers

Practicals

USMTP05, UAMTP05Practicals based on

USMT501/UAMT501, 3 6

USMT502/UAMT502

and USMT503/UAMT503

USMTPJ5,UAMTPJ5 Project 3 6

2

T.Y.B.A./T.Y.B.Sc. (CBCS)

Semester VI

Real and Complex Analysis

Course Code Unit Topics Credits L/Week

USMT601,UAMT601Unit I Sequences and series of functions

Unit II Introduction to Complex Analysis 2.5 3

Unit III Complex Power series

Algebra V

USMT602, UAMT602Unit I Normal Subgroups

Unit II Ring Theory 2.5 3

Unit III Factorisation

Metric Topology

USMT603, UAMT603Unit I Complete Metric Spaces

Unit II compact Metric spaces 2.5 3

Unit III Connected sets

Numerical Analysis-II (Elective A)

USMT6A4,UAMT6A4Unit I Interpolation

Unit II Polynomial Approximations and 2.5 3

Numerical dierentiation

Unit III Numerical Integration

Number Theory and its Applications-II (Elective B)

USMT6B4,UAMT6B4Unit I Quadratic Reciprocity

Unit II Continued Fractions 2.5 3

Unit III Pell's equation, Arithmetic Functions and

Special Numbers

Graph Theory and Combinatorics (Elective C)

USMT6C4,UAMT56C4Unit I Colorings of Graphs

Unit II Planar Graphs 2.5 3

Unit III Combinatorics

Operations Research (Elective D)

USMT6D4,UAMT6D4Unit I Linear Programming I

Unit II Linear Programming II 2.5 3

Unit III Queing Systems

Practicals

USMTP06, UAMTP06Practicals based on

USMT601/UAMT601,USMT602/UAMT602 3 6

and USMT603/UAMT603

USMTPJ6,UAMTPJ6 Project 3 6

3

Note :

1. USMT501/UAMT501, USMT502/UAMT502, USMT503/UAMT503 are com-

pulsory courses for Semester V.

2. Candidate has to opt one Elective Course from USMT5A4/UAMT5A4,

USMT5B4/UAMT5B4, USMT5C4/UAMT5C4 and USMT5D4/UAMT5D4

for Semester V.

3. USMT601/UAMT601, USMT602/UAMT602, USMT603/UAMT603 are com-

pulsory courses for Semester VI.

4. Candidate has to opt one Elective Course from USMT6A4/UAMT6A4,

USMT6B4/UAMT6B4, USMT6C4/UAMT6C4 and USMT6D4/UAMT6D4

for Semester VI.

5. Passing in theory and practical shall be separate in the compulsory courses.

6. Candidate has to do a project in the courses USMT5PR/UAMT5PR of Semester

V and USMT6PR/UAMT6PR of Semester VI.

Teaching Pattern for SY B.A./B.Sc :

1. Three lectures per week per course (1 lecture/period is of 48 minutes dura-

tion).

2. One practical of three periods per week per course (1 lecture/period is of 48

minutes duration).

3. Each project for the courses USMT5PR/UAMT5PR in Semster V and

USMT6PR/UAMT6PR in Semester VI shall have at most 08(eight) students

and the workload for each project is 1L/W. However a teacher guiding more

than one project gets 1L/W workload, irrespective of the number of projects

he/she guides.

Syllabus for Semester V & VI

SEMESTER V

Note : All topics have to be covered with proof in details (unless mentioned other-

wise) and with examples.

USMT501/UAMT501 Discrete Mathematics

Unit I: Preliminary Counting (15 Lectures)

1. Finite and innite sets, Countable and uncountable sets, examples such as N;

NN;Q;Rand open interval (0;1):

2. Addition and multiplication principle, Counting sets of pairs, two ways counting.

3. Stirling numbers of second kind, Simple recursion formulae satised by S(n;k)

4

and direct formulae for S(n;k)fork= 0;1;;n1:

4. Pigeon hole principle and its strong form, its applications to geometry, monotonic

sequences.

Reference for para 1 of unit I: Sections 2.1 and 2.4 of Discrete Mathematics &

Its Applications by Kenneth Rossen , Tata McGraw Hill.

Reference for para 2 of unit I: Sections 10.1 and 10.2 of Discrete Mathematics

byNorman L. Biggs , (Second Edition) Oxford University press

Unit II: Advanced Counting (15 Lectures)

Binomial and Multinomial Theorem, Pascal identity, examples of standard identities

such as the following with emphasis on combinatorial proofs:nX

i=0

n

i

= 2nand

rX

k=0

m

k

m

rk

=

m+n

r

;nX

k=r

k

r

=

n+ 1

r+ 1

;kX

i=0

k

i2

=

n+ 1

r+ 1

:

Permutation and combination of sets and multi-sets, circular permutations, empha-

sis on solving problems. Non-negative and positive integral solutions of equation

x1+x2++xk=n:

Principle of Inclusion and Exclusion and its applications ,derangements, explicit for-

mula fordn;various identities involving dn:

Unit III: Permutations (15 Lectures)

1. Permutation of objects, composition of permutations, results such as every per-

mutation is product of disjoint cycles, every cycle is product of transpositions, even

and odd permutations, Sn;An, rank and signature of permutation, results such as

() =()() (;2Sn); () = 1i

2. Partially ordered sets, Mobius Inversion Formula with application to deriving the

formula for Eulers phi-function '(n):,

3. Recurrence relation, denition of homogeneous, non-homogeneous, linear and

non linear recurrence relation, obtaining recurrence relation in counting problems,

solving (homogeneous as well as non-homogeneous) recurrence relation by using

iterative method, solving a homogeneous recurrence relation of second degree using

algebraic method proving the necessary result.

Reference for para 1 of unit III: A First Course in Abstract Algebra byJohn.

B. Fraleigh , third edition, Narosa Publishing House.

Recommended Text Book :

Richard Brualdi :Introductory Combinatorics , Pearson (Fourth Edition).

(Sections 2.1, 2.2, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 5.1, 5.2, 5.3, 6.1, 6.2, 6.3, 6.6, 7.1,

7.2, 7.3, 7.4, 7.5, 7.6, 8.2.)

Additional Reference Books :

1.Norman Biggs ,Discrete Mathematics , Oxford University Press.

5

2.V. Krishnamurthy ,Combinatorics Theory and Applications .

3.A. Tucker ,Applied Combinatorics , John Wiley and Sons.

USMT502/UAMT502 Algebra IV

Unit I: Quotient Spaces and Orthogonal Linear Transformations (15

Lectures)

Review of vector spaces over R;subspaces and linear transformations.

Quotient Spaces: For a real vector space Vand a subspace W;the cosetsv+W

and the quotient space V=W: First Isomorphism theorem for real vector spaces

(Fundamental theorem of homomorphism of vector spaces), dimension and basis of

the quotient space V=W whenVis nite dimensional.

Inner product spaces: Examples of inner product including the inner product

hf;gi=Z

f(x)g(x)dxonC[;], the space of continuous real valued func-

tions on [;]. Orthogonal sets and orthonormal sets in an inner product space.

Orthogonal and orthonormal bases. Gram-Schmidt orthogonalization process and

simple examples in R3,R4.

Real Orthogonal transformations and isometries of Rn:Translations and Re
ections

with respect to a hyperplane. Orthogonal matrices over R:

Equivalence of orthogonal transformations and isometries of Rnxing the origin.

Characterization of isometries as composites of orthogonal transformations and

translations.

Orthogonal transformation of R2:Any orthogonal transformation in R2is a re
ec-

tion or a rotation.

Unit II: Eigenvalues, Eigenvectors (15 Lectures)

Eigenvalues and eigenvectors of a linear transformation T:V!VwhereVis a

nite dimensional real vector space and examples, eigenvalues and eigenvectors of

nn- real matrices, linear independence of eigenvectors corresponding to distinct

eigenvalues of a linear transformation / Matrix.

Characteristic polynomial of an nn- real matrix. Result: A real number is an

eigenvalues of an nnmatrixAif and only if is a root of the characteristic

polynomial of A:Cayley-Hamilton Theorem (statement only), Characteristic roots.

Similar matrices and relation with a change of basis. Invariance of the characteristic

polynomial and (hence of the) eigenvalues of similar matrices.

Reference for Unit II :

Sections 1, 2, 3 of Chapter VIII of Introduction to Linear Algebra (Second Edi-

tion) by Serge Lang .

Recommended Text Books :

6

1.Serge Lang :Introduction to Linear Algebra , Springer Verlag.

2.S. Kumaresan :Linear Algebra A geometric approach , Prentice Hall of

India Private Limited.

Unit III: Diagonalisation (15 Lectures)

Diagonalizability of an nnreal matrix and a linear transformation of a nite dimen-

sional real vector space to itself. Denition: Geometric multiplicity and Algebraic

multiplicity of eigenvalues of an nnreal matrix and of a linear transformation.

Examples of non-diagonalisable matrices over R:

Annnreal matrixAis diagonalisable if and only if Rnhas a basis of eigenvectors

ofAif and only if the sum of dimension of eigen spaces of Aisnif and only if the

algebraic and geometric multiplicities of eigenvalues of Acoincide.

Diagonalisation of real Symmetric matrices and applications to real quadratic forms,

rank and signature of a real quadratic form, classication of conics in R2and quadric

surfaces in R3:

Recommended Text Books for Unit I & unit II :

1.S. Kumaresen ,Linear Algebra: A Geometric Approach , PHI.

2.M. Artin, Algebra , Pearson India.

3.L. smith ,Linear Algebra , Springer.

4.T. Banchoff and J. Wermer ,Linear Algebra through geometry , Springer.

Additional Reference books:

1.N.S. Gopalakrishnan ,University Algebra , Wiley Eastern Limited.

2.M. Artin ,Algebra , Prentice Hall of India, New Delhi.

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

Second edition, Foundation Books, New Delhi, 1995.

4.T.W. Hungerford ,Algebra , Springer.

5.D. Dummit & R. Foote ,Abstract Algebra, John Wiley & Sons, Inc.

USMT503/UAMT503 Topology of Metric Spaces

Note: In this course, denitions of closed set in a metric space, limit point and

closure of a subset of metric space shall be used as indicated below in Unit II.

Unit I: Metric spaces (15 Lectures)

Denition of metric space, Euclidean space Rnwith its Euclidean norm function

and the distance metric induced by it, sup metric and sum metric on RnandC

(complex numbers). Discrete metric spaces and examples such as Z:

Normed linear spaces: Denition, the distance (metric) induced by the norm, trans-

lation invariance of the metric induced by the norm. Examples of normed linear

spaces including

7

1.Rnwith sum normkk1;the Euclidean norm kk2, and the sup norm kk1:

2.C[a;b];the space of continuous real valued functions on [a;b]with norms

kk1;kk2;andkk1wherekfk1=Rb

ajf(t)jdt;kfk2=Rb

ajf(t)j2dt1=2

;

kfk1= supfjf(t)j:t2[a;b]g:

Open balls, open subsets of a metric space. Verication of the result: any open ball

of a metric space is an open subset of the metric space. Examples of open sets in

various metric spaces, structure of an open set in R:

Properties of open subsets of a metric space: the intersection of nitely many open

subsets of a metric space is an open subset of the metric space, the union of arbi-

trary collection of open subsets of a metric space is an open subset of the metric

space. Interior of a subset of a Metric space. Hausdor property of a metric space.

Subspaces of a Metric space. Product of two metric spaces. Equivalent metrics.

Distance of a point from a set, distance between two sets, diameter of a set in a

metric space.

Unit II: Sequences, closed sets, limit Points (15 Lectures)

Sequences in a metric space, convergent sequences and Cauchy sequences in a

metric space, subsequence of a sequence, examples.

Closed set in a metric space (as complement of an open set), limit point of a set

(IfAis subset of a metric space X;x2Xis a limit point of Aif each open ball of

Xwith center at xcontains a point of Aother thanx), isolated point. A closed

set contains all its limit points. Closed balls. Closure of a subset of a metric space

(closureEof a subset Eof a metric space is E[E0whereE0denotes the set of

all limit points of EinX) and properties: If Xis a metric space and EX:Then

1.Eis closed in X:

2.E=Eif and only if Eis closed in X:

3.EFfor every closed subset FofXsuch thatEF:

4.Eequals the intersection of all the closed supersets of EinX:

Boundary of a set in a metric space. Complete metric spaces.

Unit III: Continuity (15 Lectures)

-denition of continuity at a point for a function from one metric space to

another. Characterization of continuity at a point in terms of sequences, open sets.

Continuity of a function on a metric space. Characterization of continuity of a func-

tion in terms of inverse image of open sets and closed sets. Algebra of continuous

real valued functions. Uniform continuity of a function dened on a metric space:

denition and examples (emphasis on R).

Recommended Text Books :

1.W. Rudin ,Principles of Mathematical Analysis , Tata McGraw- Hill Edu-

cation in 2013.

8

2.G.F. Simmons ,Introduction to Topology and Modern Analysis , McGraw

Hill Education (India) Edition.

3.Irvin Kaplansky ,Set Theory and Metric spaces , Allyn and Bacon Inc,

Boston.

4.S. Kumaresan ,Topology of Metric spaces , Narosa.

Additional Reference Books :

1.MchealO Searc oid,Metric spaces , Springer Undergraduate Mathemat-

ics Series, 2007.

2.T. Apostol ,Mathematical Analysis , Narosa.

3.R. R. Goldberg ,Methods of Real Analysis .

4.P. K. Jain, K. Ahmed ,Metric Spaces , Narosa, New Delhi, 1996.

5.D. Somasundaram, B. Choudhary ,A rst Course in Mathematical

Analysis .

6.E. T. Copson ,Metric Spaces , Universal Book Stall, New Delhi, 1996.

USMT5A4/UAMT5A4 Numerical Analysis I (Elective A)

Note: Derivations and geometrical interpretation of all numerical methods have to

be covered.

Unit I: Errors Analysis, Transcendental and Polynomial Equations

(15 Lectures):

Measures of Errors: Relative, absolute and percentage errors. Types of errors: Inher-

ent error, Round-o error and Truncation error. Taylors series example. Signicant

digits and numerical stability. Concept of simple and multiple roots. Iterative meth-

ods, error tolerance, use of intermediate value theorem. Iteration methods based

on rst degree equation: Newton-Raphson method, Secant method, Regula-Falsi

method, Iteration Method. Condition of convergence and Rate of convergence of

all above methods

Unit II: Transcendental and Polynomial Equations (15 Lectures)

Iteration methods based on second degree equation: Muller method, Chebyshev

method, Multipoint iteration method. Iterative methods for polynomial equations;

Descarts rule of signs, Birge-Vieta method, Bairstrow method. Methods for mul-

tiple roots. Newton-Raphson method. System of non-linear equations by Newton-

Raphson method. Methods for complex roots. Condition of convergence and Rate

of convergence of all above methods.

Unit III: Linear System of Equations (15 Lectures)

Matrix representation of linear system of equations. Direct methods: Gauss elimi-

nation method. Pivot element, Partial and complete pivoting, Forward and back-

ward substitution method, Triangularization methods-Doolittle and Crouts method,

Choleskys method. Error analysis of direct methods. Iteration methods: Jacobi

iteration method, Gauss-Siedal method. Convergence analysis of iterative method.

Eigen value problem, Jacobis method for symmetric matrices Power method to de-

termine largest eigenvalue and eigenvector.

Recommended Text Books :

9

1.E. Kendall and Atkinson ,An Introduction to Numerical Analysis ,

Wiley.

2.M. K. Jain, S. R. K. Iyengar and R. K. Jain ,Numerical Methods

for Scientic and Engineering Computation , New Age International Publi-

cations.

3.S.D. Comte and Carl de Boor ,Elementary Numerical Analysis, An

Algorithmic Approach , McGraw Hill International Book Company.

4.S. Sastry ,Introductory methods of Numerical Analysis , PHI Learning.

5.F.B. Hildebrand ,Introduction to Numerical Analysis , Dover Publication,

New York.

6.S.B. James ,Numerical Mathematical Analysis , Oxford University Press,

New Delhi.

USMT5B4/UAMT5B4

Number Theory and its applications I (Elective B)

Unit I. Congruences and Factorization (15 Lectures)

Review of Divisibility, Primes and The fundamental theorem of Arithmetic.

Congruences : Denition and elementary properties, Complete residue system mod-

ulom;Reduced residue system modulo m;Euler's function ??? and its properties,

Fermat's little Theorem, Euler's generalization of Fermat's little Theorem, Wil-

son's theorem, Linear congruence, The Chinese remainder Theorem, Congruences

of higher degree, The Fermat-Kraitchik Factorization Method.

Reference for Unit I: Sections 2.1, 2.2, 2.3, 2.4, 2.5 of Niven, H. Zuckerman and H.

Montogomery, An Introduction to the Theory of Numbers, John Wiley & Sons. Inc.

and section 5.4 of David M. Burton, An Introduction to the Theory of Numbers.

Tata McGraw Hill Edition.

Unit II: Diophantine equations and their solutions (15 Lectures)

The linear equations ax+by=c:The equations x2+y2=pwherepis a prime.

The equation x2+y2=z2;Pythagorean triples, primitive solutions, the equations

x4+y4=z2andx4+y4=z4have no solutions (x;y;z)withxyz6= 0:Every

positive integer ncan be expressed as sum of squares of four integers, Universal

quadratic form x2

1+x2

2+x2

3+x2

4:

Reference for Unit II: Sections 5.1, 5.2, 5.3, 5.4, 5.5 of Niven, H. Zuckerman and H.

Montogomery, An Introduction to the Theory of Numbers, John Wiley & Sons. Inc.

Unit III: Primitive Roots and Cryptography (15 Lectures)

Order of an integer and Primitive Roots. Basic notions such as encryption (en-

ciphering) and decryption (deciphering), Crypto-systems, symmetric key cryptog-

raphy, simple examples such as shift cipher, Ane cipher, Hill's cipher, Vigenere

cipher. Concept of Public Key Crypto-system; RSA Algorithm. An application of

Primitive Roots to Cryptography.

Reference for Unit III: Elementary number theory, David M. Burton, Chapter 8 sec-

tions 8.1, 8.2 and 8.3, Chapter 10, sections 10.1, 10.2 and 10.3.

Additional Reference Books :

10

1.G. H. Hardy and E.M. Wright ,An Introduction to the Theory of

Numbers , Oxford University Press.

2.Neville Robins ,Beginning Number Theory , Narosa Publications.

3.S.D. Adhikari ,An introduction to Commutative Algebra and Number

Theory , Alpha Science International.

4.N. Koblitz ,A course in Number theory and Cryptography , Springer.

5.M. Artin ,Algebra , Prentice Hall.

6.K. Ireland, M. Rosen ,A classical introduction to Modern Number The-

ory, Springer.

7.W. Stalling ,Cryptology and network security , Prentice Hall.

USMT5C4/UAMT5C4 Graph Theory (Elective C)

Unit I: Basics of Graphs (15 Lectures)

Denition of general graph, Directed and undirected graph, Simple and multiple

graph, Types of graphs- Complete graph, Null graph, Complementary graphs, Reg-

ular graphs Sub graph of a graph, Vertex and Edge induced sub graphs, Spanning

sub graphs. Basic terminology- degree of a vertex, Minimum and maximum degree,

Walk, Trail, Circuit, Path, Cycle. Handshaking theorem and its applications, Isomor-

phism between the graphs and consequences of isomorphism between the graphs,

Self complementary graphs, Connected graphs, Connected components. Matrices

associated with the graphs Adjacency and Incidence matrix of a graph- properties,

Bipartite graphs and characterization in terms of cycle lengths. Degree sequence

and Havel-Hakimi theorem, Distance in a graph- shortest path problems.

Unit II: Trees (15 Lectures)

Cut edges and cut vertices and relevant results, Characterization of cut edge, Def-

inition of a tree and its characterizations, Spanning tree, Recurrence relation of

spanning trees and Cayley formula for spanning trees of Kn;Binary and mary

tree, Weighted graphs and minimal spanning trees.

Unit III. Eulerian and Hamiltonian graphs (15 Lectures)

Eulerian graph and its characterization Hamiltonian graph, Necessary condition for

Hamiltonian graphs using GSwhereSis a proper subset of V(G);Sucient

condition for Hamiltonian graphs- Ore's theorem and Dirac's theorem, Hamiltonian

closure of a graph, Cube graphs and properties like regular, bipartite, Connected

and Hamiltonian nature of cube graph, Line graph of graph and simple results.

Recommended Text Book :

J.A. Bondy and U.S.R. Murty ,Graph Theory with Applications , Elsevier.

Additional Reference books :

1.R. Balakrishnan and K. Ranganathan ,A Textbook of Graph Theory ,

Springer.

2.Behzad and Chartland ,Graph Theory .

3.Choudam S.A. ,Introduction to Graph Theory .

4.West D.G. ,Graph Theory . Allyn and Bacon.

11

USMT5D4/UAMT5D4

Basic Concepts of Probability and Random Variables

(Elective D)

Unit I: Basic Concepts of Probability and Random Variables

Basic Concepts: Algebra of events including countable unions and intersections,

Sigma eldF;Probability measure PonF;Probability Space as a triple (

;F;P);

Properties of Pincluding Sub-additivity.

Discrete Probability Space, Independence and Conditional Probability, Theorem of

Total Probability. Random Variable on (

;F;P)denition as a measurable func-

tion, Classication of random variables - Discrete Random variable, Probability

function, Distribution function, Density function and Probability measure on Borel

subsets of R;Absolutely continuous random variable. Function of a random vari-

able; Result on a random variable Rwith distribution function Fto be absolutely

continuous, Assume Fis continuous everywhere and has a continuous derivative

at all points except possibly at nite number of points, Result on density func-

tionf2ofR2whereR2=g(R1); hjis inverse of gover a suitable subinterval

f2(y) =nX

i=1f1(hj(y))jh0

j(y)junder suitable conditions.

Reference for Unit 1, sections 1.1-1.6, 2.1-2.5 of Basic Probability theory by

Robert Ash , Dover Publication, 2008.

Unit II: Properties of Distribution function, Joint Density function

(15 lectures)

Properties of distribution function F; F is non-decreasing, lim

x!1F(x) = 0;

lim

x!1F(x) = 0;Right continuity of F;lim

x!x0F(x) =P(fR

0):

Joint distribution, Joint Density, Results on Relationship between Joint and In-

dividual densities, Related result for Independent random variables. Examples of

distributions like Binomial, Poisson and Normal distribution. Expectation and kth

moments of a random variable with properties.

Reference for Unit II: Sections 2.5-2.7, 2.9, 3.2-3.3,3.6 of Basic Probability theory

byRobert Ash , Dover Publication, 2008.

Unit III: Weak Law of Large Numbers (15 lectures)

Joint Moments, Joint Central Moments, Schwarz Inequality, Bounds on Correla-

tion Coecient ;Result onas a measure of linear dependence, Var (Pn

i=1Ri) =Pn

i=1var(Ri) + 2P

1i

Reference for Unit III, Sections 3.4, 3.5, 3.7, 4.1-4.4 of Basic Probability theory by

Robert Ash , Dover Publication, 2008

Additional Reference Books :

12

M. Capinski ,Probability through Problems , Springer.

USMTP05, UAMTP05

Practicals for USMT501/UAMT501, USMT502/UAMT502 & USMT503/UAMT503

A. Practicals for USMT501/UAMT501 :

1. Problems based on counting principles, Two way counting.

2. Stirling numbers of second kind, Pigeon hole principle.

3. Multinomial theorem, identities, permutation and combination of multi-set.

4. Inclusion-Exclusion principle, Euler phi function.

5. Derangement and rank signature of permutation.

6. Recurrence relation.

7. Miscellaneous Theoretical Questions based on full paper.

B. Practicals for USMT502/UAMT502 :

1. Quotient spaces.

2. Orthogonal transformations,Isometries.

3. Eigenvalues, eigenvectors of nnmatrices over R;C(n= 2;3):

4. Diagonalization.

5. Orthogonal diagonalization.

6. Miscellaneous Theoretical Questions based on full paper.

C. Practicals for USMT503/UAMT503 :

1. Metric spaces and normed linear spaces, Examples.

2. Open balls, open sets in metric spaces, subspaces and normed linear spaces.

3. Limit points and closure points, closed balls, closed sets, closure of a set,

boundary of a set, distance between two sets.

4. Cauchy Sequences, completeness.

5. Continuity.

6. Uniform continuity in a metric space.

7. Miscellaneous Theoretical Questions based on full paper.

13

USMTPJ5, UAMTPJ5: Projects

A student can submit a project which shall have 20-30 typed pages, on one of the

following topics:

1. Computer implementation of rational numbers in python or C++:

R.G. Dromey, How to Solve it by Compute , Pearson Education.

2. Various Sorting Algorithms like merge sort, insertion sort, quick sort, heap

sort, bucket sort, radix sort:

R.G. Dromey, How to Solve it by Computer , Pearson Education.

3. Algorithms: Integer knapsack problem, fractional knapsack problem, back-

tracking algorithm for the n-queens problem:

-R.G. Dromey, How to Solve it by Computer , Pearson Education.

4. Normalization in databases:

Jerey D. Ullman, Principles of Database and Knowledge-base Systems ,

Volume 1.

5. Vector Fields, Integral curves, Phase
ows in the plane:

V.I. Arnold, Ordinary Dierential Equations , PHI.

6. Eigenvalues, Eigenfunctions of the vibrating string and Applications to the

Heat Equation, Dirichlet problem for the circle:

G. F. Simmons, Dierential Equations with Applications and Historical

Notes , McGRAW-Hill International.

7. Bessel Functions and the vibrating membrane:

G. F. Simmons, Dierential Equations with Applications and Historical

Notes , McGRAW-Hill International.

8. Sturm-Liouville Bounday value problems, Eigenvalues, Eigenfunctions:

G. F. Simmons, Dierential Equations with Applications and Historical

Notes , McGRAW-Hill International.

9. Continued Fractions and applications to irrational numbers:

H.S. Zuckerman and I. Niven, An Introduction to the Theory of Numbers ,

Wiley Eastern Ltd.

10. Distribution of primes:

H.S. Zuckerman and I. Niven, An Introduction to the Theory of Numbers ,

Wiley Eastern Ltd.

11. Prime Number theorem, Zeta function.

D.M. Burton, Elementary Number Theory , Tata McGraw-Hill.

12. Transcendental and algebraic numbers, Transcendence of e;Irrationality of

ande:

I.N. Herstein, Topics in Algebra , Wiley India Pvt. Limited.

13. The real numbers-a survey of constructions:

https://arxiv.org/pdf/1506.03467

14

14. Fourier series of circular functions and applications to Series of real numbers:

R.R Goldberg, Methods of Real Analysis , Oxford IBM Publications.

15. Fourier series, Orthogonal Functions, Dirichlet's problem:

G. F. Simmons, Dierential Equations with Applications and Historical

Notes , McGRAW-Hill International.

16. Pointwise convergence of Fourier series, the Gibbs Phenomenon:

R. Bhatia, Fourier series , Hindustan Book Agency.

17. Cesaro summablity and Fejer's theorem:

R. Bhatia, Fourier series , Hindustan Book Agency.

18. Construction of everywhere continuous but no-where dierentiable functions:

R.R Goldberg, Methods of Real Analysis , Oxford IBM Publications.

19. Study of uniform convergence of various sequences of real valued continuous

functions and plotting the functions of the sequences.

R.R Goldberg, Methods of Real Analysis , Oxford IBM Publications.

20. Henstock Kurzweil integration:

R.G. Bartle and D. Sherbert, Introduction to Real Analysis , Wiley India Pvt.

Ltd.

21. Symmetric matrices, Spectral theorem, quadratic forms in nvariables:

M. Artin, Algebra , Prentice Hall of India.

22. Classication of Isometries of R2:

M. Artin, Algebra , Prentice Hall of India.

23. Discrete subgroups of isometries of the plane:

M. Artin, Algebra , Prentice Hall of India.

24. Surface integrals, Line integrals, Theorem on Curl, Divergence theorem of

Gauss:

T.M. Apostol, Calculus , Volume II, Wiley India Pvt. Limited.

25. Parametrised regular surfaces in R3;tangent spaces, Orientable surfaces:

T.M. Apostol, Calculus , Volume II, Wiley India Pvt. Limited.

26. Applications of WX-Maxima plot graphs of surfaces, tangent vectors, level

sets of real valued functions f(x;y;z ):

27. Circular Permutations, Study of Sterling numbers of First Kind:

K.H. Rosen, Discrete Mathematics and its Applications (Sixth edition), Tata

McGraw Hill Publishing Company, New Delhi.

28. Generating Functions and its applications (Counting, Solving Dierential Equa-

tions): K.H. Rosen, Discrete Mathematics and its Applications (Sixth edi-

tion), Tata McGraw Hill Publishing Company, New Delhi.

29. Recurrence relations and applications:

K.H. Rosen, Discrete Mathematics and its Applications (Sixth edition), Tata

McGraw Hill Publishing Company, New Delhi.

15

30. Forbidden position problems:

K.H. Rose, Discrete Mathematics and its Applications (Sixth edition), Tata

McGraw Hill Publishing Company, New Delhi.

31. Applications of Pigeon Hole Principle:

K.H. Rosen, Discrete Mathematics and its Applications (Sixth edition), Tata

McGraw Hill Publishing Company, New Delhi.

32. Basic Logic, Poset and Lattices:

a) K. H. Rosen, Discrete Mathematics and its Applications (Sixth edi-

tion), Tata McGraw Hill Publishing Company, New Delhi(Chapter 1).

b) V. K. Khanna, Lattices and Boolean Algebras- First Concepts , Vikas

Publishing House Pvt Ltd ( Chapter 2).

33. Boolean algebra (Lattices and Algebraic Systems):

C.U. Liu and D.P. Mahapatra, Discrete mathematics , McGraw Hill.

34. Algorithms in Cryptography:

a) Kenneth H. Rosen, Discrete Mathematics and Its Applications , 7th

Edition, McGraw Hill, 2012.

b) Douglas R. Stinson, Cryptography Theory and Practice , 3rd Edition,

2005.

35. Berge Vieta and Bairstow Method, proofs and programming implementation:

S.S. Sastry, Numerical Methods: For Scientic and Engineering Computa-

tion, New Age International Publishers. See also M.K.Jain,S.R.K.Iyengar &

R.K.Jain, Numerical Methods .

36. Jordan Rational Form, Algorithmic proofs and computations:

D. S. Dummit and R.M. Foote, Abstract Algebra , Wiley India Pvt. Limited.

37. Homogeneous coordinates, transformations and computer geometry:

Computer Graphics (Special Indian Edition) (Schaum's Outline Series) 2nd

Edition.

38. Bezier curves, B-splines implementation and denition:

S.S. Sastry, Introductory Methods of Numerical Analysis , Prentice hall India.

39. Number systems in various bases:

- H. M. Antia, Numerical Methods for Scientists and Engineers , American

Mathematical Society, 2012.

40. Financial Mathematics (Theory of interest rates and Discounted cash
ow):

a) Mc Cutch eon and Scot Heinemann, An introduction to the Mathemat-

ics of Finance , Professional publishing.

b) Sheldon M.Ross, An Introduction to Mathematical Finance , Cam-

bridge University Press.

41. Financial Mathematics (Valuation of securities, Cumulative Sinking Funds):

a) Mc Cutch eon and Scot Heinemann, An introduction to the Mathemat-

ics of Finance , Professional publishing.

16

b) Sheldon M.Ross, An Introduction to Mathematical Finance , Cam-

bridge University Press.

42. Mathematical Economics (Demand and Supply Analysis, Cost and Revenue

Functions, Theory of Consumer Behaviour):

H.L. Ahuja, Principles of Micro Economics , 15th Revised Edition, S. Chand.

43. Basic Statistics (Correlation and Regression):

a) G. Gupta and D. Gupta, Fundamentals of Statistics , Vol. 1, The World

Press Pvt. Ltd., Kolkata.

b) Gupta and Kapoor, Fundamentals of Mathematical Statistics , Sultan

Chand and Sons, New Delhi.

c) Hogg, R. V. and Craig R. G., Introduction to Mathematical Statistics ,

fourth Edition, MacMillan Publishing Co., New York.

44. Implementing Statistical methods using R:

W. N. Venables, D.M. Smith and the R Development Core Team, An Intro-

duction to R, Notes on R: A Programming Environment for Data Analysis

and Graphics , Version 3.0.1 (2013-05-16), (URL: https://cran.r-project.org/doc/manuals/r-

release/R-intro.pdf).

45. Social Network Analysis:

R. A. Hanneman, M. Riddle, Introduction to Social Network Methods , Uni-

versity of California, 2005 (Published in digital form and available at http://faculty.ucr.edu/ han-

neman/nettext/index.html).

46. Basics of R programming:

W. N. Venables, D.M. Smith and the R Development Core Team, Notes on

R: A Programming Environment for Data Analysis and Graphics , Version

3.0.1 (2013-05-16), (URL: https://cran.r-project.org/doc/manuals/r-release/R-

intro.pdf).

47. Topics in Data Sciences:

C. ONeil, R. Schutt and OReilly, Doing Data Science, Straight Talk From

The Frontline , 2014.

SEMESTER VI

USMT601/UAMT601 Real and Complex Analysis

Unit I: Sequence and series of functions (15 Lectures)

Sequence of real valued functions, pointwise and uniform convergence of sequences

of real-valued functions, examples. Uniform convergence implies pointwise conver-

gence, example to show converse not true.

Series of functions, convergence of a series of functions, Weierstrass M-test. Ex-

amples.

Properties of uniform convergence: Continuity of the uniform limit of a sequence of

continuous function, conditions under which integral and the derivative of sequence

17

of functions converge to the integral and derivative of uniform limit on a closed

and bounded interval, examples. Consequences of these properties for series of

functions, term by term dierentiation and integration.

lim inf

n!1xn& lim sup

n!1xnfor a bounded sequence (xn)n2NofR:

Properties of lim sup

n!1xn=:x:

1.9a subsequence (xnk)k2Nof the sequence (xn)such thatxnk!x:

2. Ifx>x;then9n02Nsuch thatxnx8nn0:

Power series in Rcentered at origin and at some point x0inR;radius of convergence,

region (interval) of convergence, uniform convergence, term by-term dierentiation

and integration of power series, examples. Uniqueness of series representation,

functions represented by power series, classical functions dened by power series such

as exponential, cosine and sine functions, the basic properties of these functions.

Reference for Unit I :

1.R.R. Goldberg ,Methods of Real Analysis , Oxford and International Book

House (IBH) Publishers, New Delhi.

2.W. Rudin ,Principles of mathematical Analysis , Tata McGraw- Hill Edu-

cation in 2013.

3.Ajit Kumar, S. Kumaresan ,Introduction to Real Analysis , CRC Press.

Unit II: Introduction to Complex Analysis (15 Lectures)

Review of complex numbers: Complex plane, polar coordinates, exponential map,

powers and roots of complex numbers, De Moivres formula, Cas a metric space,

bounded and unbounded subsets of C;point at innity and the extended complex

plane, sketching of set in complex plane. (No question be asked).

Limit at a point, theorems on limits, convergence of sequences of complex num-

bers and results using properties of real sequences. Functions f:C!C;real

and imaginary part of functions, continuity at a point and algebra of continuous

functions.

Derivative of f:C!C;comparison between dierentiability in real and complex

sense, Cauchy-Riemann equations, sucient conditions for dierentiability, analytic

functions. If f;g are complex analytic then f+g;fg;fg andf=g are analytic.

Theorem: If f0(z) = 0 everywhere in a domain D;thenf(z)must be constant

throughout D:Harmonic functions and harmonic conjugate.

Reference for Unit II :

Sections 5,6, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 of J. W. Brown

and R. V. Churchill ,Complex variables and applications , McGraw-Hill Inter-

national, sixth edition.

Unit III: Complex power series (15 Lectures)

Contour integralR

Cf(z)dzover a contour C;the contour integralR

Cf(z)dzwhere

Cis the circlejzz0j=rinC:

Cauchy-Gursat theorem (statement only).

Principle of deformation of paths (statement only): Let C1;C2denote positively

18

oriented circles where C2is interior to C1:If a function fis analytic in the closed

region consisting of those contours and all points between them, thenR

C1fdz =R

C2fdz:

Cauchy integral formula (with proof): If f:C!Cis an analytic function, then

f(w) =1

2iZ

Cf(z)dz

zw(w2B(z0;r)whereCis the circlejzz0j=rtaken in

the positive sense.

Taylors theorem (with proof) for an analytic function.

Mobius transformations, examples.

Exponential function and its properties (without proof), trigonometric functions,

hyperbolic functions.

Power series of complex numbers and related results following from Unit I, radius

of convergence of a power series, disc of convergence of a power series, uniqueness

of series representation, examples.

Denition of Laurent series, denition of isolated singularity, statement (without

proof) of existence of Laurent series expansion in neighbourhood of an isolated

singularity, type of isolated singularities viz. removable, pole and essential dened

using Laurent series expansion, Cauchy's residue theorem (statement only), calcu-

lation of residue.

Reference for Unit III :

Sections 23, 24, 25, 30, 31, 32, 33, 39, 44, 45, 46, 47, 49, 50, 53, 54, 55, 56

ofJ. W. Brown and R. V. Churchill ,Complex variables and applications ,

McGraw-Hill International, sixth edition. Dene residue of a function at a pole using

Theorem in section 56. Statement of Cauchys residue theorem in section 54.

Additional Reference Books :

1.T. Apostol ,Mathematical Analysis , Narosa.

2.M. H. Protter and C. B. Morrey Jr. ,Intermediate Calculus .

3.T. W. Gamelin ,Complex analysis .

4.R. Courant and F. John ,Introduction to Calculus and Analysis , Vol.2.

5.W. Fleming ,Functions of Several Variables .

6.D. V. Widder ,Advanced Calculus , Dover Pub., New York.

7.S. R. Ghorpade and B. Limaye ,A course in Multivariable Calculus

and Analysis .

8.G.B. Thomas and R.L Finney ,Calculus and Analytic Geometry .

9.R.E. Greene and S.G. Krantz ,Function theory of one complex vari-

able.

USMT602/UAMT602 Algebra VI

Unit I: Normal Subgroups (15 Lectures)

Review of Groups, Subgroups, Abelian groups, Order of a group, Finite and innite

19

groups, Cyclic groups, The Center Z(G)of a groupG;Cosets, Lagranges theorem,

Group homomorphisms, isomorphisms, automorphisms, inner automorphisms.

Normal subgroups: Normal subgroups of a group, denition and examples including

center of a group, Quotient group, Alternating group An;Cycles. List of all normal

subgroups of A4;S3:

First Isomorphism theorem (Fundamental Theorem of homomorphisms of groups),

Second Isomorphism theorem, third Isomorphism theorem.

Cayleys theorem (statement only). External direct product of a group, properties of

external direct products, order of an element in a direct product, criterion for direct

product to be cyclic. The classication of groups of order upto 7:

References for unit I :

1.N. Herstein ,Topics in Algebra , Wiley india Pvt. Ltd, 2015.

2.M. Artin, Algebra , Pearson India, Fifth Edition, 2017.

Unit II: Ring Theory (15 Lectures)

Denition of a ring (the denition should include the existence of a unity ele-

ment). Properties and examples of rings including Z;Q;R;C;Mn(R);Q[X];R[X];

C[X];Z[i];Z[p

2];Z[p

5];Zn:

Commutative rings. Units in a ring. The multiplicative group of units of a ring.

Characteristic of a ring.

Ring homomorphisms. First Isomorphism theorem of rings.

Ideals in a ring, sum and product of ideals in a commutative ring.

Quotient rings. Integral domains and elds. Denition and examples. A nite

integral domain is a eld. Characteristic of an integral domain, and of a nite eld.

Construction of quotient eld of an integral domain (emphasis on Z;Q). A eld

contains a subeld isomorphic to ZporQ:

References for Unit II :

1.M. Artin, Algebra , Pearson India, Fifth Edition, 2017.

2.N.S. Gopalkrishnan ,University Algebra , New Age International, third

edition, 2015.

Unit III: Factorisation (15 Lectures)

Prime ideals and maximal ideals. Denition and examples. Characterization in

terms of quotient rings.

Polynomial rings. Irreducible polynomials over an integral domain. Unique Fac-

torization Theorem for polynomials over a eld (statement only). Divisibility in

an integral domain, irreducible and prime elements, ideals generated by prime and

irreducible elements.

Denition of a Euclidean domain (ED), Principal Ideal Domain (PID), Unique

20

Factor-ization Domain (UFD). Examples of ED including Z; F[X]whereFis a

eld, and Z[i]:An ED is a PID, a PID is a UFD.

Prime (irreducible) elements in R[X];Q[X];Zp[X]:Prime and maximal ideals in

polynomial rings. Z[X]is not a PID. Z[X]is a UFD (Statement only).

Reference for Unit III :

1.M. Artin, Algebra , Pearson India, Fifth Edition, 2017.

2.N.S. Gopalkrishnan ,University Algebra , New Age International, third

edition, 2015.

Additional Reference Books :

1.P. B. Bhattacharya, S. K. Jain, and S. R. Nagpaul ,Abstract Al-

gebra , Cambridge University Press, 1995.

2.J. B. Fraleigh ,A First course in Abstract Algebra , Narosa.

3.D. Dummit and R. Foote Abstract Algebra , John Wiley & Sons, Inc.

USMT603/UAMT603 Metric Topology

All concepts have to be taught with plenty of examples and worked out in special

case of Euclidean space, Complex plane and other metric spaces.

Unit I. Complete metric spaces (15 Lectures)

Convergent sequences, Cauchy's principle of convergence, convergent Cauchy se-

quences, Complete metric spaces. Completeness property in subspaces of a complete

metric space: Any closed subset of a complete metric space is complete.

Cantor's intersection theorem. Examples of Complete metric spaces: R;Rn;C[a;b]:

IfX;Y are complete metric spaces with metrics d1;d2respectively, then XYis

complete with metric d((x1;y1);(x2;y2)) =p

d1(x1;x2)2+d2(x2;y2)2:

Reference for unit I :

S. Kumaresan ,Topology of Metric spaces , Narosa.

Unit II: Compact metric spaces :

(a) Denition of a compact set in a metric space (as a set for which every open

cover has a nite subcover), examples. Properties such as: i) Continuous image

of a compact set is compact, ii) Compact subsets of a metric space are closed and

bounded, iii) A continuous function on a compact set is uniformly continuous.

Compactness and nite intersection property: A metric space Xis compact if and

only if for every innite family fF:2Sgof closed subsets of Xwith nite

intersection property, \2SFis not empty. Every innite, bounded subset of a

compact metric space has an accumulation point (cluster point). A compact metric

space is complete.

Characterization of compact sets in Rn:

The following are equivalent statements for a subset of Rnto compact:

1. Heine-Borel property.

21

2. Closed and boundedness property.

3. Bolzano-Weierstrass property.

4. Sequentially compactness property.

Reference for Unit II :

1.S. Kumaresan ,Topology of Metric spaces , Narosa.

2.W. Rudin ,Principles of Mathematical Analysis , Tata McGraw- Hill Edu-

cation in 2013..

Unit III. Connected sets (15 lectures)

Connected metric spaces ( a metric space which can not be represented as the

union two disjoint non-empty open subsets). Characterization of a connected space,

namely a metric space Xis connected if and only if every continuous function from

Xto the discrete metric space f1;1gis a constant function. Connected subsets

of a metric space, connected subsets of Rare intervals. A continuous image of

a connected set is connected, applications such as : i) GL(2;R);O(n;R)are not

connected, ii) graph of a real valued continuous function dened on an interval is a

connected subset of R2:

ForA;B be two connected subsets of a metric space X;i)A\B6=;impliesA[B

is connected , ii) ABAimpliesBis connected. Circle S1is a connected

subset of R2:

Denition of a path connected metric space, examples including Rn;Sn(n2N):A

path connected metric space is connected and applications including connectedness

ofRn;Cn:An example of a connected subset of R2which is not path connected

(proof not required). An open subset of Rnis connected if and only if it is path-

connected ((proof not required)).

Reference for Unit III :

1.S. Kumaresan ,Topology of Metric spaces , Narosa.

2.G.F. Simmons ,Introduction to Topology and Modern Analysis , McGraw-

Hill Education (India), 2004.

Recommended Text Books :

1.S. Kumaresan ,Topology of Metric spaces , Narosa.

2.G.F. Simmons ,Introduction to Topology and Modern Analysis , McGraw-

Hill Education (India), 2004.

3.Irvin Kaplansky ,Set Theory and Metric spaces , Allyn and Bacon Inc,

Boston.

Additional Reference Books :

1.MchealO Searc oid,Metric spaces , Springer Undergraduate Mathemat-

ics Series, 2007.

2.R.G. Goldberg ,Methods of Real Analysis , Oxford and IBH Publishing

House, New Delhi.

22

USMT6A4/UAMT6A4 Numerical Analysis II (Elective A)

N.B. Derivations and geometrical interpretation of all numerical methods with the-

orem mentioned have to be covered.

Unit I: Interpolation (15 Lectures)

Interpolating polynomials, uniqueness of interpolating polynomials. Linear, Quadratic

and higher order interpolation. Lagranges Interpolation.

Finite dierence operators: Shift operator, forward, backward and central dierence

operator, Average operator and relation between them. Dierence table, Relation

between dierence and derivatives. Interpolating polynomials using nite dier-

ences: Gregory-Newton forward dierence interpolation, Gregory-Newton backward

dierence interpolation, Stirlings Interpolation. Results on interpolation error.

Unit II: Polynomial Approximations and Numerical Dierentiation

(15 Lectures)

Piecewise Interpolation: Linear, Quadratic and Cubic. Bivariate Interpolation: La-

granges Bivariate Interpolation, Newtons Bivariate Interpolation.

Numerical dierentiation: Numerical dierentiation based on Interpolation, Nu-

merical dierentiation based on nite dierences (forward, backward and central),

Numerical Partial dierentiation.

Unit III: Numerical Integration (15 Lectures)

Numerical Integration based on Interpolation: Newton-Cotes Methods, Trapezoidal

rule, Simpsons 1/3-rd rule, Simpsons 3/8-th rule. Determination of error term for

all above methods. Convergence of numerical integration: Necessary and sucient

condition (with proof). Composite integration methods: Trapezoidal rule, Simp-

sons rule.

Recommended Text Books :

1.E. Kendall and Atkinson ,An Introduction to Numerical Analysis ,

Wiley.

2.M. K. Jain, S. R. K. Iyengar and R. K. Jain ,Numerical Methods

for Scientic and Engineering Computation , New Age International Publi-

cations.

3.S.D. Comte and Carl de Boor ,Elementary Numerical Analysis,an

Algorithmic Approach , McGraw Hill International Book Company.

4.S. Sastry ,Introductory methods of Numerical Analysis , PHI Learning.

5.F.B. Hildebrand ,Introduction to Numerical Analysis , Dover Publication,

NY.

6.J.B. Scarborough ,Numerical Mathematical Analysis , Oxford University

Press, New Delhi.

23

USMT6B4/UAMT6B4

Number Theory and its applications II (Elective B)

Unit I: Quadratic Reciprocity (15 Lectures)

Quadratic residues and Legendre symbol, Gausss Lemma, Theorem on Legendre

symbol2

p

;the result: If pis an odd prime and ais an odd integer, then

a

p

= (1)t;wheret=(p1)=2X

k=1ka

p

;Quadratic Reciprocity law. Theorem on

Legendre Symbol3

p

. The Jacobi symbol and law of reciprocity for Jacobi Sym-

bol. Quadratic Congruences with Composite moduli.

Unit II: Continued Fractions (15 Lectures)

Finite continued fractions. Innite continued fractions and representation of an ir-

rational number by an innite simple continued fraction, Rational approximations

to irrational numbers and order of convergence, Best possible approximations. Pe-

riodic continued fractions.

Unit III. Pells equation, Arithmetic function and Special numbers

(15 Lectures)

Pell's equation x2dy2=n;wheredis not a square of an integer. Solutions of

Pell's equation (The proofs of convergence theorems to be omitted).

Arithmetic functions of number theory: d(n)(or(n)); (n); k(n); !(n)and

their properties, (n)and the Mbius inversion formula.

Special numbers: Fermat numbers, Mersenne numbers, Perfect numbers, Amicable

numbers, Pseudo primes, Carmichael numbers.

Recommended Text Books :

1.N.H. Zuckerman and H. Montogomery ,An Introduction to the The-

ory of Numbers , John Wiley & Sons.

2.D.M. Burton ,An Introduction to the Theory of Numbers , Tata McGraw

Hill.

Additional Reference Books:

1.G. H. Hardy and E.M. Wright ,An Introduction to the Theory of

Numbers .

2.Neville Robins ,Beginning Number Theory , Narosa Publications.

3.S. D. Adhikari ,An introduction to Commutative Algebra and Number

Theory .

4.N. Koblitz ,A course in Number theory and Crytopgraphy , Springer.

5.M. Artin ,Algebra , Prentice Hall.

6.K.Ireland and M. Rosen ,A classical introduction to Modern Number

Theory .

7.W. Stalling ,Cryptology and network security .

24

USMT6C4/UAMT6C4

Graph Theory and Combinatorics (Elective C)

Unit I. Colorings of graph (15 Lectures)

Vertex coloring- evaluation of vertex chromatic number of some standard graphs,

critical graph. Upper and lower bounds of Vertex chromatic Number- Statement of

Brooks theorem. Edge coloring- Evaluation of edge chromatic number of standard

graphs such as complete graph, complete bipartite graph, cycle. Statement of Vizing

Theorem.

Chromatic polynomial of graphs- Recurrence Relation and properties of Chromatic

polynomials. Vertex and Edge cuts vertex and edge connectivity and the relation

between vertex and edge connectivity. Equality of vertex and edge connectivity of

cubic graphs. Whitney's theorem on 2-vertex connected graphs.

Unit II. Planar graphs (15 Lectures)

Denition of a planar graph. Euler formula and its consequences. Non planarity

ofK5; K(3; 3):Dual of a graph. Polyhedrons in R3and existence of exactly ve

regular polyhedra- (Platonic solids).

Colorability of planar graphs- 5color theorem for planar graphs, statement of 4color

theorem.

Networks and
ow and cut in a network- value of a
ow and the capacity of cut

in a network, relation between
ow and cut. Maximal
ow and minimal cut in a

network and Ford- Fulkerson theorem.

Unit III: Combinatorics (15 Lectures)

Applications of Inclusion Exclusion Principle- Rook polynomial, Forbidden position

problems.

Introduction to partial fractions and using Newtons binomial theorem for real power

nd series expansion of some standard functions.

Forming recurrence relation and getting a generating function. Solving a recurrence

relation using ordinary generating functions. System of Distinct Representatives and

Hall's theorem of SDR.

Introduction to matching, Malternating and Maugmenting path, Berge theorem.

Bipartite graphs.

Recommended Text Books :

1.J. A. Bondy and U.S.R. Murty ,Graph Theory with Applications ,

Springer, 2008.

2.R. Balkrishnan and K. Ranganathan ,Graph theory and applica-

tions , North Holland, 1982.

3.D.G. West ,Introduction to Graph theory , Pearson Modern Classics.

25

4.R. Brualdi ,Introduction to Combinatorics , Pearson Education.

Additional Reference Books :

1.M. Behzad and G. Chartrand ,Introduction to the theory of Graphs ,

Allyn and Bacon.

2.S.A. Choudam ,A First course in Graph Theory , Macmillam India Ltd.

USMT6D4/UAMT6D4 Operations Research (Elective D)

Unit I: Linear Programming-I (15 Lectures)

Prerequisites: Vector Space, Linear independence and dependence, Basis, Convex

sets, Dimension of polyhedron, Faces.

Formation of LPP, Graphical Method. Theory of the Simplex Method- Standard

form of LPP, Feasible solution to basic feasible solution, Improving BFS, Optimality

Condition, Unbounded solution, Alternative optima, Correspondence between BFS

and extreme points. Simplex Method Simplex Algorithm, Simplex Tableau.

Reference for Unit-I :

G. Hadley ,Linear Programming , Narosa Publishing.

Unit II: Linear programming-II (15 Lectures)

Simplex Method Case of Degeneracy, Big-M Method, Infeasible solution, Alternate

solution, Solution of LPP for unrestricted variable. Transportation Problem: For-

mation of TP, Concepts of solution, feasible solution, Finding Initial Basic Feasible

Solution by North West Corner Method, Matrix Minima Method, Vogels Approxi-

mation Method. Optimal Solution by MODI method, Unbalanced and maximization

type of TP.

Reference for Unit-II:

1.G. Hadley ,Linear Programming , Narosa Publishing.

2.J. K. Sharma ,Operations Research, Theory and Applications .

Unit III: Queuing Systems (15 Lectures)

Elements of Queuing Model, Role of Exponential Distribution. Pure Birth and Death

Models; Generalized Poisson Queuing Mode. Specialized Poisson Queues: Steady-

state Measures of Performance, Single Server Models, Multiple Server Models, Self-

service Model, Machine-servicing Model.

Reference for Unit III :

1.J. K. Sharma ,Operations Research, Theory and Applications .

2.H. A. Taha ,Operations Research , Prentice Hall of India.

Additional Reference Books :

1.Hillier and Lieberman ,Introduction to Operations Research .

2.R. Broson ,Schaum Series Book in Operations Research , Tata McGraw

Hill Publishing Company Ltd.

USMTP07, UAMTP07

Practicals for USMT601/UAMT601, USMT602/UAMT602 & USMT603/UAMT603

A. Practicals for USMT601/UAMT601 :

26

1. Pointwise and uniform convergence of sequence functions, properties.

2. Point wise and uniform convergence of series of functions and properties.

3. Analytic function, nding harmonic conjugate, Mobius transformations.

4. Cauchy integral formula, Taylor series, power series.

5. Limit continuity and derivatives of functions of complex variables.

6. Finding isolated singularities- removable, pole and essential, Laurent series,

calculation of residue.

7. Miscellaneous theory questions based on full paper (3 theory questions from

each unit).

B. Practicals for USMT602/UAMT602 :

1. Normal Subgroups and quotient groups.

2. Cayleys Theorem and external direct product of groups.

3. Rings, Ring Homomorphism and Isomorphism.

4. Ideals, Prime Ideals and Maximal Ideals.

5. Euclidean Domain, Principal Ideal Domain and Unique Factorization Domain.

6. Fields.

7. Miscellaneous theory questions based on full paper.

C. Practicals for USMT603/UAMT603 :

1. Completeness of R;Rn:

2. A metric space Xis complete if and only if every closed ball of Xis complete.

3. Compact sets in a metric space, Compactness in Rn(emphasis on R;R2),

properties.

4. Continuous image of a compact set.

5. Example of a closed and bounded subset of a metric space which is not

compact.

6. Connectedness, Path connectedness.

7. Continuous image of a connected set.

8. Miscellaneous Theoretical Questions based on full paper.

27

USMTPJ6, UAMTPJ6: Projects

A student can submit a project which shall have 20-30 typed pages, on one of the

following topics:

1. Apps for small devices using Python:

Chapter 7ofHead First Python by Paul Barry, O'Reilly Media, second edi-

tion.

2. Apps for small devices using Java:

Java How to Program (early objects) by Paul Deitel and Harvey Deitel,

Pearson 9th edition (2012).

3. Elliptic Curves and their uses in Cryptography, Pollard's Algorithm:

D. Hankerson, A.J. Menezes, S. Vanstone, Guide to Elliptic Curve Cryptog-

raphy , Springer.

4. Runge-Kutta methods, principle and proofs of second and fourth order com-

puter programs:

S.S. Sastry, Introductory Methods of Numerical Analysis , Prentice hall In-

dia.

5. The matrix exponential and applications to system of Dierential equations

X0=AX:

M. Artin, Algebra , Pearson India Education.

6. Iterated solutions of Picard's theorem and solutions of second order linear

ODE:

M. W. Hirsch, S. Smale and R.L. Devaney, Dierential Equations, Dynam-

ical Systems, and an Introduction to Chaos , Academic Press.

7. The Qualitative properties of the solutions of y00+P(x)y0+Q(x)y= 0;Sturm

Separation theorem:

G. F. Simmons, Dierential Equations with Applications and Historical

Notes , McGRAW-Hill International.

8. Bessel Functions, The Gamma Function and the general solution of bessel's

equation: G. F. Simmons, Dierential Equations with Applications and

Historical Notes , McGRAW-Hill International.

9. Algebraic numbers, algebraic integers:

H.S. Zuckerman and I. Niven, An Introduction to the Theory of Numbers ,

Wiley Eastern Ltd.

10. Quadratic elds ,units, primes, UFD:

H.S. Zuckerman and I. Niven, An Introduction to the Theory of Numbers ,

Wiley Eastern Ltd.

11. Structure of nite Abelian groups:

S. Lang, Algebra , Springer.

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12. The Class Equation, Application to pgroups:

M. Artin, Algebra , Prentice Hall of India.

13. The Class Equation of Icosahedral group.:

M. Artin, Algebra , Prentice Hall of India.

14. The Class Equation, classication of groups of order 12:

M. Artin, Algebra , Prentice Hall of India.

15. Construction of numbers by Ruler & Compass:

D. S. Dummit and R.M. Foote, Abstract Algebra , Wiley India Pvt. Limited.

16. Field Extensions, Cubic equations, Cardano's method:

D. S. Dummit and R.M. Foote, Abstract Algebra , Wiley India Pvt. Limited.

17. Character groups of small Order:

D. S. Dummit and R.M. Foote, Abstract Algebra , Wiley India Pvt. Limited.

18. Finite Division ring is a feild and sum of two squares:

I.N. Herstein, Topics in Algebra , Wiley India Pvt. Limited.

19. Study of Polya theory of Counting:

K.H. Rosen, Discrete Mathematics and its Applications , Tata McGraw Hill

Publishing Company, New Delhi,(Sixth edition).

20. Hall's Marriage Theorem, Graph theory & Applications:

K.H. Rosen, Discrete Mathematics and its Applications , Tata McGraw Hill

Publishing Company, New Delhi,(Sixth edition).

21. Ramsey numbers:

K.H. Rosen, Discrete Mathematics and its Applications , Tata McGraw Hill

Publishing Company, New Delhi,(Sixth edition).

22. Axiom of choice, Zorn's Lemma:

Set theory related topics, Schaum series. See also S. Lang, Analysis II.

23. Introduction to Cryptography:

a) Kenneth H. Rosen, Discrete Mathematics and Its Applications, 7th Edi-

tion, McGraw Hill, 2012.

b) Cryptography Theory and Practice, 3rd Edition, Douglas R. Stinson,

2005.

24. Separable metric spaces, study of completions of C[a;b]under norms such as

sup-norm,L1norm,L2norm:

R.R Goldberg, Methods of Real Analysis , Oxford IBM Publications.

25. Study of Baire spaces and application to limit of a sequence of real valued

continuous functions dened on R:

J. R. Munkres, Topology , Pearson Education India.

26. Completion of Metric spaces:

J. R. Munkres, Topology , Pearson Education India.

27. Maximum principle for analytic functions and applications:

Lars Ahlfors, Complex Analysis , McGraw Hill Education (India) Private Lim-

ited, 2013.

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28. Exponential function ez;Epimorphism Theorem, eix=c(x) +is(x), study of

circular functions c(x);s(x)and identication with Trigonometric functions:

R. Remmert, Classical Topics in Complex Function Theory , Springer.

29. Plotting regions under Mobius Transformations:

S. Lang, Complex Analysis , Springer.

30. Study of Mobius Transformations, cross-ratio, Applications:

J.B. Conway, Functions of One Complex Variable I , Mcgraw Hill.

31. Radius of Convergence of power series, Abel's Limit theorem and applications:

T.M Apostol, Mathematical Analysis , Narosa.

32. Dixon's proof of Cauchy's theorem ( Homology version) and deduction of

simply connected version:

S. Lang, Complex analysis , Springer. see also: Lars Ahlfors, Complex Anal-

ysis, McGraw Hill Education (India) Private Limited.

33. Classication of Isometries of R3:

J.T. Smith, Methods of geometry .

34. Discrete subgroups of isometries of the plane:

M. Artin, Algebra , Prentice Hall of India.

35. Tiling and Crystallographic groups:

M. Berger, Geometry I , Springer.

36. Group of symmetries of regular polyhedra:

R. Hartshorne, Geometry: Euclid and Beyond , Springer.

37. Jordan blocks of matrices and application to solving a system of linear ODE:

M. W. Hirsch, S. Smale and R.L. Devaney, Dierential Equations, Dynam-

ical Systems, and an Introduction to Chaos , Academic Press.

38. Alternating k-tensors on a nite dimensional real vector space, orientation

and volume elements and theory of dierential forms:

M. Spivak, Calculus on manifolds , W.A. Benzamin Inc.

39. Dierential forms, Stokes' theorem and applications to Green's theorem, Gauss'

Divergence theorem:

M. Spivak, Calculus on manifolds , W.A. Benzamin Inc.

40. Locally compact metric spaces, One-Point Compactication, One-Point com-

pactication of Cis homeomorphic to the unit sphere S2R3:

J. R. Munkres, Topology , Pearson Education India.

41. First Fundamental Groups of a metric space, computation of 1(S1;1) :

J. R. Munkres, Topology , Pearson Education India.

42. Covering spaces of a Metric space, examples:

J. R. Munkres, Topology , Pearson Education India.

43. Path-homotopy relation, Simply Connected metric Spaces, Examples:

J. R. Munkres, Topology , Pearson Education India.

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44. Simplicial complexes in R2;R3singular chains, Homology groups H0;H1:

J. R. Munkres, Elements of Algebraic Topology , Addison-Wesley Publishing.

45. Homology groups:

John B. Fraleigh, A First Course in Abstract Algebra , Pearson Education,

India.

46. A non-computational proof of Cayley-Hamilton theorem, canonical isomor-

phism with double dual Tensor products:

S. Lang, Introduction to Linear Algebra , Springer Verlag.

47. Topics in Projective Geometry:

R. Artzy, Linear geometry , Addison-Wesley.

48. Topics in Non-Euclidean Geometries:

R. Artzy, Linear Geometry , Dover Publications.

49. Special relativity:

W. H. Greub, Linear Algebra , Springer.

50. LU factorization using Gaussian elimination:

S.S. Sastry, Numerical Methods: For Scientic and Engineering Compu-

tation , New Age International Publishers.

51. Error estimates of proofs and implementation of Trapezoidal rule, simpson

rule, Romberg method adaptive integration:

S.S Sastry, Numerical methods: For Scientic and Engineering Computa-

tion, New Age International Publishers.

52. Discrete Fourier transform, Fast Fourier transform:

S.S Sastry, Numerical methods: For Scientic and Engineering Computa-

tion, New Age International Publishers.

53. Topics in Automata Theory:

K.L.P. Mishra and N. Chandrasekaran, Theory of Computer Science, Au-

tomata, Languages and Computation (Third Edition), Prentice- Hall of India

Pvt. Ltd.

54. Operations Research (Game Theory and Quality Control):

a) Kanti Swarup, P. K. Gupta and Man Mohan, Operation Research , Sul-

tan Chand and Sons.

b) Taha, Operations Research: introduction , Pearson.

55. Operations Research (Integer L.P.P. and Inventory models):

a) P. K. Gupta, Man Mohan and Kanti Swarup, Operation Research , Sul-

tan Chand and Sons.

b) Taha, Operations Research: introduction , Pearson.

56. Operations Research and Markov Chains:

a) Kanti Swarup, P. K. Gupta and Man Mohan, Operation Research , Sul-

tan Chand and Sons.

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b) Taha, Operations Research: introduction , Pearson.

57. Discrete and Continuous probability distributions:

W.G. Cochran, Sampling techniques , third Edition, Wiley Eastern Ltd., New

Delhi.

Scheme of Examination

I. Semester End Theory Examinations :

There shall be a Semester-end external Theory examination of 100 marks for all

the courses of Semester V and VI- except for the two project courses USMTPJ5/UAMTPJ5,

USMTPJ6/UAMTPJ6 - to be conducted by the University.

1. Duration: The examinations shall be of 3Hours duration.

2. Theory Question Paper Pattern:

a) There shall be FIVE questions. All the questions shall be compulsory.

The rst question Q1 shall be of objective type for 20 marks based on

the entire syllabus.

The next four questions Q2, Q3, Q4, Q5 shall be of 20 marks each.

The questions Q2, Q3, Q4 shall be based on the units I, II , III respec-

tively.

The question Q5 shall be based on the entire syllabus.

b) The questions Q2,Q3,Q4,Q5 shall have internal choices within each ques-

tion. Including the choices, the marks for each question shall be 30-32.

c) The questions Q2,Q3,Q4,Q5 may be subdivided into sub-questions as a,

b, c, d & e, etc and the allocation of marks depends on the weightage

of the topic.

d) The question Q1 may be subdivided into 10 sub-questions of 2 marks

each.

II. Semester End Examinations Practicals :

There shall be a Semester-end practical examinations of three hours duration and

150 marks for each of the courses USMTP05/UAMTP05 of Semester V and

USMTP06/UAMTP06 of semester VI.

In semester V, the Practical examinations for USMT501/UAMT501 ,

USMT502/UAMT502, USMT503/UAMT503 are conducted together by the college.

Similarly in semester VI, the Practical examinations for USMT601/UAMT601 ,

USMT602/UAMT602, USMT603/UAMT603 are conducted together by the college.

Question Paper pattern: The question paper shall have three parts A,B, C. Every

part shall have three questions of 20 marks each. Students to attempt any two

questions from each part.

For each course USMT501/UAMT501, USMT502/UAMT502, USMT503/UAMT503,

USMT601/UAMT601, USMT602/UAMT602, and USMT603/UAMT603 marks for

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journal and viva are as follows:

Journals : 5 marks

Viva : 5 marks

Each Practical of every course of Semester V & VI shall contain 10 (ten) problems

out of which minimum 05 (ve) have to be written in the journal.

Practical Part A Part B Part C Marks duration

Course out of

Questions Questions Questions

USMTP05 from from from 120 3 hours

USMT501 USMT502 USMT503

UAMT501 UAMT502 UAMT503

Questions Questions Questions

USMTP06 from from from 120 3 hours

USMT601 USMT602 USMT603

UAMT601 USMT602 UAMT603

III. Evaluation of Project work

( courses: USMTPJ5/UAMTPJ5 & USMTPJ6/UAMTPJ6):

The evaluation of the Project submitted by a student shall be made by a Committee

appointed by the Head of the Department of Mathematics of the respective college.

The presentation of the project is to be made by the student in front of the com-

mittee appointed by the Head of the Department of Mathematics of the respective

college. This committee shall have two members, possibly with one external referee.

The Marks for the project are detailed below:

Contents of the project : 40 marks

Presentation of the project : 30 marks

Viva of the project : 30 marks.

Total Marks= 100 per project per student.

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