MASTERS OF MATHEMATICS
SEMESTER
1
MAT101-ALGEBRA
UNIT-1:
Group Theory: Definition
of a Group –Some Examples of Groups-some
Preliminary Lemmas-Subgroups-A Counting Principle –Normal Subgroups and
Quotient Groups- Homomorphisms - Automorphisms
–Cayley’s theorem-Permutation groups .(2.1 to 2.10 prescribed book(1)).
UNIT-2:
Group Thoery Continued :Another
counting principle-Sylows’s theorem-direct products-finite Abelian groups (2.11
to2.14 of the prescribed book (1)).
UNIT-3:
Ring
theory:Definitions and examples of rings –some special classes of
rings-Homomorphisms-Ideals and Quotient rings of an integral domain.(3.1 to 3.8
of the prescribed book (1)).
UNIT-4:
Ring theory continued: Euclidean rings-A particular Euclidean
ring,Polynomials over the rational field-Polynomial rings over commutative
rings.(3.9 to 3.11 of the prescribed book (1)).
UNIT-5:
Vector Spaces:Elememtary
Basic concepts –Linear Independence and
basics –Dual spaces (4.1 to 4.3 of prescribed book (1)).
PRESCRIBED
BOOK:I.N.HERSTEIN ,Topics in Algebra ,2 edition,WILEY EASTERN
LIMITED .NEW DELHI(1988).
REFERENCE
BOOK:BHATTACHARYA P.B .,JAIN.SK.,NAGPAUN S.R.”Basic Abstract
Algebra “,Cambridge press 2 edition .
MAT102- REAL ANALYSIS-I
UNIT-I:
Continuity:
Limits
of functions, continuous functions,
continuity and compactness, continuity and connectedness. Discontinuities,
Monotone functions, Infinite limits and limits at infinity. (4.1 to 4.34 of
chapter 4)
UNIT
II
Differentiation:
Derivative
of a real function, Mean value theorems, The continuity of derivatives, L’
Hospital’s rule, Derivatives of higher Order, Taylor’s theorem, Differentiation
of vector valued functions. (5.1 to 5.19 of chapter 5)
UNIT
III
Riemann
Stieltjes integral: Definition and Existence of the integral,
Properties of the integral, integration and Differentiation, integration of
vector valued functions, Rectifiable curves. (6.1 to 6.27 of chapter 6)
UNIT
IV
Sequences
and series of functions:
Discussion of
main problem, Uniform convergence, Uniform convergence and continuity, ,
Uniform convergence and integration ,
Uniform convergence and Differentiation.(7.1 to 7.18 of chapter 7)
UNIT
V
Sequences
and series of functions:
Equi continuous
family of functions- Weierstrass theorem and Stone’s generalization.( .(7.19 to
7.33 of chapter 7)
PRESCRIBED BOOK: WALTER
RUDIN , Principles of Mathematical Analysis, third Edition, Tata Mc. Graw Hill.
REFERENCE BOOKS: 1. TOM. M. APOSTOL, Mathematical Analysis,
second Edition, Narosa Pub.,2002.
2. D.
SOMASUNDARAM, B. CHOUDARY, A First Course in Mathematical Analysis , Narosa
Publishing House.
MAT103- ORDINARY DIFFERENTIAL EQUATIONS
UNIT
I
Second
order linear equations: Introduction, The general solution of
the homogeneous equation, the use of a known solution to find another, the
homogeneous equation with constant coefficients, the method of undetermined
coefficients, The method of variation of parameters.(sections 14 to 19 of
chapter 3 of [1])
UNIT
II
Power
series solutions and special functions: Introduction, A
review of Power series, series solutions of first order equations, Second order
Linear equations- Ordinary points, Regular singular points, Regular singular
points(continued),( sections 26 to 30 of chapter 5 of [1])
UNIT
III
Gauss’s hyper
geometric equation , Linear systems, Homogeneous Linear systems with constant
coefficients, The method of successive approximations, Picards theorem (section
31 of chapter 5 of [1], sections 55 to 56 of chapter 10 of [1], sections 68 and
69 of chapter 13 of [1])
UNIT
IV
Some
special functions of Mathematical Physics: Legendre
polynomials, Bessels functions, The Gamma function, Properties of Bessel
functions (sections 44 to 47 of chapter 8 of [1])
UNIT
V
Laplace
Transformations:Introduction, A few remarks on theory, Applications to
differential equations, Derivatives and Integrals of Laplace transforms,
Convolutions.(sections 48 to 53 of chapter 9 of [1])
PRESCRIBED
BOOK: [1] G.F. SIMMONS, DIFFERENTIAL EQUATIONS WITH APPLICATIONS
AND Historical Notes, Tata McGraw Hill, 2003.
MAT104-TOPOLOGY
UNIT
1
METRIC
SPACES: Definitions and some examples, Open sets, Closed sets,
Convergence, completeness and Baire’s
theorem, Continuous mapping.(Section 9 to 13 of chapter-2)
UNIT
2
TOPOLOGICAL
SPACES: The Definition and some examples, Elementary concepts, Open
bases and Open subbases, Weak topologies.(Section 16 to 19 of chapter 3)
UNIT
3
COMPACTNESS:
Compact
spaces, Products of spaces, Tychonoff’s theorem and locally compact spaces,
compactness for metric spaces, Ascoli’s
theorem. (sections 21 to 25 of chapter-4)
UNIT
4
SEPERATION:
T1
spaces and Hausdorff Spaces, completely regular spaces and normal spaces,
Uryshon’s lemma and Tietze extension Theorem. (sections 26 to 28 of chapter 5)
UNIT
5
CONNECTEDNESS:
Connected
spaces, The components of a space, Totally Disconnected spaces. ( sections 31
to 33 of chapter 6)
PRESCRIBED
BOOK : G.F. SIMMONS , Introduction to Topology and Modern
Analysis, Mc. Graw Hill Book Company, Newyork International Student edition.
REFERENCE
BOOK: 1.JAMES
DUGUNDJI, Topology, Universal book stall,1990
2. JOHN L
KELLY, Genereal Topology, Springer, 2005.
MAT 105- DISCRETE
MATHEMATICS
UNIT – I
LOGIC: Computer
Representation of sets, Mathematical Introduction, Matrices, Logic,
Tautology,Normal Forms, Logical Inferences, Predicate Logic, Universal
Quantifiers, Rules of Inference. ( chapter 1 of [3])
UNIT- II
FINITE MACHINES : Introduction,
State Tables and state diagrams, Simple Properties, Dynamics, Behaviour And
Minimization. (Sections 5.1 to 5.5 of chapter 5 of [1])
UNIT –III
LATTICES: Properties and Examples of Lattices,
Distributive Lattices, Boolean Algebras.
(Sections 1 to 3 of Chapter 1 of [2])
UNIT –IV
LATTICES CONTINUED: Boolean
Polynomials, Ideals, filters and Equations, Minimal Forms of Boolean
Polynomials, (Sections 4,5,6 of chapter-1 of [2])
UNIT –V
APPLICATION OF LATTICES: Switching Circuits, Applications of Switching
Circuits, More Applications of Banach Algebras (sections 7,8 of chapter-2 of
[2])
PRESCRIBED BOOKS [1] JAMES L FISHER, Applications Orientede
algebra” IEP, Dun-Downplay Pub.1977.
[2] R.LIDL AND
G. PILZ, Applied abstract algebra, Second edition, Springer, 1998.
[3]
RM.SOMASUNDARAM, Discrete Mathematical Structures, Prentice Hall of India, 2003
REFERENCE BOOK: J.P.TREMBLAY
AND R.MANOHAR, Discrete Mathematical Structures with Applications to Computer
science, Tata Mc.Graw Hill, 2002.
II-SEMESTER
MAT 201- GALOIS THEORY
UNIT – I
MODULES: Definitions and
examples; Submodules and direct sums; R-Homomorphisms and Quotient Modules;
Completely reducible modules.(Sections 1 to 4 of chapter 14 of [1])
UNIT-II
ALGEBRAIC EXTENSION OF FIELDS: Irreducible Polynomials and Eisenstein’s
criterion; Adjunction of Roots; Algebraic extensions; Algebraically closed
Fields, (sections 1 to 4 of chapter 15 of [1])
UNIT-III
NORMAL AND SEPERABLE EXTENSIONS: Splitting
fields; Normal extensions; multiple roots; finite fields; Seperable
extensions.(sections 1 to 5 of chapter 16 of [1])
UNIT-IV
GALOIS THEORY: Automorphism
groups and fixed fields; Fundemental theorem of Galois Theory; Fundemental
Theorem of Algebra (sections 1 to 3 of chapter 17 of [1])
UNIT-V
APPLICATIONS OF GALOIS THEORY TO CLASSICAL PROBLEMS: Roots
of Unity and cyclotomic polynomials; Cyclic extensions; Polynomials solvable by
radicals; Ruler and compass constructions. (sections 1,2,3,5 of chapter 18 of
[1])
PRESCRIBED BOOK: [1].
BHATTACHARYA P.B., JAIN S.K.,
NAGPAUL S.R., Basic Abstract Algebra,
Second edition, Cambridge Press.
REFERENCE BOOKS: 1.
JOSEPH ROTMAN, Galois Theory, Second Edition, Springer, 1998
2. ARTIN
M, Algebra, PHI, 1991
MAT 202- REAL ANALYSIS
–II
UNIT-I
SOME SPECIAL FUNCTIONS: Power
series, The exponential and logarithmic functions, the trigonometric functions,
The algebraic completeness of the complex field, Fourier Series. (Sections 8.1
to 8.15 of Chapter 8)
UNIT-II
FUNCTIONS OF SEVERAL VARIABLES: Linear Transformations, Differentiation,
Contraction Principle ( sections 9.1 to 9.23 of chapter 9)
UNIT –III
FUNCTIONS OF SEVERAL VARIABLES (Continued): Inverse function theorem, Implicit Function
Theorem, The Rank Theorem, determinants, Derivatives of Higher order and
differentiation of integrals ( sections 9.24 to 9.43 of chapter 9)
UNIT-IV
INTEGRATION OF DIFFERENTIAL FORMS:
Integration, Primitive mappings, partitions of unity, change of variables,
differential forms. ( sections 10.1 to 10.25 of chapter 10)
UNIT-V
INTEGRATION OF DIFFERENTIAL FORMS (CONTINUED): simplexes and chains, Stoke’s Theorem, closed
forms and exact forms. (sections 10.26 to 10.41 of chapter 10)
PRESCRIBED BOOK: WALTER
RUDIN , Principles of Mathematical Analysis, third Edition, Tata Mc. Graw Hill.
REFERENCE BOOKS: 1. TOM. M. APOSTOL, Mathematical Analysis,
second Edition, Narosa Pub.,2002.
2. D.
SOMASUNDARAM, B. CHOUDARY, A First Course in Mathematical Analysis , Narosa
Publishing House.
MAT 203- PARTIAL
DIFFERENTIAL EQUATIONS
UNIT- I
FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS: Curves and Surfaces, Genesis of first order
partial differential equations, Classification of integrals, Linear Equations
of the first order, pfaffian Differential Equations, Compatible systems,
Charpit’s method. (sections 1.1 to 1.7 of chapter 1 of [1])
UNIT-II
Jacobi’s
Method, Integral surfaces through a given curve, Second order partial
differential equations, Genesis of second order partial differential equations,
classification of second order partial differential equations (sections
1.8&1.9 of chapter 1of[1], sections 2.1 to 2.2 of chapter 2 of[1])
UNIT-III
One Dimensional
Wave Equation, Vibrations of an infinite string, vibrations of a semi-infinite
string, vibrations of a string of finite length, Riemann’s method, Vibrations
of a string of finite length( Method of separation of variables ). (sections 2.3.1 to 2.3.5 of chapter 2 of
[1])
UNIT-IV
LAPLACES
EQUATION: Boundary Value Problems,
Maximum and Minimum Principles, The Cauchy problem, The Dirichlet Problem for
the upper half plane, The Neumann Problem for the upper half plane, The
Dirichlet interior problem for a circle, The Dirichlet exterior problem for a
circle, The Neumann problem for a
circle, The Dirichlet problem for a
Rectangle.. (sections 2.4.1 to 2.4.9 of chapter 2 of[1])
UNIT-V
Harnacks theorem, Laplaces Equation – Green’s
Function, The Dirichlet problem for a
Half Plane, The Dirichlet problem for a
circle, Heat Conduction- Infinite rod case, Heat conduction-Finite rod case,
Duhamel’s principle, Wave Equation, Heat
Conduction Equation. (sections 2.4.10 to 2.4.13, 2.5.1 to 2.5.2, 2.6.1 to 2.6.2
of chapter 2 of[1])
PRESCRIBED BOOK: [1] T.AMARANATH, An Elementary course in
partial Differential Equations, second edition, Narosa Publishing House, 2003
Reference Book: [2]. SNEEDON IAN, Elements of Partial Differential
Equations, Tata Mc Graw Hill, 1987.
[3] :
K.Sankararao, Introduction to partial differential equations,PHI,2003
MAT 204- NUMERICAL METHODS
WITH “C”
UNIT-1
C-BASICS: C- Character set, data
types, Variables, Constants, Expressions, structure of C Program, Operators and
their precedence and associativity,Basic input and output statements, control
structures, simple programmes in C using all the operators and control
structures
FUNCTIONS: Concept Of a
Function, Parameters and how they are passed, automatic variables, Recursion,
scope and extent of variables ,writing programs using recursive and non-
recursive functions. (1.4,1.7,1.11,1.12 of chapter 1, 2.2,2.3,2.4of chapter
2,3.1,3.2,3.3 of chapter 3 and 5.1,5.2,5.3 of chapter 5 of [1])
UNIT-II
ARRAYS AND STRINGS: Single
and Multi-dimensional Arrays, Character array as a string, Functions on
strings, Writing C programs using arrays and for string manipulation.
POINTERS: Pointers declarations, Pointers expressions,
Pointers as parameters to functions, Pointers and arrays, Pointers arithmetic
STRUCTURES AND UNIONS: Declaring
and using Structures, Operations on Structures, arrays of Structures, User
defined data types, Pointers to structures. (4.1 to 4.6 of chapter 4, 6.1 to
6.8 of chapter 6, chapter 9 & chapter 10 of [1])
UNIT-III
INTERPOLATION AND APPROXIMATION:INTRODUCTION,LAGRANGE
and newton interpolation,finite difference operators,interpolating polynomials
using finite differences,hermite interpolation.(section 4.1 to 4.5 of chapter 4
of [2]).
Unit-IV
Numerical differentiation and integration:INTRODUCTION,numerical
differentiation,numerical Integration,methods based on interpolation,methods
based on undetermined coefficients,composite
integration methods.{section 5.1,5.2,5.6,5.7,5.8,5.9 of chapter 5 of [2]
UNIT-V
ORDINARY DIFFERENTIAL EQUATIONS : Introduction,numerical
method,single step methods,multi step methods.(sections 6.1 to 6.4 of chapter 6
of [2]).
Prescribed books:[1]
AJAY MITTAL,C PROGRAMING A PRACTICAL APPROACH PEARSON,.
[2]
M.K.JAIN,S.R.K.IYANGAR AND R.K.JAIN ,NUMERICAL METHODS FOR SCIENTIFIC AND
ENGINEERING COMPUTAYION,THIRD EDITION ,NEW AGE INTERNATIONAL PVT LTD,NEW DELHI
1997.
MAT205C GRAPH THEORY
UNIT-1
INTRODUCTION:
What is graph,finite and infinite graphs,incidence and degree,isolated vertex
and pendent vertex and null graph
Paths and circuits:
Isomorphism,subgraphs,a puzzle with multi coloured cubes,walks paths and
circuits,connected graphs disconnected graphs,components,euler
graphs,operations on graphs,More on euler graphs ,Hamiltonion paths and
circuits,travelling-salesman problem
(chapters 1&2 of (1))
UNIT-2
Tress and Fundamental Circuits:
Tress,some properties pf tress, pendent vertices in a tree , distances and
centers in a tree, rooted and binary tress, on counting trees, spanning trees,
fundamental circuits, finding all spanning trees of a graph, spanning trees in
a weighed graphs. (chapter 3 of (1))
UNIT-3
Cut sets and cut vertices:
Cut sets, some properties of cut set,All cut sets in a Graph, Fundamental
circuits and cut sets, connectivity and seperability, network flows 1 isomorphism 2-isomorphism . ( chapter 4 of
(1))
UNIT-4
Planar and dual graphs: combinatorial
Vs Geometric graphs, planar graphs, kuratowsk’s two graphs,different
representations of a planar graph, detection of planarity, geometric dual
.(section 1 to 6 of chapter 5 of (1))
UNIT-5
Vector spaces of a graph:
Sets with one operation,sets with two operations,Modular arithmetic and galois
field, vector and vector spaces,vector space associated with the graph,basis
vectors of graph,circuits and cutset subspaces.(section 1 to 7 of chapter 6 of
{1})
Prescribed books:[1]
NARSING DEO,graph theory with applications to engineeringand computer
science,prentice hall of India pvt.,New Delhi 1993.
REFERENCE BOOK:BONDY J.A.AND
U.S.R MURTHY,graph theory with applications,North Holland
III SEMESTER
MAT 301
–RINGS AND MODULES
UNIT-I :
Fundamental Concepts of Algebra
Rings and related Algebraic
systems; Subrings, Homomorphisms, Ideals.
(Sections 1.1, 1.2 of chapter -1)
UNIT-II :
Fundamental Concepts of Algebra
Modules, Direct products and
Direct sums; Classical Isomorphism Theorems.
(Sections 1.3, 1.4 of chapter 1)
UNIT-III :
Selected Topics on Commutative
Rings
Prime ideals in Commutative
Rings; Prime ideals in Special Commutative Rings.
(Sections 2.1, 2.2 of Chapter 2)
UNIT-IV :
Selected Topics on Commutative
Rings
The Complete Ring of Quotients of
a Commutative Ring; Ring of quotients of Commutative Semi Prime Rings.
(Sections 2.3, 2.4 of Chapter 2)
UNIT-V:
Selected Topics on Commutative
Rings: Prime Ideal Spaces (Section 2.5 of chapter 2)
Appendices: Functional
Representations ( Appendix 1: Proposition 1 to Proposition 9)
PRESCRIBED BOOK: J. Lambek, “Lectures
on Rings and Modules”, A Blasidell Book in Pure and Applied Mathematics.
REFERENCE
MAT-302-
COMPLEX ANALYSIS
UNIT-I
The Complex Number system: The real
numbers, The Field of Complex Numbers, The Complex plane, Polar representation
and roots of Complex numbers, Lines and Half planes in the Complex plane, The
extended plane and it’s spherical representation.
Elementary Properties and
Examples of Analytic Functions: Power series, Analytic
functions. Analytic functions as mappings, Mobius transformations. (Chapters I
and III).
UNIT-II
Complex Integration: Power
series representation of Analytic functions, Zeros of an Analytic function, The
Index of a closed curve ( Sections 2 , 3 and 4 of chapter IV)
UNIT-III
Complex Integration :, Cauchy's
Theorem and Integral formula, The homotopic version of Cauchy’s theorem and
simple connectivity, Counting zeros , The open mapping theorem. The Goursat’s
Theorem (Sections 5 to 8 of Chapter IV)
UNIT-IV
Singularities:
Classification of Singularities, Residues , The Argument Principle.(ChapterV)
UNIT-V
The Maximum Modulus Theorem: The
Maximum Principle, Schwarz’s Lemma, Convex functions and Hadamard’s three
circle Theorem, Phragmen-Lindelof theorem.(Chapter VI).
PRESCRIBED BOOK: John B.
Conway, “Functions of one Complex Variable”, Second
Edition, Springer, International
Student Edition , Narosa publishing House.
REFERENCE BOOKS:
1. James ward
Brown and Ruel V. Churchill, “Complex Variables and Applications”, sixth
Edition, Mc Graw Hill International Editions.
2.
S. Ponnusamy, Foundations of Complex Analysis , Narosa Publishing House
MAT 303 – FUNCTIONAL ANALYSIS
UNIT-I
Review of Properties of Metric
spaces, Vector spaces, Normed spaces, Banach spaces, Further properties of
Normed spaces, Finite Dimensional normed spaces and Sub spaces, Compactness and
Finite Dimension.
(Chapter-1 and 2.1 to 2.5 of
Chapter 2)
UNIT-II
Linear Operators, Bounded and
Continuous Linear Operators, Linear Functionals, Linear Operators and
Functionals on Finite Dimensional Spaces, Normed Spaces of Operators,
Dual space.
(2.6 to 2.10 of Chapter 2 )
UNIT-III
Zorn’s Lemma, Hahn Banach
Theorem, Hahn Banach Theorem for Complex Vector Spaces and Normed Spaces,
Applications to Bounded Linear Functonals of C[a,b], Adjoint Operators,
Reflexive spaces.
(4.1 to 4.6 of Chapter 4)
UNIT- IV
Category Theorem, Uniform
Boundedness Theorem, Strong and Weak Convergence, Convergence of Sequences of
Operators and Functionals, Open Mapping Theorem, Closed Linear Operators,
Closed Graph Theorem
(Sections 4.7, 4.8, 4.9, 4.12 and
4.13 of Chapter 4).
UNIT- V
Banach Fixed Point Theorem,
Application of Banach’s Theorem to Linear Equations, Application of Banach’s
theorem to differential Equations, Application of Banach’s theorem to Integral
Equations.
(Chapter 5)
PRESCRIBED BOOK: Erwin
Kreyszig, Introductory Functional analysis with Applications, John Wiley
& Sons.
REFERENCE
BOOK: M. Thamban Nair, Functional
Analysis- A First Course, PHI
MAT 304 A –
LATTICE THEORY
UNIT –I
Partly Ordered Sets:
Set Theoretical Notations,
Relations, Partly Ordered Sets, Diagrams, Special Subsets of a Partly Ordered
Set, Length, Lower and Upper Bounds, The Minimum and Maximum Conditions, The
Jordan–Dedekind Chain Condition, Dimension Functions.
( Sections 1 to 9 of chapter I)
UNIT – II
Lattices in General:
Algebras, Lattices, The Lattice
Theoretical Duality Principle, Semi Lattices, Lattices as Partly Ordered Sets,
Diagrams of Lattices, Sub Lattices, Ideals, Bound Elements of a Lattice, Atoms
and Dual Atoms, Complements, Relative Complements, Semi Complements,
Irreducible and Prime Elements of a Lattice, The Homomorphism of a Lattice,
Axiom Systems of Lattices.
( Sections 10 to 21 of chapter
II)
UNIT – III
Complete Lattices:
Complete Lattices, Complete Sub
Lattices of a Complete Lattice, Conditionally Complete Lattices, Compact
Elements and Compactly Generated Lattices, SubAlgebra Lattice of an Algebra,
Closure Operations, Galois Connections, Dedekind Cuts, Partly Ordered Sets as
Topological Spaces.
( Sections 22 to 29 of chapter
III)
UNIT – IV
Distributive and Modular
Lattices:
Distributive Lattices, Infinitely
Distributive and Completely Distributive Lattices, Modular Lattices,
Characterization of Modular and Distributive Lattices by their Sublattices,
Distributive Sublattices of Modular Lattices, The Isomorphism Theorem of Modular
Lattices, Covering Conditions, Meet Representation in Modular and Distributive
Lattices.
(Sections 30 to 36 of chapter IV)
UNIT-V
Boolean Algebras:
Boolean Algebras, De Morgan
Formulae, Complete Boolean Algebras, Boolean Algebras and Boolean Rings, The
Algebra of Relations, The Lattice of Propositions, Valuations of Boolean
Algebras.
(Sections 42 to 47 of chapter VI)
PRESCRIBED BOOK: Gabor Szasz,
Introduction to Lattice Theory, Acadamic press
REFERENCE
BOOK: G.Birkhoff, Lattice Theory, Amer.
Math.Soc.
MAT 304 B –
SEMI GROUPS
UNIT – I
Basic Definitions, Monogenic
Semigroups, Ordered Sets, Semilattices and Lattices, Binary Relations,
Equivalences.
( Sections 1 to 4 of Ch. I)
UNIT – II
Congruences, Free Semigroups,
Ideals and Rees Congruences, Lattices of Equivalences and Congruences.
( Sections 5 to 8 of Ch. I )
UNIT - III
Introduction, The equvivalences L
,R , H, J and D , The structure of D - Classes, Regular
Semigroups.
(Chapter II )
UNIT –IV
Introduction, Simple and 0 –
Simple Semigroups, Principle Factors, Rees’s Theorem, Primitive Idempotents.
(Sections 1 to 3 of Chapter III )
UNIT –V
Congruences on Completely 0 –
Simple semigroups, The Lattice of Congruences on a Completely 0 – Simple
Semigroup, Finite Congruence- Free Semigroups.
(Sections 4 to 6 of Chapter III )
PRESCRIBED BOOK: J.M. Howie, An
Introduction to Semigroup Theory, Academic Press,
REFERENCE
BOOK: A.H. Clifford, G.B. Preston, The
Algebraic Theory of Semigroups, American Mathematical Society.
MAT 305 A –
THEORY OF COMPUTER SCIENCE - I
UNIT-I
Mathematical Preliminaries:
1.1 Sets, Relations and Functions
1.2 Graphs and Trees
1.3 Strings and their properties
1.4 Principle of Induction
The Theory of Automata:
2.1 Definition of an Automation
2.2 Description of a Finite
Automation
2.3 Transition Systems
2.4 Properties of Transition
Functions
2.5 Acceptability of a String by
a Finite Automation
2.6 Nondeterministic Finite State
Machines
2.7 The Equivalence of DFA and
NDFA
2.8 Mealy and Moore Models
2.9 Minimization of Finite Automata
(Chapters 1 and 2)
UNIT-II
Formal Languages :
3.1 Basic
Definitions and Examples
3.2 Chomsky
Classification of Languages
3.3
Languages and their relation
3.4
Recursive and Recursively Enumerable Sets
3.5
Operations on Languages
3.6 Languages and Automata
(Chapter 3)
UNIT-III
Regular Sets and Regular Grammars
:
4.1 Regular
Expressions
4.2 Finite
Automata and Regular Expressions
4.3 Pumping
Lemma for Regular Sets
4.4
Application of Pumping Lemma
4.5 Closure
Properties of Regular Sets
4.6 Regular Sets and Regular
Grammars
UNIT -IV
Context-free Languages :
5.1
Context-free Languages and Derivation Trees
5.2
Ambiguity in Context-free Grammars
5.3
Simplification of Context-free grammars
5.4 Normal
Forms for Context-free grammars
5.5 Pumping
lemma for Context-free Languages
5.6 Decision Algorithms for
Context-free Languages
(Chapter 5)
UNIT-V
Pushdown Automata:
6.1 Basic
Definitions
6.2
Acceptance by pda
6.3 Pushdown
Automata and Context-free languages
6.4 Parsing and Pushdown automata
(Chapter 6)
PRESCRIBED BOOK: KLP Mishra
& N.Chandrasekharan, Theory of Computer Science (Automata,Languages and
Computation), Prentice Hall of India.
REFERENCE BOOKS : 1. Hopcroft J.E
& Ullman J.D., Introduction to Automata Theory, Languages and
Computation, Narosa Publishing House, 1987.
2. E.V. Krishna Murthy , Introductory
Theory of Computer Science, Affiliated East –West Press., New Delhi, 1984.
MAT 305 B --
LINEAR PROGRAMMING
UNIT – I
Mathematical Background : Lines and
hyperplanes: Convex sets, Convex sets and hyperplanes, Convex cones. [Sections
2.19 to 2.22 of Chapter 2of [1] ].
Theory of the simplex method : Restatement
of the problem, Slack and surplus Variables , Reduction of any feasible
solution to a basic feasible solution, Some definitions and notations ,
Improving a basic feasible solution, Unbounded solutions, Optimality
conditions, Alternative optima, Extreme points and basic feasible solutions.
[Sections 3.1, 3.2, 3.4 to 3.10
of Chapter 3 of [1] ]
UNIT –II
Detailed development and
Computational aspects of the simplex method: The
Simplex method, Selection of the vector to enter the basis, degeneracy and
breaking ties, Further development of the transportation formulas, The initial
basic feasible solution –artificial variables, Tableau format for simplex
computations, Use of the tableau format, conversion of a minimization problem
to a maximization problem, Review of the simplex method , Illustrative
examples.
[Sections 4.1 to 4.5, 4.7 to 4.11
of Chapter 4 of [1] ].
UNIT –III
Transportation problem: Introduction,
properties of the matrix A, The Simplex Method and transportation
problems, Simplifications resulting from all yijαβ = ± 1 or 0, The
Transportation Problem Tableau, Bases in the transportation Tableau, The
Stepping-Stone algorithm, Determination of an initial basic feasible solution,
Alternative procedure for computing zij –cij; duality.
[Sections 9.1 to 9.7 & 9.10,
9.11 of Chapter 9 of [1] ].
UNIT –IV
The Assignment problem : Introduction,
Description and Mathematical statement of the problem, Solution using the
Hungarian method, The relationship between Transportation and Assignment
problems, Further treatment of the Assignment problem, The Bottleneck Assignment
problem.
(Chapter 6 of [2] )
UNIT V
Further Discussion of the Simplex
Method: The two phase Method for Artificial variables, phase-I, Phase-II,
Numerical examples of the two phase method.
(Sections 5.1 to 5.4 of Chapter
-5 of [1] ]
PRESCRIBED BOOKS:
[1] G.Hadley, Linear
Programming, Narosa publishing House
[2] Benjamin Lev and Howard
J.Weiss, Introduction to Mathematical Programming
Edward Arnold Pub. London, 1982.
IV SEMESTER
MAT 401 – NON COMMUTATIVE RINGS
UNIT I
Classical theory of Associative
rings : Primitive Rings, Radicals, Completely reducible modules.
[Sections 3.1, 3.2 ,3.3 of
Chapter 3]
UNIT II
Classical theory of Associative
rings: Completely reducible rings, Artinian and Noetherian rings,
[Sections 3.4, 3.5 of Chapter 3]
UNIT III
Classical theory of Associative
rings: On lifting idempotents, Local and Semi perfect rings.
[Sections 3.6, 3.7 of Chapter 3]
UNIT IV
Injectivity and Related concepts:
Projective modules, Injective modules
[Sections 4.1, 4.2 of Chapter 4]
UNIT V
Injectivity and Related concepts:
The complete ring of quotients, Rings of endomorphisms of
Injective modules.
[Sections 4.3,4.4 of Chapter 4]
PRESCRIBED BOOK: J. Lambek, Lectures
on Rings and Modules, A Blasidell book in Pure and Applied Mathematics.
REFERENCE
BOOK: Thomas W. Hungerford, Algebra , Springer
publications
MAT 402 – MEASURE AND INTEGRATION
UNIT-I
Lebesgue Measure:
Introduction, Outer measure , Measurable sets and
Lebesgue measure, A nonmeasurable
set, Measurable functions, Littlewood’s three principles (Chapter 3)
UNIT-II
The Lebesgue Integral: The
Riemann Integral, The Lebesgue Integral of a
bounded function over a set of
finite measure, The Integral of a non- negative function, The general Lebesgue
Integral. ( Sections 4.1 to 4.4 of Chapter 4).
UNIT-III
Differentiation and Integration:
Differentiation of monotone functions, Functions of bounded variation,
Differentiation of an Integral, Absolute continuity. ( Sections 5.1 to 5.4 of
Chapter 5)
UNIT-IV
Measure and Integration: Measure
spaces, Measurable functions, Integration, General Convergence theorems, Signed
Measures, The Radon-Nikodym theorem. (Sections 11.1 to 11.6 of Chapter 11)
UNIT-V
Measure and Outer Measure: Outer
Measure and Measurability , The Extension theorem, Product measures.
(Sections 12.1,12.2 & 12.4 of
Chapter 12 ).
PRESCRIBED BOOK: H.L. Royden,
Real Analysis, Third Edition, Pearson pub.
REFERENCE BOOKS : [1] P.R.Halmos,
Measure Theory, Springer-Verlag, 1974
[2]. V.I. Bogachev,
Measure Theory, Springer –Verlag, 1997
MAT 403 – OPERATOR THEORY
UNIT –I
Inner product spaces, Hilbert
Space, Further properties of Inner product spaces, Orthogonal Complements and
Direct sums, Orthonormal sets and sequences, Series related to Orthonormal
sequences and sets.
( Sections: 3.1 to 3.5 of Chapter
3)
UNIT – II
Total orthonormal sets and
sequences, Legendre, Hermite and Laguerre polynomials, Representation of
functionals on Hilbert Spaces, Hilbert-Adjont Operator, Self-Adjoint, Unitary and
Normal operators.
( Sections: 3.6 to 3.10 of
Chapter 3)
UNIT –III
Spectral theory in finite
dimensional normed spaces, Basic concepts, Spectral properties of Bounded
Linear Operators, Further properties of resolvent and Spectrum.
( Sections :7.1 to 7.4 of Chapter
-7)
UNIT –IV
Banach Algebras, Further
properties of Banach Algebras, Compact linear operators on Normed spaces,
Further properties of compact linear operators, Spectral properties of Compact
linear operators on Normed spaces,
(Sections 7.6 , 7.7 of Chapter 7
& Sections 8.1 to 8.3 of Chapter -8)
UNIT –V
Further Spectral properties of
Compact linear operators, Operator equations involving Compact linear
operators, Further Theorems of Fredholm type , Fredholm alternative.
(Sections 8.4 to 8.7 of Chapter
-8)
PRESCRIBED BOOK: Erwin
Kreyszig, Introductory Functional analysis with Applications, John Wiley
& Sons.
REFERENCE
BOOK: M. Thamban Nair, Functional
Analysis- A First Course, PHI
MAT 404 A – ALGEBRAIC CODING
THEORY
UNIT –I
Introduction to Coding Theory: Introduction,
Basic assumptions, Correcting and Detecting error patterns, Information Rate,
The Effects of error Correction and Detection, Finding the most likely codeword
transmitted, Some basic algebra, Weight and Distance, Maximum likelihood
decoding, Reliability of MLD.
(Section 1.1 to 1.10 of Chapter
1)
UNIT – II
Introduction to Coding Theory : Error-
detecting Codes, Error – correcting Codes
Linear Codes : Linear Codes
, Two important subspaces , Independence, Basis, Dimension, Matrices, Bases for
C= <S> and C┴
(Sections 1.11, 1.12 of chapter 1
& Sections 2.1 to 2.5 of chapter 2).
UNIT – III
Linear Codes : Generating
Matrices and Encoding , Parity – Check Matrices, Equivalent Codes, Distance of
a Linear Code, Cosets, MLD for Linear Codes, Reliability of IMLD for Linear
Codes.
(section 2.6 to 2.12 of chapter
2)
UNIT –IV
Perfect and Related Codes: Some bounds
for Code, Perfect Codes, Hamming Codes , Extended Codes, The extended Golay
Code, Decoding the extended Golay Code, The Golay code, Reed – Mullar Codes,
Fast decoding for RM (1,m).
(Chapter 3)
UNIT –V
Cyclic Linear Codes : Polynomials
and Words , Introduction to Cyclic codes, Polynomials encoding and decoding,
Finding Cyclic Codes, Dual Cyclic Codes.
(Chapter 4)
PRESCRIBED BOOK: D.G.
Hoffman, D.A. Lanonard , C.C. Lindner, K. T. Phelps,C. A. Rodger, J.R.Wall, CODING
THEORY- THE ESSENTIALS , Marcel Dekker Inc.
REFERENCE BOOK: J.H. Van
Lint, Introduction to coding Theory , Springer Verlag. .
MAT 404 B – FUZZY SETS AND
APPLICATIONS
UNIT-1
From Classical (Crisp) sets to
Fuzzy sets: Introduction, Crisp Sets: An overview,
Fuzzyset: Basic types, Fuzzy sets: Basic Concepts, Characteristics and
significance of the paradigm shift
Fuzzy sets versus Crisp sets: Additional
Properties of α-cuts, Representations of Fuzzy sets, Extension principle for
Fuzzy sets
(Chapters 1 and 2 ).
UNIT – II
Operations on Fuzzy sets: Types of
Operations, Fuzzy Compliments, Fuzzy Intersections: t-Norms, Fuzzy unions: t-Conorms,
Combinations of operations, Agreegation Operations (Chapter 3 ).
UNIT- III
Fuzzy Arithmetic: Fuzzy
Numbers, Linguistic Variables, Arithmetic Operations on Intervals, Arithmetic
Operations on Fuzzy numbers, Lattice of fuzzy numbers, Fuzzy equations
(Chapter 4 ).
UNIT-IV
Fuzzy Relations: Crisp
versus fuzzy relations, Projections and Cylindric Extensions, Binary Fuzzy
Relations, Binary Relations on a Single set, Fuzzy Equivalence Relations, Fuzzy
Compatibility Relations,Fuzzy Ordering Relations, Fuzzy Morphisms, Sup – i
Compositions of Fuzzy Relations, Inf- ωi Compositions of fuzzy Relations
(Chapter 5)
UNIT-V
Fuzzy Logic: Classical
Logic: an Over View, Multivalued Logics, Fuzzy Propositions, Fuzzy Quantifiers,
Linguistic Hedges, Inference from conditional Fuzzy Propositions, Inference
from conditional and qualified propositions, Inference from Quantified
propositions.
(Chapter 8)
PRESCRIBED BOOK: G.J.Klir
& B.YUAN “Fuzzy sets and Fuzzy Logic, Theory and Applications”
Prentice - Hall of India Pvt. Ltd., New Delhi., 2001.
REFERENCE BOOK : H.J.
Zimmermann, Fuzzy set Theory and its Applications, Allied Publishers.
MAT 404 C – NEAR RINGS
UNIT-1
The Elementary Theory of
Near-Rings: .
(a) Fundamental definitions and
properties: Near-rings, N-groups, Substructures,
Homomorphisms and Ideal-like
concepts, Annihilators, Generated objects. .
(b) Constructions: Products,
Direct sums & Subdirect products.
(c) Embeddings: Embedding in M(
), More beds.
UNIT-11
Ideal Theory:
(a) Sums: (1) Sums and direct
sums (2) Distributive sums.
(b) Chain conditions
(c) Decomposition theorems
(d) Prime ideals (1) Products of
subsets (2) Prime ideals (3) Semiprime ideals
(e) Nil and nil potent.
UNIT-III
Elements of the structure theory
:
(a) Types of N-groups
(b) Change of the near-ring
(c) Modularity
(d) Quasi-regularity
(e) Idempotents
(f) More on Minimality.
UNIT-IV
Primitive Near-Rings:
(a) General (1) Definitions and
elementary results (2) The centralizer (3) Independence and density
(b) 0-Primitive near-rings
(c) 1-Primitive near-rings
(d) 2-Primitive near-rings
(1) 2-Primitive near-rings
(2) 2-primitive near-rings with
identity.
UNIT-V
Radical Theory: (a)
Jacobson-type radicals: Common Theory, (1) Definitions and Characterizations of
the radicals (2) Radicals of related near-rings(3) Semi simplicity.
b) Jacobson – type radicals:
Special Theory c) The Nil Radical d) The Prime Radical
PRESCRIBED BOOK: Gunter Pilz,
Near-Rings: The Theory and its Applications, Revised Edition 1983,
North-Holland Publishing Company, AMSTERDAM.
MAT 405 A – THEORY OF COMPUTER
SCIENCE - II
UNIT-I
Turing Machines and Linear
Bounded Automata: Turing Machine Model. Representation of
Turing Machines, Language Acceptability by Turing machines, Design of Turing
Machines, Universal Turing machines and Other Modifications.
(Sections 7.1 to 7.5 of [1])
UNIT-II
Turing Machines and Linear
Bounded Automata: The Model of Linear Bounded Automaton,
Turing Machines and Type 0 Grammars, Linear Bounded Automata and Languages,
Halting Problem of Turing Machines, NP-Completeness.
(Sections 7.6 to 7.10 of [1])
UNIT-III
LR(k) Grammars: LR(k)
Grammars, Properties of LR(k) Grammars, Closure properties of Languages.
(Sections 8.1 to 8.3 of [1])
UNIT-IV
Computability:
Introduction and Basic Concepts, Primitive Recursive Functions, Recursive
Functions, Partial Recursive Functions and Turing Machines.
(Sections 9.1 to 9.4 of [1])
UNIT-V
Propositions and Predicates:
Propositions (Or Statements), Normal Forms of Well-formed Formulas, Rules of
Inference for Propositional Calculus (Statement Calculus), Predicate Calculus,
Rules of Inference for predicate Calculus.
(Sections 10.1 to 10.5 of [1])
PRESCRIBED BOOK: KLP Mishra
& N.Chandrasekharan, Theory of Computer Science (Automata Languages and
Computation), Prentice Hall of India, 1999.
REFERENCE BOOKS : 1. Hopcroft J.E
& Ullman J.D., Introduction to Automata Theory, Languages and
Computation, Narosa Publishing House, 1987.
2. E.V. Krishna Murthy , Introductory
Theory of Computer Science, Affiliated East –West Press., New Delhi, 1984
MAT 405 B – OPERATIONS RESEARCH
UNIT –I
Duality theory and its
Ramifications: Alternative formulations of linear
programming problems, Dual linear programming problems, Fundamental properties
of dual problems, Other formulations of dual problems, Unbounded solution in
the primal, The dual simplex algorithm –an example. Post optimality problems,
Changing the price vector, Changing the requirements vector, Adding variables
or constraints
(Sections 8.1 to 8.7 & 8.10
of Chapter 8 of [1] ).
UNIT –II
The Revised simplex method: Introduction,
Revised simplex method: Standard form I, Computational procedure for standard
form I, Revised simplex method: Standard form II, Computational procedure for
standard form II, Initial identity matrix for phase –I , Comparison of the
simplex method and Revised simplex method.
( Sections 7.1 to 7.6 &7.8 of
Chapter 7 of[1] ).
UNIT –III
Game theory: Game theory
and Linear programming, Introduction, Reduction of a game to a linear
programming problem, Conversion of a linear programming problem to a game
problem. (Sections 11.12 to 11.14 of Chapter 11 of [1] and Section 10.3 of
Chapter10 of [2] )
Integer programming: Introduction,
Gomory’s cut, Balas Implicit Enumeration technique. (Sections 7.1,7.2 and 7.4
of Chapter 7 of [2]).
UNIT IV
Job Sequencing: Introduction,
Classification, Notations and Terminologies, Assumptions, Sequencing Problems:
Sequence for n jobs through two machines, Sequence for n jobs through three
machines, Sequence for 2 jobs through m machines, Sequence for n jobs through m
machines
( Sections 12.1 to 12.5 of
chapter 12 of [3])
UNIT V
Dynamic Programming: Introduction,
Characteristics of Dynamic Programming problem, Deterministic Dynamic
Programming: Dynamic Programming approach to Shortest Route Problem, Dynamic
Programming approach to Resource Allocation: Equipment, Replacement, Cargo
loading, and capital budgeting. Dynamic Programming approach to linear
programming, Stochastic Dynamic Programming. ( sections 6.1 to 6.4 of chapter 6
of [3])
PRESCRIBED BOOKS:
[1] G.Hadley “ Linear
programming” Addison Wesley Publishing Company.
[2] Benjamin Lev and Howard J.
Weiss “ Introduction to Mathematical Programming” Edward Arnold Pub,
London, 1982.
[3] Rathindra p. Sen, Operations
Research- Algorithms and Applications, PHI
REFERENCE BOOK:Nita H.Shah,
Ravi M. Gor, Hardik Soni, “ Operations Research”, PHI
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