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2. ATNA UNIVERSITY

COURSE OF STUDY
M.A/M.Sc, MATHEMATICS
SEMESTER- 1, 11, HE & IV
CHOICE BASED CREDIT SYSTEM (CBCS)

(To be effective from 2018-2019)

3. M.A. /M.Se. Mathematics
SCHEME OF EXAMINATION

Passing of Examination and Promotion Rule

Course in Mathematics shall be of two academie sessions comprising of |
FOUR SEMESTERS. Fach academic session shall consist of two Semesters ~ 1 & II from July
to December and Semester & IV from January to June.

Each theory paper respective of ther nature and credits shal be of 100 marks out of whieh the
performance of a student in cach paper will be assessed on the basis of Continuous Internal
‘Assessment (CIA) of 30 marks and the End Semester Examination (ESE) consisting of 70
marks.

‘The components of CÍA shall be

(a) Two Mid Semester Written Tests of one hour duration each 15 Marks
(6) Seminariquiz 5 Marks
(©) Assignment 5 Marks,
(& Punctuality & Conduct 5 Marks
Total 30 Marks

1. There shall be no supplementary examination in any of he Semester Course (LI, I & IV),

2. A student who has appeared at Ihe CIA and altended the required minimum percentage (75%)
‘of the attendance in theory shall be permitted to appear in the End Semester Examination
(ESE),

3. To be declared passe in ESE in any subject, a students must secure atleast 45% marks in each

paper separately.
A student has to socure minimum 45% marks in CIA of any paper. In cae, a student fails to
secure minimum 50% marks in CIA of any paper, hehe will be declared fail in that paper.
Students shall have to reappeur in that paper and in CIA examination alo in the same semester
of next academic session

If students fil to secure minimum 50% marks in C
{alin that paper. Students shall have to reappear
academic session.

of any paper his result willbe declared as
‘that paper in the same semester of next

A promoted candidate, if he has passed in CIA but fils in theory paper/papers, he/she shall
retain hise CIA award and wil reappear inthe theory paper only of the semester whenever
available. However, if a candidate is declared fail in any End Semester Fxamination, shall
‘etain nothing and will have to redo the course work of filed semester again and he has to
appear again in CIA as-well-as in theory paper.

4. Ifa candidate passes in at least two paper in hisher First, Second and third End Semester
Examination, helshe shall be promoted to next higher semester. But he/she will have to lear
their m. page In te net ex semen ein of ita Ma EE

ern CRT er

4. available. Even ¡a student is promoted to fourth semester his final result will only be declared
‘shen heshe has cleared al their backlog pa

Final result of M.Sc, will be published o
minimum qualifying marks

Student shall he awarded Grade Point (GP) at the end of each semester examination and
‘Cumulative Grade Point (CGP) at the end of final End Semester Examination in 10 point
scoring system.

x he/she has cleared all the 16 paper securing

Declaration of Result
The following grading system shall be used by teacher! Examination department

Leiter Grad

Pereentage Range | Number of

nn | Grade
9010 in [Outstanding
jo | Eeellemt

Lessihanas |

A student shall be declared to have passed and promoted to the next semester when helshe
um B or above grade in the semester examination covering continuous evaluation, mid term
and end term examina

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5. Syllabus of M.A/M.Sc. (Mathematics) Semester I

PAPER I (MAT CC 01

Al

stract Algebra

Abstract Algebra

Prerequisites: Introduction to Group. Hlementary Properties of Group, Finite Group,

and subgroup, Cyelie Group, Permutation Group, Properties of Permutations,

rings, integral Domains, Characteristic of rings.

Homomorphism: Group actions. Sylow theorems, Normal and subnormal. series
group. Jordan- Holder Theorem, Solvable groups,

coup. Nilpotent groups.

Unit

composition series of a
‘commutator subgroup of a

ideals, Kemel of ring

Unit 2: Ring homomorphism. isomomhism. quotient rings
al ideal, Euclidean

shim, principal ideal ring and domain, prime and mai

hom

domain.

«tension, splitting field of

ental

brag and tase

Unit 3: Extension Fields à

sarable extension, normal extension, constructible

Polynomial, separable and in

real number

Unit 4 Cyclie Modules, simple Modules, semi-simple Modules, Schurs Lemma, Free

Modules
of equations of degree $ by

Unit 5: Solution of equa

cia Vols & II Naross Publication House
and R.M. Foote:= Abstract Algebra

\

ate
AA a we

DS. Damm
versity Algebra

6. PAPER II (MAT CC- 02)

Real Analysis,

Real Analysis

Unit

«quences and se

es o funcions, pointwise and uniform convergence, Cauchy
criterion for uniform convergence, Weiersrass-M tes, Abel's and Dirihlet’s

test for uniform convergence.

Unit

Uniform convergence and differentiation, Weierstrass approximation theorem
Power series, Uniqueness theorem for power series, Able and Tauber's theorem.

Unit

nition and examples of Riemann-Stelje's integral Property of integral,
Integration and differentiation, the fundamental theorem of Caleulus, Integration

Of vector valued function rectifable curves.

Unit

+ Funetions of several vara

>, linear transformation, Derivatives in an open subset
‘of R* chain rule partial derivatives, interchange of order of differentiation,

deriv

e. of higher orders, Taylor's theorem.

were function bcorem, Implicit function theorem, Jacobian, Fxtremum
Problems with constrains, Lagrange multiplier methods ferentiation of
Integrals, partition of unity, Differential forms, Stoke's theorem.

Referene

1. W Rudin Principles of Mathematical Analysis
2. T.M. Apostal ; Mathematical Analysis

3. LP. Natanson ; Theory of function of Real Variable
4, HLL, Royden : Real Analysis

Brzide QUE Sos

7. APER III (MAT CC-03

incar Algebra

Linear Algebra
Unit 1: Finite dimensional vector spaces: Lincar transformations and their matrix.

representations rank; systems oflincar equations, eigenvalues and eigenvectors,

minimal polynomial, Cayley-Hamilton Theorem, diagonalization

‘Unit 2: Hermitian, SkewHlermitian and unitary matrices; Finite dimensional inner product

spaces, Gram-Schmidt orthonormalization process, self-adjoint operators

Unit à Similarity of linear transformations, Invariant subspaces. reduction 10

angular forms. Nilpotent transformations, Index of Nilpotency. invariants of a
Nilpotent transformations, primary decomposition theorem, Jourdan blocks and
Jordan forms rational canonical form

sn

inca form. algebra of bilinear form Matrix of bilinear forms, degenerate and
Non-degenerae bilinear forms, Alternating bilinear forms.

Unit $: Symmetrie and Skew-symmetrie bilincar forms, Quadratic form, law of Inertia,

Sylvester's theorem, Hermitian forms definite Forms.

References:

1. KiB Data: Mates and Linear Algebra
2. 8. Lipsehutz- Linear Algebr
3. Hoflman and Kunze: Linear Algebra

À -
die ana, fa

Jhaum's outline series

ae
DASS
oS

8. PAPER IV (MAT CC-04)

Diserete Mathematics

Diserete Mather

Graph Theory

Unit 1 Det

ion of graphs. paths, circuits and subgraphs, induced subgraphs, degree

plana graphs and ther properties, Tees and simple
applications of graphs

Lattice Theory

Unit 2: Latics as partially ordered sets and their properties, latices as algebraic system,
Sub latices, direct products and Homomorphisms of Lattices some special latices

es Complete latices, complemented

ces and disribuive latices,

Boolean Algebra

Unit: Boolean algebra as a complemented distributive lattice, Boolean rings,
identification of Boolean algebra and Boolean rings, sub-algebra and generators.

{Unit 4 Boolean homomorphism and ring homomorphism ideals in a Boolean algebra and
Dual ideals, Fundamental theorem of homomorphism and Stone's representation
‘theorem for Boolean algebras and Boolean rings, simple application to cletrical
network, solvability of Boolean equations and ogical puzzles.

Combinatories

Unit : Permuta

and combinations, partitions, pigeonhole principle, inlusion-exelusion
principle, generating function, recurrence relations
References:

1. Ku. Rosen + Discreto Mathematics and its applications.
2. S.Lipschutz and M. Lipson: Discrete Mathematics

3. C.L Liu: Elements of Diserete Mathematics

4
5

E.Mendelson : Boolean Algebra and Switching Cire
Kolman, Bushi and Ross :- Diserete Mathematical Structure

dur Chiat, SRE

9. Syllabus of M.A/M.Sc (Mathematics) Semester II
PAPER V (MAT CC-05)

General Advanced Mathematics

Unit tz Elementary st theory, finito, countable and uncount
number system a a complete orders field, Archimedean property. supremum, infimum
Furry Set Theory:

Unit 1: Fuzzy Sets Versus Crisp ets, Rasi definitions, types, properties and representations
of Fuzzy sets, Convex Fuzzy set, Bases operation on Fuzzy set. u- Cuts, Decompositions
‘tension principles and Simple applications

thoorem. €
of Fury

omplements, t-norm and eonorms,

n of graphs path. circuits and subgraphs, induced subgraphs, degree of a
vertex, connectivity, planar graphs and their properties, Trees and simple applications of

Number Theory:

Unit IV + Divisblity Theory In the Integers: Division Algorithim, the Greatest Common
Divisor. The Fuclidean Algorithm, The Diophantine Equations axıby = €, Fundamental
Theoem of

References:

1. Kolman, Bushi and Ross te Mathematical Structure
2. Pundir And Pundir- Fuzzy Sets & thei Application,

3. GuKlir RB. Yuan: Fuzzy sets
4
s

Graph Theory °F, Harare, Addison Wesley
Baker, A concise introduction o the Theory of Numbers

seh ee ee
w

agota

10. PAPER VI (MAT CC-06
Complex Analysis

Complex Analysis:

Unit 1: Algebra of complex numbers, the complex plane, polynom

is. power series

transcendental functions such as exponential, trigonometric and. hyperbolic

functions, Analyie functions, Cauchy-Riemann equations.

Unit 2 : Contour tera formula, Liowsile’s

‘theorem,

Schwarz’s Lemma, Laurent

Unit 3 2 Taylor's thooren

Maximum modulus Prine

Series, Isolated singularities. Meromorphie function. Mitag-LefMler’s theorem The
argument principle. Roucbe's theorem, fundamental theorem of algebra, Power
Unit 4 + Residues, Cauchy's residue theorem, Evaluation of integral, Branches of any

valued functions with special refer

10 ang loge and Bilinear transformations,
cir properties und classifications, definition and examples of conformal
‘mappings. Mobius Transformatio

References +

1. J.B. Conway ¿Functions of one Complex Variables

LV. Ablfors = Complex Analysis

ere Ski

ae Gr el

11. PAPER MAT CC-07)
Differential and Integral Equation

Differential and Integral Equations

Unit 1: Initial Value problem and the equivalent integral equation, n order equation ind

dimension as a first onder system. Concepts of local existence existence and

uniqueness of solution with examples

{Unit 2: Integral Equations and their classifications Eigen values and eigen functions.
Fredholm Integral equations of Second Kind, Iterative Scheme and method of

successive approximations.

Unit 3: Ascoli-Arzla theorem, theorem on convergence of solutions of a
family of Initial value problems, Picard-Lindelof theorem, Peano's

‘existance theorem Corollris, Kamke's convergence theorem,

Unit 4: Gronwal’s inequality, maximal and minimal solution, Differential inequali

Uniqueness theorem, Nagumo’s and Osgood's criteria, successive approximations.

References :

1. P. Hartman : Ordinary Differential Equation

4. 8.G.Mikhlin + Linear lo

ral Equations.

5. R.P-Kanwal at Theory and Techniques

Bree er Ir LS
€

12. PAPER VIII (MAT CC-08

Measure Theory

Measure theory:

Uni

Lebesgue outer measure, Measurable sets Messuribility, Measurable functions.

surtt, non- measurable sets.

Unit 2 Integration of non-negative functions, the ge

oral integral

negation of series,

Riemann and L.cbesgue integral

The Four Derivatives funcion of bounded variation, Lebesgue diferen

Theorems, Differentiation and Integration.
Unit 4: Measure and outer measure, extension of measures uniqueness of extension

Completion of a measure, measurable spaces,

gration with respect 104

{Unit The L!spaces, convex functions, Jensen inequality Holder's and Minkowshi's

Inequalities, completeness of” «spaces, convergence in measure, Almost

uniform Convergence.

Reference

1. Gide Barra: Measure Theory and Integration

2. PK. Jain and VP Gi

Lehesque Measure and I

ration

3. LK, Ram An Introduction to Measure and Integration

4. PLR, Halmos- Meas

Theory

u, pr NG
Berka Oe 18 Las

°

13. PAPER IX (MAT CC-09
Topology

Unit ition and examples of topological spaces, closed sets, dense subsets,

Neighbourhood, interior, exterior, boundary and accumulation points. Derived

‘Sets, Bases and subbases, Subspaces and Relative topology:

{Unit 2: Continuous functions and homeomorphism, chracterisaion of continuity in

Terms of open ses, closed sets and closure. First and second countable

topological spaces Lindclos theorem, separable Spaces, second countability and
separability.

Unit 3 + Separation axioms To. Tı and Ty spaces and their basic properties,

compactness, Continuous function and compact sets, basic properties of

‘compactness and Finite intersection property.

{Unit 4: Connectedness, continuous function and connected sets characterization of
Connsetsdness in terms ofa discrete to point space, connectedness on real
line.

Unit 5; Reyular and Normal spaces Ts and T, spaces, charactersations and basic

properties, Urysohn’s lemma and Tietze extension Theorems.

References

1. G.F:Simmons- Introduction to Topology and Modern Analysis

2. K.Kha - Functional Analysis, Advanced General Topology

3. Futton:- Algebraic Topology First Course

BIT
deat, OLY “skis

14. PAPER X (MAT CC-10)
Number Theory

Number Theory

Unica

Divisiili, GC and L.C.M, Primes, Fermat numbers, congruences and resides, theorems oF
Euler Fermat and Wilson, solutions of congruence, near congruences, Chinese remainder theorem,
Unit

Arthmeia functions gfe, ¡4 and Anand of), Mocbis inversion formu, congruences of higher
<eqree,congrueness of prime power moduli and prime modus, power residue

Unis
Quadra esd, Legendre symbols, lemma of Guns and reciprocity law: Jacobi symbols Fary
series, ational approximation Hrwitz ore, rational numbers, irationaliy of and x,
Representation ofthe eal numbers by devimals.

v

Finite continued factions, simple continued factions infinite simple continued fractions, periodic

continued faction, approximation by convergence, best possible approximation, el equations,

Lagrange four sphere theorem

Reference:

1. theory of Numbers. G H Hardy and E M Wright, Oxford Science Publications, 2003.

2. Induction tothe Theory of Numbers, Niven and I $ Zuckerman, John Wiley Sons, 1960.
3. Elementary Number Theory, D M Burton, Tata McGraw Hill Publishing Howse,

2006

A. Higher ArlhmeticH. Davenport, Cambridge University Pres, 1909.

5. Invoduction to Analytic Number Theory, TM. Apestal, Narosa Publishing House

mn ary See
¥

15. Syllabus of M.A/M.

e (Mathematics) Semester IIT

PAPER XI (MAT CC-11)

Functional Analysis

Functional Analysis

Unit 1: Normed linear spaces, Banach spaces and examples, Quotié

space of normed linear
Spaces and its completeness, equivalent noms, Ries Lemma, Basie properties of

Finite dimensional normed linear spaces and compactness

Unit2 : Weak convergence and bounded linear transformation, normed linear spaces of
bound linear transformations, dual spaces with examples, uniform oundness

theorem and some ofits consequences.

Unit Open mapping

heorem und closed graph theorem, Hahn- Banach Theorem on rel

linear spaces, complex liner spaces and normed linear spaces, Reflexive spaces.

nit 4: Inver product spaces, Ries lemma on Hilbert space, orthonormals ses and
Parseval's identity structure of Hilbert spaces, Projection theorem Riesz

Representation Theoren.

Unit: Adjoint of an operator ona Hilbert space, Reflexivity of Hiller spaces, Sef
adjoint Operators. positive operator, Projection, Normal and unitary operators

References

1. 6 Simmons: Introduc

1 to Topology and Modern Analysis

2. KKJha: Functional Analysis. Advanced General Topology

ino
tu rd LE

16. PAPER XII (MAT CC-12
Fluid Dynamics

Fluid Mechanies :

Unit 1: Lagrangian and Eulerian methods, Equation of Continuity, Boundary Surfaces,

am lines, Path lines and Streak lines, velocity potential, rotational and rotational motions,

vortex lines

{Unit 2: Lagrange's and Euler's equations of motion, Bernoulli's theorem, equation ofmotion

by lux method, eq

tion refered to moving axis, implusive ations.

Unit 3: Irotational Motion in two dimension, steam function, complex velocity
potential, sources, sinks, doublets and their images, conformal mapping

Milne-Thompson circle theorem.
Unit 4: Two dimensional irottional motion produced by motion of a circular,

‘coaxial and elliptic cylinders in an infinite mass of liquid, kinetic energy of

liquid, Theorem of Blasius, motion of a sphere through a liquid at rest at

infinity, liquid streaming past a fixed sphere, Equation of motion of a

sphere, Stoke's stream function

{Unit Vortex motion and its elementary properties, Kelvin's proof of permanence,

References
1. EChorton + A text Book of Fluid Dynamics

2. MD. Raisinghania:- Fluid Dynamics

mi ete SEE

17. PAPER XIII (MAT CC-13)

Classical Mechanics (Rigid Dynamics)

Unit

jeneralised Co-ordinates, Holonomie and Non Holonomie systems,
‘equations of motion, energy equations for conservative fields

Unit 2: Hamilton’s canonical equations, Rouths equations, Hamilton Principle,

Principle of Least Action.
Unit: Small Osilaions, normal Co ordinates, normal mode of vibration,

Units:

‘ntact transformations, Lagrange brackets and Poisson brackets, the most

general infinitesimal contact transformation, Hamilton-Jacobi equation

{Unit 5: Motivating problem of Calulus of variation, Fule- Lagrange equation

shortest distance, minimum surfaces of revolution, Brachistochrone problem.
References

1. AS, Ramsey = Dynamics Par Il

2, SL. Loney’ + Dynamics of parie and rigid bodies

.
y
en His os

18. PAPER XIV (MAT CC-14)

Optimization Techniques

‚car Programming

simplex method for unrestricted variable, Two phase method. Dual simplex method,

Unie
ing Upper Bound technique, Interior point algorithm, Linear

Parametric L
Goal pro

Integer programing, Branch and bound technique, Gomory's algorithm.

ar progra
«sing

¡ear programming

€ and multivariable unconstrained optimization, Kubn- Tucker condition for

constrained optimization. Wolfe's and Beale’s methods.

ame theory. Two person Zero sum games with mixed strategies, Graphical

solution by expressing as a linear program

Unit 5: Inventory theory. Di

lot size model, FOQ

production runs of equal length EOQ modele Shortages not allowed, shortages

allowed,

References:

1. 1L.A Tala; Operations Research An Introduction

2. Kami Swarup, P.K.Gupia and Man Mohan: Operations Res

S. Hira Operations Research

oe exe Ss

19. PAPER XV (MAT CC-15

Differential Geometry

Unit

than are en

Curves in spaces, parameters u

ts, tangent principal normal.
Binormal and three fundamental planes, Curvature and torsion of space curves, Seret-Frenet
formulae, Fundamental theorem on spaces curves élices, spherical, indiatrix, Involuts and

volutes, Bertrand curves

Unit 2: Representation of surfaces, Curves on surfaces in R? spaces, tangent plane and

Normal, Envelope, characteristic and edge of regression, developable surface of revolution,

directions on a surface,

Unit 3: Parametric curves, an

betwen them, frst order and second order magnitudes,

principal directions and lines of curvature, Normal Curvature, Euler's theorem and Meunier’s

(heorem. Theorem of Beltrami and Enneper, Gauss Characteristic equation, Maina

Codazzi

Er

Unit 4 Conjugate directions, Isometric fines, asymptotic lines and Geodesics- their
‘equations and properies, curvarure and torsion. their structures on surfaces of revolution,

Bonnes theorem, Clara's theorem and Dupin’s indicar.

1. CE. Wesherbum- Differential Geometry In Three Dimension

2. LA thorpe : Elementary Topies in Differential Geometry

3. A. Gray: Differential Geometry of three dimensions, Cambridge University Press

we OUT e

20. List of Elective Paper (MAT EC-01 & MAT EC-02)
1.Fuzzy sets and their application
2.Mathematical Methods
3.Operational Research
4. Theory of Relativity
5.Galois Theory.
6.Advanced Topology
7.Banach Algebras
8.Commutative Algebra
9. Programming in C

(2 i } [23

21. set and their applications

1 Fuzzy Sets Versus Crisp sets, Basie ns types, properties and

representations of Fuzzy sets, Convex Fuzzy sets, Basics operation on Fuzzy set, a- Cas,

Decompositions thearem, Complements. - norm and -semorms, Extension principles

and Simple applications of Fuzzy ses.

v

Fuzzy logics An overview of classical logic, Multvalued logics, Fuzzy

propos

ions, fuzzy quantifiers, Linguistic variable and hedges, inference from conditional

{fuzzy propositions the compositional rule of inference

Unit 3 : Approximate Reasoning An overview of fuzzy expert system. Fuzzy

implication and their secetion Muliconditional approximate reasoning the role a fuzzy

Unit An introduction to Fuzzy control

ers controller, Fuzzy rule base Fuzzy

inference engine Fuzzification, Defuzzifiation and the various defuzzifiation method

(The centre of maxima and the

san of maxima method).

Uni

Decision making in Fuzzy Environment Individual decision making,
Multiperson decision making, Mulirteria decision making, Multistage decision making.

Fuzzy ranking methods, Fuzzy linear programming

u

Mise Application specially in social science, Biological Science and engineering

el

sory and mathematical statistics,

References:

1. GA. Klieand i

fun 2 Fuzzy sets and Fuzay Logies

2. HS. Zimmermann,

jury set theory and its Applications.

3. GU. Klirand TA, Folger + Fuzzy Seu

Uncertinty and Information.

44. Pundir and Pundir - Fuzzy sets und their applications.

DE ES

22. 2.Mathematical Methods
Unit 1: Orthogonalistion, Bessel’ Inequality, Mean error minimization,completeness

ion, Weierstrass approximation theorem, polynomials of Legendre, Hermite and

Bessel, generating Function, orthogonality, recurrence relation and Rodrique formula

Unit 2 : Partial Differential Equation and properties, concept of well posed problems,
Reduction of PD.E in two independent variables tothe canonical forms, classification
into elliptic, hyperbolic and parabolic equations, Laplace's equations in cartesian,

cylindrical and spherical co-ordinates, Equipotential surfaces, Interior and exterior

Dirihlet problem, the Maximum. Minimum property, solutions and Uniqueness,

Dirichlet’ problem for a circle, fundamental properties of Harmonie function

Unit 3 : Wave equation in one dimension and two dimension, vibrations of struck and

plucked string with fixed ends,

nogencous rectangular and circular membranes,

eigen vibrations, D'Alember's solution of one dimensional wave equation. One

<ime-nsional Diffusion equation & solution of initial value problem by integral

transform

Unit 4: Tensors- Transformations of Co-ordinates, contravariat and covariant vectors
Symmetric and skew-symmetric tensos, addition and multiplication of tensors,
Contraction and composition of tensors, Quotient law.

Unit $ : Reciprocal symmetric tensors ofthe second order, Christoffel’ symbols, covariant
‘derivative of a contravariat vector, Co varian derivative of a covariant vector,
covariant derivatives of tensors, cul of a vector, Divergence ofa covariant vector,
Laplacian ofa scalar invariant

References:
1. LN. Sneddon:- Elements of Parial Differential Equations
2. R. Courant and D, Hilbert Methods of Mathematical Physics Vol I & Vol I

CE. Weutherbum :- Riemannian Geometry and Tensor calculus

4. Smimov and 1ychonoff; - Pani! Differential Equations

aa Cr ss

23. 3. Operations Research

Uni

Queuing Thoory- Poisson probability law, Distribution of inter arrival time

Distribution of time between successive arias, Differential difference equation of MM 1

a ]FIEO M MIES NIBIFO.M MIC: #[FIFO M MIC: NIFIFO.

u

Information Theory: Description of communi

tion system, Mathematical definition
‘of information. Axiomatie approach to information, Measures of uncertainty, Entropy In two.

dimensions: property, conditional entropy.

Unit O

nel capacity Efficiency and redundaney, Encoding. Fano-encoding procedure,

[Necessary and sufficient condition, average length of encoded message.

Unit 4: Replacement Model- introduction concepts of present value, replacement of

‘whose maintenance cost increase with

sand value of money also changes,

Replacement o items tha fail completely individual and group replacement policy
Unit 5: Sequencing - N jobs and 2 machines, N jobs and 3 machines, N jobs M machines.

References:
1 HLA. Taha + Operations Research: An Introduction
2. Kant Swarup, P.K.Gupia and Man Mohan: Operations Research

3. P.K.Gupts and DS. Mira + Operations Research

La aid ke
dre +

24. Unit 2:

Unit 3:

Unit 4

2 Cosmology

4.Theory of Relativity

Principle of equivalen and general covariance, Pistia Feld

General theory of Retain

uations and its Newtonian approximation

Schwarz Schild external Slaton and its soropie form, Binkhof theorem, planetary ois

and analogous of Kepler's laws in general relativity

Advance of prihelion ofa planet, Boning of ight ays a gravitational filé,
Gratton shit of spectral lines Ense theory

[Energy Momentum of sensor of perl Mid, Schwarz Schill intra solution, Energy

Momentum tensor of a electromagnetic field, Einstein Maxwell equation, Reisner
Norstrom Solution

cin modii Gel equation wih cosmologin! term sti cosmological

sof inc and DesSinr, thir er

mod in properties and comparison with the actual

References:

3

4

CH. Weatherburm An Introduction o Riemannian Geometry and the tensor calculos

AD. Faldngton += The Mathematical theory of Relativity
Goyal and Gps Thot oF Relativity

Adler, Mazi, M, Schiffer

rodocion 10 Genera Ratt

LSynge + Spt ity & General thon of Relativity

o (Es
u, AAN oe

wi NA

theory of Rea

25. 5.Galois Theory

Unit 1: Rings, examples of tng ideals, prime and maximal ideals. Integral domains,
Euelidcan Domains, Principal deal Domains and Unique Factorizations Domains
Polynomial rings over UFD's,

Unit 2: Fields, Characteristic and prime subfield, field extensions init, algebraic and
finitely generate field extensions, algebraic closures.

Unit 3 Spliting fields, normals extension, Multiple ros. Finite field, separable
Extension

Unit 4 Galois group. Fundamental Theorem of Galois Theory. Solvability by radicals
Galois theoem on solvability. Cyeli and abelian extensions. Classical ruler and Compass

Referen

1. DS. Dummit
2. Joseph Rotman, Galois Theory

3. N.Jacobson, Basic Algebra 1, 2° ed, Hindustan Publising Co 1984
3. S.Lang, Algebra L 1! Edition, Addison Wesley, 2005

Ind RM. Foot

Abstract Algebra

nh
Sects CRU or

26. 6. Advanced Topology

Unit 1: Countably compact spaces, sequentially compact spaces, totally bounded! metre
spaces

Unit 2: Lebesgue’ covering lemma, spaces of continuous functions, Arzela-Ascoli
theorem, Weierstras

approximation theory

Unit 3: Stone Weierstrass's theorem, metizable spaces and meztriztion theorems,
uniform spaces, topology of uniform spaces

Uniform continuity. uniform mettizable topological spaces, metriable uniform

me properties of completely regular spaces, the Stone-Chech ompactfietion,

Reference:

1. $. Willard: General Topology. Addison - Wesley 1970
2. SW. Davis: Topology. TMH 2006

3. K.K. Ma: Advanced General Topology, Nav Bharat Prakashan, Patna
4. GF, Simmons: An introduction to Topology and Modem analysis

EE

27. 7. Banach Algebras

lementary properties and Examples of Banach Algebras, Kal quotients, the
um ofan element, dependence of spectrum on algebra, Ahelian Banach Algebras,

Unit
spe
Unit 2: Elementary properties of C*-Algebras and examples, Abelian Algebras and
Functional ealeulus, positive elements

Unit 3: Ideas and quotients, representations of C*-Algcbras and the Gelfand Naimark

Unit 43 Spectral me «presentations of Abelian C*-Algebras, Special theorem,

+ Topologies on BH) the double commutant rem and Abelian Von - Neumann

Uni
Ales
Reference:

1. J.B. Conway: A course in Functional Analysis, springer 1990

2. RV. Kadison and LR. Ri Is ofthe theory of Operator Algebras,
AMS 1997

3, G Murphy: C* -Algebras and Operator theory, Academic Press 1990

ens (a

prose: Fundam

28. 8. Commutative Algebra

Unit 1: Ring and ring homomorphishns. ideas, quotient rings, Zero divisors, Nilpotent
elements units, prime idcals and maximal ideals, Nil Radical and Jacobson Radical
‘Operations on ideals, extension and contraction.

Unit 2: Modules and module homomorphishms, sub-modules, quotient mod
Operations on subemodules, Direct sum and products, Finitely generated modales, exact
sequences.

Unit 3: Tensor product of modules, restriction and extension of scalars, exactness
properties of tensor product, Algebras, Tensor product of algebras,

Unit 4: Local properties, extended and contracted ideals in
decompositions. integral dependence, the going up theorem,
domains, he going-down theorem, chain conditions,

ng of fractions, primary
rally closed integral

Unit
Dedck

Primary decompositions in Nocthcrian ring, Artin rings, der
vd domains, Fractional ideals

valuation ring.

Reference:

1. MI. Atiyah and LG, Macdonald: Introduction to Com
Wesley

2, 1. Matsumura: Commutative ring theory, Camb, Univ. Press
3. N.S. Gopala Krishnan- Commutative algebra

4. $. Lang: Algebra, spring

5. DP. Pati, Pati, Storch: Introduction Lo Algebraic Geometry and Con
Altra, Anshan Publishers

, An ES
bra Obi, Codes

ve
es

29. 9. Programming in C

‘Theory
1. Introduction o programming languages, € language and is features.
2. Understanding of Stueture of Programme in €

3. Basic datatypes. Library in C

44 Operators and expression in €

5 ns use for input and ouput in €

6 ional branching in C, use of then,

7 Looping in €. use of for loop, while loop, do-while loop, nested loops
8

Practical

1

a

5. Convert a number o any given base

6. Generate fest perfect numbers

7

x

9

10. Inverse ofa matrix

References

1. Y.kanitkar: Lets €.
2. Rober laore: C programmi
E. Balaguruswami: Programmin

sanannenenesarens Eugen

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