COURSE OFFERED



                   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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