## Mathematics T Y B Sc syllabus2018 11_1 Syllabus Mumbai University by munotes

## Page 2

(UNIVERSITY OF MUMBAI)

Syllabus for: T.Y.B.Sc./T.Y.B.A.

Program: B.Sc./B.A.

Course: Mathematics

Choice based Credit System (CBCS)

with eect from the

academic year 2018-19

## Page 3

2

SEMESTER V

Multivariable Calculus II

Course Code UNIT TOPICS Credits L/Week

USMT 501, UAMT 501I Multiple Integrals

2.5 3 II Line Integrals

III Surface Integrals

Linear Algebra

USMT 502 ,UAMT 502I Quotien spaces and Orthogonal

2.5 3Linear Transformations

II Eigen values and Eigen vectors

III Diagonalisation

Topology of Metric Spaces

USMT 503/UAMT503I Metric spaces

2.5 3 II Sequences and Complete metric spaces

III Compact Sets

Numerical Analysis I(Elective A)

USMT5A4 ,UAMT 5A4I Errors Analysis

2.53II Transcendental and Polynomial

& Equations

III Linear System of Equations

Number Theory and Its applications I (Elective B)

USMT5B4 ,UAMT 5B4I Congruences and Factorization

2.5 3 II Diophantine equations and their

& solutions

III Primitive Roots and Cryptography

Graph Theory (Elective C)

USMT5C4 ,UAMT 5C4I Basics of Graphs

2.5 3 II Trees

III Eulerian and Hamiltonian graphs

Basic Concepts of Probability and Random Variables (Elective D)

USMT5D4 ,UAMT 5D4I Basic Concepts of Probability and

2.5 3 Random Variables

II Properties of Distribution function,

Joint Density function

III Weak Law of Large Numbers

PRACTICALS

USMTP05/UAMTP05Practicals based on

3 6 USMT501/UAMT 501 and

USMT 502/UAMT 502

USMTP06/UAMTP06Practicals based on

3 6USMT503/ UAMT 503 and

USMT5A4/ UAMT 5A4 OR

USMT5B4/ UAMT 5B4 OR

USMT5C4/ UAMT 5C4 OR

USMT5D4/ UAMT 5D4

## Page 4

3

SEMESTER VI

BASIC COMPLEX ANALYSIS

Course Code UNIT TOPICS Credits L/Week

USMT 601, UAMT 601I Introduction to Complex Analysis

2.5 3II Cauchy Integral Formula

III Complex power series, Laurent series and

isolated singularities

ALGEBRA

USMT 602 ,UAMT 602I Group Theory

2.5 3 II Ring Theory

III Polynomial Rings and Field theory

Homomorphism

Topology of Metric Spaces and Real Analysis

USMT 603 / UAMT 603I Continuous functions on

2.5 3Metric spaces

II Connected sets

Sequences and series of functions

Numerical Analysis II(Elective A)

USMT6A4 ,UAMT 6A4I Interpolation

2.53II Polynomial Approximations and

Numerical Dierentiation

III Numerical Integration

Number Theory and Its applications II (Elective B)

USMT6B4 ,UAMT 6B4I Quadratic Reciprocity

2.5 3 II Continued Fractions

III Pell's equation, Arithmetic function

& and Special numbers

Graph Theory and Combinatorics (Elective C)

USMT6C4 ,UAMT 6C4I Colorings of Graphs

2.5 3 II Planar graph

III Combinatorics

Operations Research (Elective D)

USMT6D4 ,UAMT 6D4I Basic Concepts of Probability and

2.5 3Linear Programming I

II Linear Programming II

III Queuing Systems

PRACTICALS

USMTP07/ UAMTP07Practicals based on

3 6 USMT601/UAMT 601 and

USMT 602/UAMT 602

USMTP08/UAMTP08Practicals based on

3 6USMT603/ UAMT 603 and

USMT6A4/ UAMT 6A4 OR

USMT6B4/ UAMT 6B4 OR

USMT6C4/ UAMT 6C4 OR

USMT6D4/ UAMT 6D4

## Page 5

4

Note: 1 . USMT501/UAMT501, USMT502/UAMT502, USMT503/UAMT503 are compul-

sory 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 compulsory

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.

Teaching Pattern for T.Y.B.Sc/B.A.

1. Three lectures per week per course (1 lecture/period is of 48 minutes duration).

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

duration).

Scheme of Examination

I.Semester End Theory Examinations: There will be a Semester-end external Theory

examination of 100 marks for each of the courses USMT501/UAMT501, USMT502/UAMT502,

USMT503 and USMT5A4 OR USMT5B4 OR USMT5C4 OR USMT 5D4 of Semester V

and USMT601/UAMT601, USMT602/UAMT602, USMT603 and USMT6A4 OR USMT6B4

OR USMT 6C4 OR USMT 6D4 of semester VI to be conducted by the University.

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

2. Theory Question Paper Pattern:

a) There shall be FIVE questions. The rst question Q1 shall be of objective type

for 20 marks based on the entire syllabus. The next three questions Q2, Q2, Q3

shall be of 20 marks, each based on the units I, II, III respectively. The fth

question Q5 shall be of 20 marks based on the entire syllabus.

b) All the questions shall be compulsory. The questions Q2, Q3, Q4, Q5 shall have

internal choices within the questions. 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 100

marks for each of the courses USMTP05/UAMTP05 of Semester V and USMTP06/UAMTP06

of semester VI.

In semester V, the Practical examinations for USMTP05/UAPTP05 and USMTP06/UAMTP06

are conducted by the college.

In semester VI, the Practical examinations for USMTP07/UAMTP07 and USMTP08/UAMTP08

are conducted by the University.

## Page 6

5

Question Paper pattern:

Paper pattern: The question paper shall have two parts A, B.

Each part shall have two Sections.

Section I Objective in nature: Attempt any Eight out of Twelve multiple choice ques-

tions. (83 = 24 Marks)

Section II Problems: Attempt any Two out of Three. (8 2 = 16 Marks)

Practical Part A Part B Marks duration

Course out of

USMTP05/UAMTP05 Questions from Questions from 80 3 hours

USMT501/UAMT501 USMT502/UAMT502

USMTP06/UAMTP06 Questions from Questions from 80 2 hours

USMT503/UAMT503 USMT504/UAMT504

USMTP07/UAMTP07 Questions from Questions from 80 3 hours

USMT601/UAMT601 USMT602/UAMT602

USMTP06/UAMTP08 Questions from Questions from 80 2 hours

USMT603/UAMT603 USMT604/UAMT604

Marks for Journals and Viva:

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

USMT504/UAMT504, USMT601/UAMT601, USMT602/UAMT602 USMT603/UAMT603, and

USMT604/UAMT604:

1. Journals: 5 marks.

2. Viva: 5 marks.

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

which minimum 05 (ve) have to be written in the journal. A student must have a certied

journal before appearing for the practical examination.

SEMESTER V

MULTIVARIABLE CALCULUS II

Course Code: USMT501/UAMT501

ALL Results have to be done with proof unless otherwise stated.

Unit I-Multiple Integrals (15L)

Denition of double (resp: triple) integral of a function and bounded on a rectangle (resp:box).

Geometric interpretation as area and volume. Fubini's Theorem over rectangles and any closed

bounded sets, Iterated Integrals. Basic properties of double and triple integrals proved using

the Fubini's theorem such as

(i) Integrability of the sums, scalar multiples, products, and (under suitable conditions) quo-

tients of integrable functions. Formulae for the integrals of sums and scalar multiples of

integrable functions.

## Page 7

6

(ii) Integrability of continuous functions. More generally, Integrability of functions with a

\small set of (Here, the notion of \small sets should include nite unions of graphs of

continuous functions.)

(iii) Domain additivity of the integral. Integrability and the integral over arbitrary bounded

domains. Change of variables formula (Statement only).Polar, cylindrical and spherical

coordinates, and integration using these coordinates. Dierentiation under the integral

sign. Applications to nding the center of gravity and moments of inertia.

References for Unit I:

1. Apostol, Calculus, Vol. 2, Second Ed., John Wiley, New York, 1969 Section 1.1 to 11.8

2. James Stewart , Calculus with early transcendental Functions - Section 15

3. J.E.Marsden and A.J. Tromba, Vector Calculus, Fourth Ed., W.H. Freeman and Co., New

York, 1996.Section 5.2 to 5.6.

Unit 2: Line Integrals (15L)

Review of Scalar and Vector elds on Rn, Vector Dierential Operators, Gradient, Curl, Diver-

gence.

Paths (parametrized curves) in Rn(emphasis on R2andR3), Smooth and piecewise smooth

paths. Closed paths. Equivalence and orientation preserving equivalence of paths. Denition of

the line integral of a vector eld over a piecewise smooth path. Basic properties of line integrals

including linearity, path-additivity and behavior under a change of parameters. Examples.

Line integrals of the gradient vector eld, Fundamental Theorem of Calculus for Line Inte-

grals, Necessary and sucient conditions for a vector eld to be conservative. Greens Theorem

(proof in the case of rectangular domains). Applications to evaluation of line integrals.

References for Unit II:

1. Lawrence Corwin and Robert Szczarba ,Multivariable Calculus, Chapter 12.

2. Apostol, Calculus, Vol. 2, Second Ed., John Wiley, New York, 1969 Section 10.1 to

10.5,10.10 to 10.18

3. James Stewart , Calculus with early transcendental Functions - Section 16.1 to 16.4.

4. J.E.Marsden and A.J. Tromba, Vector Calculus, Fourth Ed., W.H. Freeman and Co., New

York, 1996. Section 6.1,7.1.7.4.

Unit III: Surface Integrals (15 L)

Parameterized surfaces. Smoothly equivalent parameterizations. Area of such surfaces.

Denition of surface integrals of scalar-valued functions as well as of vector elds dened on a

surface.

Curl and divergence of a vector eld. Elementary identities involving gradient, curl and diver-

gence.

Stokes Theorem (proof assuming the general from of Greens Theorem). Examples.

Gauss Divergence Theorem (proof only in the case of cubical domains). Examples.

References for Unit III:

## Page 8

7

1. Apostol, Calculus, Vol. 2, Second Ed., John Wiley, New York, 1969 Section 1.1 to 11.8

2. James Stewart , Calculus with early transcendental Functions - Section 16.5 to 16.9

3. J.E.Marsden and A.J. Tromba, Vector Calculus, Fourth Ed., W.H. Freeman and Co., New

York, 1996 Section 6.2 to 6.4.

Other References :

1. T Apostol, Mathematical Analysis, Second Ed., Narosa, New Delhi. 1947.

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

New York, 1989.

3. W. Fleming, Functions of Several Variables, Second Ed., Springer-Verlag, New York, 1977.

4. M.H. Protter and C.B.Morrey Jr., Intermediate Calculus, Second Ed., Springer-Verlag,

New York, 1995.

5. G.B. Thomas and R.L Finney, Calculus and Analytic Geometry, Ninth Ed. (ISE Reprint),

Addison- Wesley, Reading Mass, 1998.

6. D.V.Widder, Advanced Calculus, Second Ed., Dover Pub., New York. 1989.

7. A course in Multivariable Calculus and Analysis., Sudhir R.Ghorpade and Balmohan Li-

maye, Springer International Edition.

Linear Algebra

Course Code: USMT502/UAMT502

Unit I. Quotient Spaces and Orthogonal Linear Transformations (15L)

Review of vector spaces over R, sub spaces and linear transformation.

Quotient Spaces: For a real vector space Vand a subspace W, the cosets v+W

and the quotient space V=W , First Isomorphism theorem of real vector spaces (fundamental

theorem of homomorphism of vector spaces), Dimension and basis of the quotient space V=W ,

whenVis nite dimensional.

Orthogonal transformations: Isometries of a real nite dimensional inner product space,

Translations and Re
ections with respect to a hyperplane, Orthogonal matrices over R, Equiv-

alence of orthogonal transformations and isometries xing origin on a nite dimensional inner

product space, Orthogonal transformation of R2, Any orthogonal transformation in R2is a re-

ection or a rotation, Characterization of isometries as composites of orthogonal transformations

and translation. Characteristic polynomial of an nnreal matrix. Cayley Hamilton Theorem

and its Applications (Proof assuming the result A(adjA) =Infor annnmatrix over the

polynomial ring R[t].

Unit II. Eigenvalues and eigen vectors (15L)

Eigen values and eigen vectors of a linear transformation T:V!V, where V is a nite

dimensional real vector space and examples, Eigen values and Eigen vectors of n n real ma-

trices, The linear independence of eigenvectors corresponding to distinct eigenvalues of a linear

transformation and a Matrix. The characteristic polynomial of an n real matrix and a linear

transformation of a nite dimensional real vector space to itself, characteristic roots, Similar

## Page 9

8

matrices, Relation with change of basis, Invariance of the characteristic polynomial and (hence

of the) eigen values of similar matrices, Every square matrix is similar to an upper triangular

matrix. Minimal Polynomial of a matrix, Examples like minimal polynomial of scalar matrix,

diagonal matrix, similar matrix, Invariant subspaces.

Unit III: Diagonalisation (15L)

Geometric multiplicity and Algebraic multiplicity of eigen values of an nnreal matrix, An

nnmatrixAis diagonalizable if and only if has a basis of eigenvectors of Aif and only if

the sum of dimension of eigen spaces of Ais n if and only if the algebraic and geometric multi-

plicities of eigen values of Acoincide, Examples of non diagonalizable matrices, Diagonalisation

of a linear transformation T:V!V, whereVis a nite dimensional real vector space and

examples. Orthogonal diagonalisation and Quadratic Forms. Diagonalisation of real Symmet-

ric matrices, Examples, Applications to real Quadratic forms, Rank and Signature of a Real

Quadratic form, Classication of conics in R2and quadric surfaces in R3. Positive denite and

semi denite matrices, Characterization of positive denite matrices in terms of principal minors.

Recommended Books.

1. S. Kumaresan, Linear Algebra: A Geometric Approach.

2. Ramachandra Rao and P. Bhimasankaram, Tata McGrawHillll Publishing Company.

Additional Reference Books

1. T. Bancho and J. Wermer, Linear Algebra through Geometry, Springer.

2. L. Smith, Linear Algebra, Springer.

3. M. R. Adhikari and Avishek Adhikari, Introduction to linear Algebra, Asian Books Private

Ltd.

4. K Homan and Kunze, Linear Algebra, Prentice Hall of India, New Delhi.

5. Inder K Rana, Introduction to Linear Algebra, Ane Books Pvt. Ltd.

## Page 10

9

Course: Topology of Metric Spaces

Course Code: USMT503/UAMT503

Unit I: Metric spaces (15 L)

Denition, examples of metric spaces R;R2,Euclidean space Rnwith its Euclidean, sup and

sum metric, C(complex numbers), the spaces l1and l2of sequences and the space C[a;b], of

real valued continuous functions on [ a;b]. Discrete metric space.

Distance metric induced by the norm, translation invariance of the metric induced by the norm.

Metric subspaces, Product of two metric spaces. Open balls and open set in a metric space,

examples of open sets in various metric spaces. Hausdor property. Interior of a set. Properties

of open sets. Structure of an open set in IR. Equivalent metrics.

Distance of a point from a set, between sets ,diameter of a set in a metric space and bounded

sets. Closed ball in a metric space, Closed sets- denition, examples. Limit point of a set,

isolated point, a closed set contains all its limit points, Closure of a set and boundary of a set.

Unit II: Sequences and Complete metric spaces (15L)

Sequences in a metric space, Convergent sequence in metric space, Cauchy sequence in a metric

space, subsequences, examples of convergent and Cauchy sequence in nite metric spaces, Rn

with dierent metrics and other metric spaces.

Characterization of limit points and closure points in terms of sequences, Denition and exam-

ples of relative openness/closeness in subspaces. Dense subsets in a metric space and Separability

Denition of complete metric spaces, Examples of complete metric spaces, Completeness prop-

erty in subspaces, Nested Interval theorem in R, Cantor's Intersection Theorem, Applications

of Cantors Intersection Theorem:

(i) The set of real Numbers is uncountable.

(ii) Density of rational Numbers(Between any two real numbers there exists a rational number)

(iii) Intermediate Value theorem: Let : [ a;b]Rbe continuous, and assume that f(a) andf(b)

are of dierent signs say, f(a)<0 andf(b)>0. Then there exists c2(a;b) such that

f(c) = 0.

Unit III: Compact sets 15 lectures

Denition of compact metric space using open cover, examples of compact sets in dierent metric

spaces R;R2;Rn, Properties of compact sets: A compact set is closed and bounded, (Converse

is not true ). Every innite bounded subset of compact metric space has a limit point. A

closed subset of a compact set is compact. Union and Intersection of Compact sets. Equivalent

statements for compact sets in R:

(i) Sequentially compactness property.

(ii) Heine-Borel property: Let be a closed and bounded interval. Let be a family of open

intervals such that Then there exists a nite subset such that that is, is contained in the

union of a nite number of open intervals of the given family.

(iii) Closed and boundedness property.

(iv) Bolzano-Weierstrass property: Every bounded sequence of real numbers has a convergent

subsequence.

## Page 11

10

Reference books:

1. S. Kumaresan, Topology of Metric spaces.

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

3. Expository articles of MTTS programme

Other references :

1. W. Rudin, Principles of Mathematical Analysis.

2. T. Apostol. Mathematical Analysis, Second edition, Narosa, New Delhi, 1974

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

4. R. R. Goldberg Methods of Real Analysis, Oxford and IBH Pub. Co., New Delhi 1970.

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

6. W. Rudin. Principles of Mathematical Analysis, Third Ed, McGraw-Hill, Auckland, 1976.

7. D. Somasundaram, B. Choudhary. A rst Course in Mathematical Analysis. Narosa, New

Delhi

8. G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hi, New York,

1963.

9. Sutherland. Topology.

Course: Numerical Analysis I (Elective A)

Course Code: USMT5A4/UAMT5A4

N.B. Derivations and geometrical interpretation of all numerical methods have to be covered.

Unit I. Errors Analysis and Transcendental & Polynomial Equations (15L)

Measures of Errors: Relative, absolute and percentage errors. Types of errors: Inherent error,

Round-o error and Truncation error. Taylors series example. Signicant digits and numerical

stability. Concept of simple and multiple roots. Iterative methods, error tolerance, use of in-

termediate 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 (15L)

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

tipoint iteration method. Iterative methods for polynomial equations; Descarts rule of signs,

Birge-Vieta method, Bairstrow method. Methods for multiple roots. Newton-Raphson method.

System of non-linear equations by Newton- Raphson method. Methods for complex roots. Con-

dition of convergence and Rate of convergence of all above methods.

Unit III. Linear System of Equations (15L)

Matrix representation of linear system of equations. Direct methods: Gauss elimination method.

## Page 12

11

Pivot element, Partial and complete pivoting, Forward and backward substitution method, Tri-

angularization methods-Doolittle and Crouts method, Choleskys method. Error analysis of di-

rect 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 determine largest eigenvalue and eigenvector.

Recommended Books

1. Kendall E. 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 Engi-

neering Computation, New Age International Publications.

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

McGrawHillll International Book Company.

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

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

6. Scarborough James B., Numerical Mathematical Analysis, Oxford University Press, New

Delhi.

Course: Number Theory and its applications I (Elective B)

Course Code: USMT5B4 / UAMT5B4

Unit I. Congruences and Factorization (15L)

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

Congruences : Denition and elementary properties, Complete residue system modulo m, Re-

duced residue system modulo m, Euler's function and its properties, Fermat's little Theorem,

Euler's generalization of Fermat's little Theorem, Wilson's theorem, Linear congruence, The

Chinese remainder Theorem, Congruences ofHillgher degree, The Fermat-Kraitchik Factoriza-

tion Method.

Unit II. Diophantine equations and their solutions (15L)

The linear equations ax+by=c. The equations x2+y2=p;wherepis a prime. The equa-

tionx2+y2=z2, Pythagorean triples, primitive solutions, The equations x4+y4=z2and

x4+y4=z4have no solutions ( x;y;z) withxyz6= 0. Every positive integer ncan be expressed

as sum of squares of four integers, Universal quadratic forms x2+y2+z2+t2. Assorted examples

:section 5.4 of Number theory by Niven- Zuckermann-Montgomery.

Unit III. Primitive Roots and Cryptography (15L)

Order of an integer and Primitive Roots. Basic notions such as encryption (enciphering) and

decryption (deciphering), Cryptosystems, symmetric key cryptography, Simple examples such as

shift cipher, Ane cipher,Hillll's cipher, Vigenere cipher. Concept of Public Key Cryptosystem;

RSA Algorithm. An application of Primitive Roots to Cryptography.

Reference for Unit III:

Elementary number theory, David M. Burton, Chapter 8 sections 8.1, 8.2 and 8.3, Chapter 10,

sections 10.1, 10.2 and 10.3

## Page 13

12

Recommended Books

1. Niven, H. Zuckerman and H. Montogomery, An Introduction to the Theory of Numbers,

John Wiley & Sons. Inc.

2. David M. Burton, An Introduction to the Theory of Numbers. Tata McGrawHillll Edition.

3. G. H. Hardy and E.M. Wright. An Introduction to the Theory of Numbers. Low priced

edition. The English Language Book Society and Oxford University Press, 1981.

4. Neville Robins. Beginning Number Theory. Narosa Publications.

5. S.D. Adhikari. An introduction to Commutative Algebra and Number Theory. Narosa

Publishing House.

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

7. M. Artin, Algebra. Prentice Hall.

8. K. Ireland, M. Rosen. A classical introduction to Modern Number Theory. Second edition,

Springer Verlag.

9. William Stalling. Cryptology and network security.

Course: Graph Theory (Elective C)

Course Code: USMT5C4 / UAMT5C4

Unit I. Basics of Graphs (15L)

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

of graphs- Complete graph, Null graph, Complementary graphs, Regular 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 the-

orem and its applications, Isomorphism 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, Dijkstra's algorithm.

Unit II. Trees (15L)

Cut edges and cut vertices and relevant results, Characterization of cut edge, Denition of a

tree and its characterizations, Spanning tree, Recurrence relation of spanning trees and Cayley

formula for spanning trees of Kn , Algorithms for spanning tree-BFS and DFS, Binary and

m-ary tree, Prex codes and Human coding, Weighted graphs and minimal spanning trees -

Kruskal's algorithm for minimal spanning trees.

Unit III. Eulerian and Hamiltonian graphs (15L)

Eulerian graph and its characterization- Fleury's Algorithm-(Chinese postman problem), Hamil-

tonian graph, Necessary condition for Hamiltonian graphs using G- S where S is a proper subset

of V(G), Sucient condition for Hamiltonian graphs- Ore's theorem and Dirac's theorem, Hamil-

tonian 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.

## Page 14

13

Recommended Books.

1. Bondy and Murty Grapgh, Theory with Applications.

2. Balkrishnan and Ranganathan, Graph theory and applications.

3. West D G. , Graph theory.

Additional Reference Book.

1. Behzad and Chartrand Graph theory.

2. Choudam S. A., Introductory Graph theory.

Course: Basic Concepts of Probability and Random Variables (Elective D)

Course Code: USMT5D4 / UAMT5D4

Unit I. Basic Concepts of Probability and Random Variables.(15 L)

Basic Concepts: Algebra of events including countable unions and intersections, Sigma eld F;

Probability measure PonF, Probability Space as a triple (

;F;P), Properties of Pincluding

Subadditivity. Discrete Probability Space, Independence and Conditional Probability, Theorem

of Total Probability. Random Variable on (

;F;P) Denition as a measurable function, Clas-

sication of random variables - Discrete Random variable, Probability function, Distribution

function, Density function and Probability measure on Borel subsets of R, Absolutely contin-

uous random variable. Function of a random variable; Result on a random variable R with

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 function f2ofR2whereR2=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 (15L) Prop-

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

x!1F(x) = 1;lim

x!1F(x) = 0, Right

continuity of F;lim

x!x0F(x) =P(fR < x og;P(fR=xog) =F(xo)F(x0). Joint distribution,

Joint Density, Results on Relationship between Joint and Individual 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 by Robert Ash, Dover Publication,

2008.

Unit III. Weak Law of Large Numbers

Joint Moments, Joint Central Moments, Schwarz Inequality, Bounds on Correlation Coecient

,Result onas a measure of linear dependence, VarnX

i=1Ri

=nX

i=1Var(Ri)+2nX

i=1i

## Page 15

14

law of Large numbers.

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. Marek Capinski, Probability through Problems, Springer.

Course: Practicals (Based on USMT501 / UAMT501 and USMT502 / UAMT502)

Course Code: USMTP05 / UAMTP05

Suggested Practicals (Based on USMT501 / UAMT501)

1. Evaluation of double and triple integrals.

2. Change of variables in double and triple integrals and applications

3. Line integrals of scalar and vector elds

4. Greens theorem, conservative eld and applications

5. Evaluation of surface integrals

6. Stokes and Gauss divergence theorem

7. Miscellaneous theory questions on units 1, 2 and 3.

Suggested Practicals (Based on USMT502 / UAMT502)

1. Quotient Spaces, Orthogonal Transformations.

2. Cayley Hamilton Theorem and Applications

3. Eigen Values & Eigen Vectors of a linear Transformation/ Square Matrices

4. Similar Matrices, Minimal Polynomial, Invariant Subspaces

5. Diagonalisation of a matrix

6. Orthogonal Diagonalisation and Quadratic Forms.

7. Miscellaneous Theory Questions

Course: Practicals (Based on USMT503 / UAMT503 and USMT5A4 /

UAMT5A4 OR USMT5B4 / UAMT5B4 OR USMT5C4 / UAMT5C4 OR

USMT5D4 / UAMT5D4)

Course Code: USMTP06 / UAMTP06

Suggested Practicals USMT503 / UAMT503:

1. Examples of Metric Spaces, Normed Linear Spaces,

2. Sketching of Open Balls in IR2, Open and Closed sets, Equivalent Metrics

3. Subspaces, Interior points, Limit Points, Dense Sets and Separability, Diameter of a set,

Closure.

## Page 16

15

4. Limit Points ,Sequences , Bounded , Convergent and Cauchy Sequences in a Metric Space

5. Complete Metric Spaces and Applications

6. Examples of Compact Sets

7. Miscellaneous Theory Questions

Suggested Practicals on USMT5A4 / UAMT5A4

The Practicals should be performed using non-programmable scientic calculator. (The use of

programming language like C or Mathematical Software like Mathematica, Matlab, MuPad, and

Maple may be encouraged).

1. Newton-Raphson method, Secant method, Regula-Falsi method, Iteration Method

2. Muller method, Chebyshev method, Multipoint iteration method

3. Descarts rule of signs, Birge-Vieta method, Bairstrow method

4. Gauss elimination method, Forward and backward substitution method,

5. Triangularization methods-Doolittles and Crouts method, Choleskys method

6. Jacobi iteration method, Gauss-Siedal method

7. Eigen value problem: Jacobis method for symmetric matrices and Power method to de-

termine largest eigenvalue and eigenvector

Suggested Practicals based on USMT5B4 / UAMT5B4

1. Congruences.

2. Linear congruences and congruences of Hilgher degree.

3. Linear diophantine equation.

4. Pythagorean triples and sum of squares.

5. Cryptosystems (Private Key).

6. Cryptosystems (Public Key) and primitive roots.

7. Miscellaneous theoretical questions based on full USMT5B4 / UAMT5B4.

Suggested Practicals based on USMT5C4 / UAMT5C4

1. Handshaking Lemma and Isomorphism.

2. Degree sequence and Dijkstra's algorithm

3. Trees, Cayley Formula

4. Applications of Trees

5. Eulerian Graphs.

6. Hamiltonian Graphs.

## Page 17

16

7. Miscellaneous Problems.

Suggested Practicals based on USMT5D4 / UAMT5D4

1. Basic concepts of Probability (Algebra of events, Probability space, Probability measure,

combinatorial problems)

2. Conditional Probability, Random variable (Independence of events. Denition, Classica-

tion and function of a random variable)

3. Distribution function, Joint Density function

4. Expectation of a random variable, Normal distribution

5. Method of Indicators, Weak law of large numbers

6. Conditional density, Conditional expectation

7. Miscellaneous Theoretical questions based on full paper

SEMESTER VI

BASIC COMPLEX ANALYSIS

Course Code: USMT501/UAMT501

Unit I: 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

sets, point at innity-extended complex plane, sketching of set in complex plane (No questions

to be asked).

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

ison between dierentiability in real and complex sense, Cauchy-Riemann equations, sucient

conditions for dierentiability, analytic function, f;ganalytic then f+g;fg;fg andf=gare

analytic, chain rule.

Theorem: If f(z) = 0 everywhere in a domain D, thenf(z) must be constant throughout D

Harmonic functions and harmonic conjugate.

Unit II: Cauchy Integral Formula (15 Lectures)

Explain how to evaluate the line integralZ

f(z)dzoverjzz0j=rand prove the Cauchy integral

formula : If fis analytic in B(z0;r) then for any winB(z0;r) we havef(w) =1

2iZf(z)

zwdz;

overjzz0j=r.

Taylors theorem for analytic function , Mobius transformations: denition and examples

Exponential function, its properties, trigonometric function, hyperbolic functions.

Unit III: Complex power series, Laurent series and isolated singularities. (15

Lectures)

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

gences, disc of convergence, uniqueness of series representation, examples.

## Page 18

17

Denition of Laurent series , Denition of isolated singularity, statement (without proof) of ex-

istence of Laurent series expansion in neighbourhood of an isolated singularity, type of isolated

singularities viz. removable, pole and essential dened using Laurent series expansion, examples

Statement of Residue theorem and calculation of residue.

Reference:

1. J.W. Brown and R.V. Churchill, Complex analysis and Applications : Sections 18, 19, 20,

21, 23, 24, 25, 28, 33, 34, 47, 48, 53, 54, 55 , Chapter 5, page 231 section 65, dene residue

of a function at a pole using Theorem in section 66 page 234, Statement of Cauchys residue

theorem on page 225, section 71 and 72 from chapter 7.

Other References:

1. Robert E. Greene and Steven G. Krantz, Function theory of one complex variable

2. T.W. Gamelin, Complex analysis

Course: Algebra

Course Code: USMT602 / UAMT602

Unit I. Group Theory (15L)

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

Cyclic groups, The Center Z(G) of a group G, Cosets, Lagranges theorem, Group homomor-

phisms, isomorphisms, automorphisms, inner automorphisms (No questions to be asked)

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

of a group, Quotient group, Alternating group An, Cycles. Listing normal subgroups of A4;S3.

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

Isomorphism theorem, third Isomorphism theorem, Cayleys theorem, 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, Classication of groups of order 7.

Unit II. Ring Theory (15L)

Motivation: Integers & Polynomials.

Denitions of a ring (The denition should include the existence of a unity element), zero di-

visor, unit, the multiplicative group of units of a ring. Basic Properties & examples of rings,

including Z;Q;R;C;Mn (R);Q[X];R[X];C[X];Z[i];Z[p

2];Z[p5];Zn.

Denitions of Commutative ring, integral domain (ID), Division ring, examples. Theorem such

as: A commutative ring R is an integral domain if and only if for a;b;c2Rwitha6= 0 the

relationab=acimplies that b=c. Denitions of Subring, examples. Ring homomorphisms,

Properties of ring homomorphisms, Kernel of ring homomorphism, Ideals, Operations on ideals

and Quotient rings, examples. Factor theorem and First and second Isomorphism theorems for

rings, Correspondence Theorem for rings: ( If f:R!R0is a surjective ring homomorphism,

then there is a 11 correspondence between the ideals of R containing the ker fand the ideals

of R. Denitions of characteristic of a ring, Characteristic of an ID.

Unit III. Polynomial Rings and Field theory (15L)

Principal ideal, maximal ideal, prime ideal, the characterization of the prime and maximal ideals

## Page 19

18

in terms of quotient rings. Polynomial rings, R[X] when R is an integral domain/ Field. Divisi-

bility in Integral Domain, Denitions of associates, irreducible and primes. Prime (irreducible)

elements in R[X];Q[X];Zp[X]. Eisensteins criterion for irreducibility of a polynomial over Z.

Prime and maximal ideals in polynomial rings. Denition of eld, subeld and examples, char-

acteristic of elds. Any eld is an ID and a nite ID is a eld. Characterization of elds in

terms of maximal ideals, irreducible polynomials. Construction of quotient eld of an integral

domain (Emphasis on Z;Q). A eld contains a subeld isomorphic to ZporQ.

Recommended Books

1. P. B. Bhattacharya, S. K. Jain, and S. R. Nagpaul, Abstract Algebra, Second edition,

Foundation Books, New Delhi, 1995.

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

3. N. Herstein. Topics in Algebra, Wiley Eastern Limited, Second edition.

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

5. J. B. Fraleigh, A First course in Abstract Algebra, Third edition, Narosa, New Delhi.

6. J. Gallian, Contemporary Abstract Algebra, Narosa, New Delhi.

Additional Reference Books:

1. S. Adhikari, An Introduction to Commutative Algebra and Number theory, Narosa Pub-

lishing House.

2. T.W. Hungerford. Algebra, Springer.

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

4. I.S. Luthar, I.B.S. Passi. Algebra, Vol. I and II.

5. U. M. Swamy, A. V. S. N. Murthy Algebra Abstract and Modern, Pearson.

6. Charles Lanski, Concepts Abstract Algebra, American Mathematical Society

7. Sen, Ghosh and Mukhopadhyay, Topics in Abstract Algebra, Universities press

Course: Topology of Metric Spaces and Real Analysis

Course Code: USMT603/ UAMT603

Unit I: Continuous functions on metric spaces (15 L) Epsilon-delta denition of con-

tinuity at a point of a function from one metric space to another. Characterization of continuity

at a point in terms of sequences, open sets and closed sets and examples, Algebra of continuous

real valued functions on a metric space. Continuity of composite continuous function. Con-

tinuous image of compact set is compact, Uniform continuity in a metric space, denition and

examples (emphasis on R). Let (X;d) and (Y;d) be metric spaces and f:X!Ybe continu-

ous. If (X;d) is compact metric, then f:X!Yis uniformly continuous.

Contraction mapping and xed point theorem, Applications.

## Page 20

19

Unit II: Connected sets: (15L)

Separated sets- Denition and examples, disconnected sets, disconnected and connected metric

spaces, Connected subsets of a metric space, Connected subsets of R. A subset of Ris connected

if and only if it is an interval. A continuous image of a connected set is connected. Character-

ization of a connected space, viz. a metric space is connected if and only if every continuous

function from Xtof1;1gis a constant function. Path connectedness in Rn, denition and

examples. A path connected subset of Rn is connected, convex sets are path connected. Con-

nected components. An example of a connected subset of Rn which is not path connected.

Unit III : Sequence and series of functions:(15 lectures)

Sequence of functions - pointwise and uniform convergence of sequences of real- valued func-

tions, examples. Uniform convergence implies pointwise convergence, example to show converse

not true, series of functions, convergence of series of functions, Weierstrass M-test. Examples.

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

function, conditions under which integral and the derivative of sequence of functions converge to

the integral and derivative of uniform limit on a closed and bounded interval. Examples. Conse-

quences of these properties for series of functions, term by term dierentiation and integration.

Power series in Rcentered at origin and at some point in R, 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.

References for Units I, II, III:

1. S. Kumaresan, Topology of Metric spaces.

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

3. Robert Bartle and Donald R. Sherbert, Introduction to Real Analysis, Second Edition,

John Wiley and Sons.

4. Ajit Kumar, S. Kumaresan, Introduction to Real Analysis

5. R.R. Goldberg, Methods of Real Analysis, Oxford and International Book House (IBH)

Publishers, New Delhi.

Other references :

1. W. Rudin, Principles of Mathematical Analysis.

2. T. Apostol. Mathematical Analysis, Second edition, Narosa, New Delhi, 1974

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

4. R. R. Goldberg Methods of Real Analysis, Oxford and IBH Pub. Co., New Delhi 1970.

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

6. W. Rudin. Principles of Mathematical Analysis, Third Ed, McGraw-Hill, Auckland, 1976.

7. D. Somasundaram, B. Choudhary. A rst Course in Mathematical Analysis. Narosa, New

Delhi

## Page 21

20

8. G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hi, New York,

1963.

9. Sutherland. Topology.

Course: Numerical Analysis II (Elective A)

Course Code: USMT6A4 / UAMT6A4

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

mentioned have to be covered.

Unit I. Interpolation (15L)

Interpolating polynomials, Uniqueness of interpolating polynomials. Linear, Quadratic andHill-

gher 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 dierences Gregory-Newton forward dierence interpolation, Gregory-Newton backward

dierence interpolation, Stirlings Interpolation. Results on interpolation error.

Unit II. Polynomial Approximations and Numerical Dierentiation (15L)

Piecewise Interpolation: Linear, Quadratic and Cubic. Bivariate Interpolation: Lagranges Bi-

variate Interpolation, Newtons Bivariate Interpolation. Numerical dierentiation: Numerical

dierentiation based on Interpolation, Numerical dierentiation based on nite dierences (for-

ward, backward and central), Numerical Partial dierentiation.

Unit III. Numerical Integration (15L)

Numerical Integration based on Interpolation. Newton-Cotes Methods, Trapezoidal rule, Simp-

son's 1/3rd rule, Simpson's 3/8th rule. Determination of error term for all above methods.

Convergence of numerical integration: Necessary and sucient condition (with proof). Com-

posite integration methods; Trapezoidal rule, Simpson's rule.

Reference Books

1. Kendall E, 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 Publications.

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

McGrawHillll International Book Company.

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

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

6. Scarborough James B., Numerical Mathematical Analysis, Oxford University Press, New

Delhi.

## Page 22

21

Course: Number Theory and its applications II (Elective B)

Course Code: USMT6B4 / UAMT6B4

Unit I. Quadratic Reciprocity (15 L)

Quadratic residues and Legendre Symbol, Gausss Lemma, Theorem on Legendre Symbol2

p

,

the result: If pis an odd prime and ais an odd integer with ( a;p) = 1 then

a

p

= (1)twheret=p1

2X

k=1hka

pi

, Quadratic Reciprocity law. Theorem on Legendre Symbol

3

p

:The Jacobi Symbol and law of reciprocity for Jacobi Symbol. Quadratic Congruences with

Composite moduli.

Unit II. Continued Fractions (15 L)

Finite continued fractions. Innite continued fractions and representation of an irrational num-

ber by an innite simple continued fraction, Rational approximations to irrational numbers and

order of convergence, Best possible approximations. Periodic continued fractions.

Unit III. Pells equation, Arithmetic function and Special numbers (15 L)

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. Spe-

cial numbers: Fermat numbers, Mersenne numbers, Perfect numbers, Amicable numbers, Pseudo

primes, Carmichael numbers.

Recommended Books

1. Niven, H. Zuckerman and H. Montogomery. An Introduction to the Theory of Numbers.

John Wiley & Sons. Inc.

2. David M. Burton. An Introduction to the Theory of Numbers. Tata McGraw-Hill Edition.

3. G. H. Hardy and E.M. Wright. An Introduction to the Theory of Numbers. Low priced

edition. The English Language Book Society and Oxford University Press, 1981.

4. Neville Robins. Beginning Number Theory. Narosa Publications.

5. S. D. Adhikari. An introduction to Commutative Algebra and Number Theory. Narosa

Publishing House

6. .N. Koblitz. A course in Number theory and Crytopgraphy. Springer.

7. M. Artin. Algebra. Prentice Hall.

8. K. Ireland, M. Rosen. A classical introduction to Modern Number Theory. Second edition,

Springer Verlag.

9. William Stalling. Cryptology and network security.

## Page 23

22

Course: Graph Theory and Combinatorics (Elective C)

Course Code: USMT6C4 /UAMT6C4

Unit I. Colorings of graph (15L)

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, com-

plete 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 graph (15L)

Denition of planar graph. Euler formula and its consequences. Non planarity of K5;K(3; 3).

Dual of a graph. Polyhedran in R3and existence of exactly ve regular polyhedra- (Platonic

solids) Colorabilty of planar graphs- 5 color theorem for planar graphs, statement of 4 color

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 (15L)

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, M alternating and M

augmenting path, Berge theorem. Bipartite graphs.

Recommended Books.

1. Bondy and Murty Grapgh, Theory with Applications.

2. Balkrishnan and Ranganathan, Graph theory and applications. 3 West D G. , Graph

theory.

3. Richard Brualdi, Introduction to Combinatorics.

Additional Reference Book.

1. Behzad and Chartrand Graph theory.

2. Choudam S. A., Introductory Graph theory. 3 Cohen, Combinatorics.

Course: Operations Research Elective D)

Course Code: USMT6D4 / UAMT6D4

Unit I. Linear Programming-I (15L)

Prerequisites: Vector Space, Linear independence and dependence, Basis, Convex sets, Dimen-

sion of polyhedron, Faces.

## Page 24

23

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

LPP, Feasible solution to basic feasible solution, Improving BFS, Optimality Condition, Un-

bounded solution, Alternative optima, Correspondence between BFS and extreme points. Sim-

plex Method Simplex Algorithm, Simplex Tableau.

Reference for unit I

1. G. Hadley, Linear Programming, Narosa Publishing, (Chapter 3).

Unit II. Linear programming-II (15L)

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

Solution of LPP for unrestricted variable. Transportation Problem: Formation of TP, Con-

cepts of solution, feasible solution, Finding Initial Basic Feasible Solution by North West Corner

Method, Matrix Minima Method, Vogels Approximation Method. Optimal Solution by MODI

method, Unbalanced and maximization type of TP.

Reference for Unit II

1. G. Hadley, Linear Programming, Narosa Publishing, (Chapter 4 and 9).

2. J. K. Sharma, Operations Research, Theory and Applications, (Chapter 4, 9).

Unit III. Queuing Systems (15L)

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

Generalized Poisson Queuing Mode. Specialized Poisson Queues: Steady- state Measures of Per-

formance, 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. Richard Broson, Schaum Series Book in Operations Research, Tata McGrawHill Publishing

Company Ltd.

Course: Practicals (Based on USMT601 / UAMT601 and USMT602 / UAMT602)

Course Code: USMTP07 / UAMTP07

Suggested Practicals (Based on USMT601 / UAMT601):

1. Limit continuity and derivatives of functions of complex variables,

2. Steriographic Projection , Analytic function, nding harmonic conjugate,

3. Contour Integral, Cauchy Integral Formula ,Mobius transformations

4. Taylors Theorem , Exponential , Trigonometric, Hyperbolic functions

5. Power Series , Radius of Convergence, Laurents Series

## Page 25

24

6. Finding isolated singularities- removable, pole and essential, Cauchy Residue theorem.

7. Miscellaneous theory questions.

Suggested Practicals (Based on USMT602 / UAMT602)

1. Normal Subgroups and quotient groups.

2. Cayleys Theorem and external direct product of groups.

3. Rings, Subrings, Ideals, Ring Homomorphism and Isomorphism

4. Prime Ideals and Maximal Ideals

5. Polynomial Rings

6. Fields.

7. Miscellaneous Theoretical questions on Unit 1, 2 and 3.

Course: Practicals (Based on USMT603 / UAMT603 and USMT6A4 / UAMT6A4

OR USMT6B4 / UAMT6B4 OR USMT6C4 / UAMT6C4 OR USMT6D4 / UAMT6D4)

Course Code: USMTP08 / UAMTP08

Suggested practicals Based on USMT603 / UAMT603:

1 Continuity in a Metric Spaces

2 Uniform Continuity, Contraction maps, Fixed point theorem

3 Connected Sets , Connected Metric Spaces

4 Path Connectedness, Convex sets, Continuity and Connectedness

5 Pointwise and uniform convergence of sequence functions, properties

6 Point wise and uniform convergence of series of functions and properties

7 Miscellaneous Theory Questions

Suggested Practicals based on USMT6A4 / UAMT6A4

The Practicals should be performed using non-programmable scientic calculator. (The use

of programming language like C or Mathematical Software like Mathematica, Matlab, MuPad,

and Maple may be encouraged).

1 Linear, Quadratic andHillgher order interpolation, Interpolating polynomial by Lagranges

Interpolation

2 Interpolating polynomial by Gregory-Newton forward and backward dierence Interpola-

tion and Stirling Interpolation.

3 Bivariate Interpolation: Lagranges Interpolation and Newtons Interpolation

4 Numerical dierentiation: Finite dierences (forward, backward and central), Numerical

Partial dierentiation

## Page 26

25

5 Numerical dierentiation and Integration based on Interpolation

6 Numerical Integration: Trapezoidal rule, Simpsons 1/3rd rule, Simpsons 3/8th rule

7 Composite integration methods: Trapezoidal rule, Simpsons rule.

Suggested Practicals based on USMT6B4 / UAMT6B4

1. Legendre Symbol.

2. Jacobi Symbol and Quadratic congruences with composite moduli.

3. Finite continued fractions.

4. Innite continued fractions.

5. Pells equations and Arithmetic functions of number theory.

6. Special Numbers.

7. Miscellaneous Theoretical questions based on full USMT6B4 / UAMT6B4.

Suggested Practicals based on USMT6C4 / UAMT6C4

1. Coloring of Graphs

2. Chromatic polynomials and connectivity.

3. Planar graphs

4. Flow theory.

5. Inclusion Exclusion Principle and Recurrence relation.

6. SDR and Mathching.

7. Miscellaneous theoretical questions.

Suggested Practicals based on USMT6D4 / UAMT6D4

All practicals to be done manually as well as using software TORA / EXCEL solver.

1. LPP formation, graphical method and simple problems on theory of simplex method

2. LPP Simplex Method

3. Big-M method, special cases of solutions.

4. Transportation Problem

5. Queuing Theory; single server models

6. Queuing Theory; multiple server models

7. Miscellaneous Theory Questions.

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX