COURSE STRUCTURE
Pre-requisite for the Course : Mathematics Courses of the B.A./B.Sc. first two years/Engineering Mathematics Course of B.E. as taught in Indian Universities.
Part – I (Duration : One Year)
MIM 101 Real and Complex Analyses
MIM 102 Algebra – I
MIM 104
Programming in C with ANSI features
MIM 105 Computer Architecture, and Data Structures using C
MIM 106 Practicals related to Papers MIM 104 and MIM 105 ( Data Structure Using C )
REAL ANALYSIS:
Revision : Standard topology on R, structure of open sets, cantor set, lim sup, limits.
Unit-1
Algebra and s - algebra of sets, s - algebra of Borel sets, Lebesgue outer measure on R, Mesurable sets, Lebesgue Measure, measurable function, Littelwood’s three principles, Egoroff’s theorem. Integral of a simple function, Lebesgue integral of bounded functions, bounded convergence theorem.
Unit-2
Integral of nonnegative functions, general Lebesgue integral, Fatou’s lemma, monotone convergence theorem,
Lebesgue’s convergence theorem, convergence in measure. Differentiation of monotone functions, functions of bounded variation, differentiation of an integral, absolutely continuous functions and indefinite integrals.
COMPLEX ANALYSIS :
Bird’s Eye-view : Elementary properties of complex numbers. The function argo. Complex plane with Euclidean metrics, subsets, [a, b], [a, b), (a, b), (a, b], D (a, r), D (R, r), c(a, r), Ñ(R, r), `Ñ(R, r), A(a: R, r), and A(a: R, r)
Domains: Convex and Star-like domains, polygon ally connected domains Continuity of modulus and argo. functions, differentiability of complex functions, regular, analytic and entire functions, singularity.
The Cauchy-Riemann equations, harmonic functions and harmonic
conjugate.
Unit-3
Infinite series of complex numbers, double sequences of complex numbers, power series functions, the exponential function, branches of log.
Arcs, oriented arcs, simple closed curves and oriented simple closed curves, the Jordan curve theorem.
CONTOUR INTEGRATION
Cauchy’s theorem for triangular contours, Cauchy’s theorem for star like domains, applications.
Unit-4
Cauchy’s integral formulae, Taylor’s and Laurent’s expansions, isolated singulari Tatylor’s expansion and properties of zeros, entire, functions and meromorphic functions, convergence in A(D)., Cauchy’s residue theorem. The Course in units 1 and 2 is covered by “Real Analysis” by H.L. Royden, Macmillan Pub. Co. – 3rd Ed. : Ch. 1, Ch. 2, Ch. 3, Ch. 4, Ch.5
The course in units 3 and 4 is covered by the following chapters of “ The Elements of Complex Analysis” J. Duncan & John Wiley & sons, London.
Reference Books :
(1) “Theory of Functions of a Real Variable” – by I.N. Natansen, Fnedrik Pub. Co., 1964.
(2) “Measure Theory” - by P.R. Halmos, East and West Press.
(3) “Introduction to Real Variable Theory” – by S.C. Saxena and S.N. Shah Prentice Hall of India, 1980.
(4)
“Real and Complex Analysis”, Rudin, W., 2nd Edition, Tata McGraw -Hill Publishing Co. Ltd.,
New Delhi, 1974.
(5) “Complex Analysis” Deshpande, J.V. Tata McGraw-Hill Publishing Co. Ltd., New Delhi, 1986
(6) “Complex Analysis Ahlfors, L.V., 3rd Edition, McGraw-Hill Kogakush Ltd., Tokyo, 1979.
(7) “Functions of One Complex Variable” Conway, J.B., 2nd Edition, Narosa Publishing House, New Delhi, 1982.
Unit-1 Revision : Group, subgroup, Lagrange’s theorem, normal subgroup. Homomorphism, Cayley’s theorem, Conjugacy on a group, Cauchy’s theorem, Sylow’s theorem, direct product, finite abelian groups.
Unit-2
Revision : Ring homomorphism, quotient ring, primes and maximal ideals and corresponding quotient rings. Prime and maximal ideals, Imbedding of an integral domain, Euclidean ring, polynomial rings, the Eisenstein Criterion for irreducibility, unique factorization domain.
Unit-3 Revision : Linear spaces, dimensions, dual spaces, linear transformations and matrix representations. Modules, Schurs’s lemma, algebra of linear transformations, canonical forms, characteristic roots, matrices triangular form, nilpotent transformations.
Unit-4 Jordan canonical form, rational form, trace and transpose, determinants, hermitian, unitary and normal transformations real quadratic forms.
Note : The course is indicated by “Topics in Algebra” by I.N. Herston, John Wiley and Sons Inc., 2nd Edition, Article – 2.7 to 2.14, 3.5 to 3.11, 4.5 and the articles of Chapter 6.
Reference Books :
(1) “ Basic Abstract Algebra” by Bhattacharya, Jain and Nagpal, 2nd Edition.
(2) “ Algebra” by S.Mcclane and G.Birkhoff, 2nd Edition,
(3) “ Basic Algebra” by N.Jacbson,Hind.Pub.Corp.1984.
(4)
“ A first course in Abstract Algebra” by John Fraleigh (3rd Edition), Narossa Publishing House, New Delhi.
Unit-1
Topological Spaces and Continuous Functions: Topological spaces, basis for a topology, the product space IIXi (
for finitely many topological spaces Xi ), subspace, order topology, closed sets, limit points, continuous functions, the
metric topology.
Unit-2 Connectedness and Compactness: connected spaces, connected sets in the real line, components and path-components, locally connected spaces, compact spaces, compact sets in the real line, limit-point compactness, locally compact spaces, one-point compactification.
Unit-3 Countability and Separation Axioms: First countable space, second countable space, separable space, Lindeloff space, Hausdorff space, regular space, completely regular space, normal space, Urysohn’s lemma, and partition of unity.
Unit-4
Complete Metric Spaces: Complete metric space, compactness in metric spaces, point-wise and compact convergence, the compact open topology, Ascoli’s theorem, Bair spaces.
Note:
The course is covered by “Toplogy – a first course” – by J.R.Munkres, 2nd Edition Prentice – Hall of India , 2002.
All results and examples are to be excluded which use the concept of the product topology of a collection of
infinitely many topological spaces.
Reference Books
(1) “ General Topology” – by S. Willard, Addison Wesley, 1970.
(2) “Topology” – by J. Dugundji, Prentice – Hall of India, 1975.
(3) “Aspects of Topology” – by C.O. Christonson and W.I. Voxman, Marcel Dekker Inc., 1977.
(4)
“General Topology” – by J.L. Kelley, D. Van Nostraml, 1950.
An Overview of programming. Programming Language. Classification. C Essentials-Program
Development. Function. Anatomy of a C Functions. Variables and Constants. Expressions Assignment Statements. Formatting Source Files. Continuation Character. The Preprocessor.
Scalar Data Types-Declarations. Different Types of Integers. Different kinds of Integer Constants.
Floating-Point Types. Initialization. Mixing Types. Explicit Conversions-Casts. Enumeration Types. The Void Data Type. Typedefs. Finding the Address of an object. Pointers.
Control Flow-Conditional Branching. The Switch Statement. Looping. Nested Loops. The break and continue Statements. The goto statement. Infinite Loops.
Operators and Expressions-Precedence and Associability. Unary Plus and Minus operators. Binary Arithmetic Operators. Arithmetic Assignment Operators. Increment and Decrement Operators. Comma Operator. Relational Operators. Logical Operators. Bit-Manipulation Operators.
Bit wise Assignment Operators. Cast Operator. Size of Operator. Conditional Operator (?:). Memory Operators.
Arrays and Pointer-Declaring an Array. Arrays and Memory. Initializing Arrays. Encryption and Decryption. Pointer Arithmetic. Passing Pointers as Function Arguments. Accessing Array Elements through Pointers. Passing Arrays as Function Arguments. Sorting Algorithms. Strings. Multidimensional Arrays. Arrays of Pointers. Pointers to Pointers.
Storage Classes-Fixed vs. Automatic Duration. Scope. Global variables. The register specifier. ANSI rules for the syntax and Semantics of the storage-class keywords. Dynamic Memory Allocation.
Structures and Unions-Structures. Linked Lists. Unions. enum Declarations.
Functions-Passing Arguments. Declarations and Calls. Pointers to Functions. Recursion. The main ( ) Function. Complex Declarations.
The C Preprocessor-Macro Substitution. Conditional Compilation. Include Facility. Line Control.
Input and Output-Streams. Buffering. The <Stdio.h> Header File. Error Handling. Opening and Closing a File. Reading and Writing Data. Selecting an I/O. Method. Unbuffered I/O Random Access. The standard Library for Input/Output.
Recommended Text
1. Peter A. Darnell and Philip E. Margolis, C: A software Engineering Approach, Narosa Publishing House (Springer International Student Edition), 1993.
Reference
1. Samuel P. Harkison and Gly L. Steele Jr., C: A Reference Manual, 2nd Edition, Prentice Hall, 1984.
2. Brain W. Kernighan & Dennis M. Ritchie, The C Programme Language, 2nd Edition, (ANSI features) Prentice
Hall, 1989.
(a) Computer Architecture
Internal structure and Organization of computer. Von-Neumann model and its limitations. Taxonomy of
architectures- Flynn’s classification. Hwang-Brigg’s modification. Handler’s classification.
Uniprocessors-Register machine. Stack machine. Language directed architecture.
Pipelining. Pipelined Arithmetic units. Example systems. RISC and CISC controversy. Overview of 286, 386, 486 and 586 processors.
Vector Processors-Vector and array processing. Vectorization. Vector code optimization. Example systems like STAR-100, CYBER-205.
Multiprocessors-Functional Structures. Interconnection networks. Parallel memory organization. System. System deadlock and Protection. Multiprocessor Scheduling Algorithms. Example System. Parallel architectures.
(b) Data Structures Using C
Introduction to the concepts of an abstract Data Structure and an implementation.
Stacks and their C implementation. Infix, Postfix, and Prefix notations.
Recursion, its Applications, and its implementation. Queues, Priority queues and linked lists and their
implementation using an array of available nodes as well as dynamic storage.
Trees-Binary trees. Binary tree representations. The Huffman algorithm. Representing lists as binary trees. Trees and their applications.
Sorting-General background. Exchange Sorts. Selection and tree sorting. Insertion sorts . Merge and Radix sorts.
Internal and external searching.
Graphs-A flow problem. Graph traversal and spanning forests.
References :
1. H.S. Stone, Introduction to Computer Architecture, Galgotia.
2. J.P. Hayes, Computer Architecture and Organization, McGraw-Hill.
3. K. Hwang & F.A. Briggs, Computer Architecture & Parallel Processing, McGraw-Hill.
4. P.M. Kogge, The Architecture of Pipelined Computers, McGraw-Hill.
5. J.L. Hennessy & D.A. Patterson, Computer Architecture : A Quantitative Approach, Morgan Kauffmann.
6. J.G. Mayers, Advances in Computer Architecture, John Wiley.
7. Aaron M. Tenenbaum, Yedidyah Langsam & Moshe J. Augenstein, Data Structures using C, Prentice-Hall of
India Pvt. Ltd. New Delhi, 1994.
Part – II (Duration: One Year)
MIM 201 Any two of the following :
(a) Algebra – II Field Theory
Unit-1 Extensions of field, the transcendence of e, roots of polynomials construction with straightedge and compass.
Unit-2 The fixed field of a group of automophisms, the theorem on symmetric polynomials, the Galois group of a polynomial, the fundamental theorem of Galois theory, solvability by radicals Galois group over the rationals, finite fields, Wedderburn’s theorem on finite division rings. Note: The topics from Algebra of units 1 and 2 are roughly covered by chapter 5 ( all articles ) and chapter 7 ( 7.1 and 7.2 only ) of the book entitled “ Topics in Algebra ” by I. N. Herstein 2nd Edition. Wiley Eastern Ltd., 1975.
(b)
Topology – II Advance Topics
Unit-1 The quotient topology, the product and box topologies, the uniform topology, the metrezability and an infinite
product of R, the product topology and Husdorff, regular, normal spaces, Tietz extension theorem, Uryshon’s metrization theorem, Stone-Cech compactification, topological dimension : Lebesgue’s covering dimension Rm and compact M- Manifold.
Unit-2
Homotopy of paths, the fundamental group, covering spaces the fundamental groups of the circle, the punctured plane, the n-sphere Sn and some surfaces, the fundamental theorem of algebra.
Note: The topics from Topology of units 1 and 2 are roughly covered by chapters 2 ( articles 2.8, 2.9, 2.10, 2.11 only ), some theorems of chapters 3 and 4, chapter 5 (5.1, 5.2, 5.3), Chapter 7 (7.9 only) and Chapter 8 (8.1 to 8.9 only) of James R. Munke’s: “ Topology, A first course ”, 2nd Prentice – Hall of India Pvt. Ltd., New Delhi. 2002.
(c)
Discrete Mathematical Structures - 1
Formal Logic-Statements. Symbolic Representation and Tautologies. Quantifiers, Predicates and Validity. Propositional
Logic.
Semigroups & Monoids, Relations and Ordering, Transitive Closure of a relation, Functions. Definitions and Examples of Semigroups and Monoids (Including those pertaining to concatenation operation). Homomorphism of Semigroups and
Monoides. Congruence relation. Quotient Semigroups. Subsemigroup and submonoids. Direct products. Basic
homomorphism Theorem.
Lattices-Lattices as partially ordered sets. Their properties. Lacttices as Algebraic systems. Sublattices, Direct products, and Hommorphisms. Some Special Lattices e.g., Complete Complemented and Distributive Lattices.
Boolean Algebras-Boolean Algebras as Lattices. Various Boolean Identities. The Switching Algebra example. Subalgebras, Direct Products and Homomorphisms. Join-irreducible elements, Atoms and Minterms. Boolean Forms and Their Equivalence. Minterm Boolean Forms, Sum of Products Canonical Forms. Minimization of Boolean Functions. Applications of Boolean Algebra to Switching Theory (using AND, OR & NOT gates). The Karnaugh Map method.
Graph Theory-Definition of (undirected) Graphs, Paths, Circuits, Cycles, & Subgraphs. Induced Subgraphs. Degree of a vertex. Connectivity. Planar Graphs and their properties. Trees. Euler’s Formula for connected Planar Graphs.
Complete & Complete Bipartite Graphs. Kuratowski’s Theorem and its use. Spanning Trees, Cut-sets, Fundamental Cut-sets, and Cycles. Minimal Spanning Trees and Kruskal’s Alogrithm. Matrix Representations of Graphs. Euler’s
Theorem on the Existence of Eulerian Paths and Circuits. Directed Graphs. Indegree and Outdegree of a Vertex.
Weighted undirected Graphs. Dijkstra’s Algorithm. Strong Connectivity & Warshall’s Algorithm. Directed Trees. Search Trees. Tree Traversals.
(d)
Discrete Mathematical Structures - 2
Introductory Computability Theory-Finite State Machines and their transition Table Diagrams. Equivalence of Finite State Machines. Reduced Machines. Homomorphism. Finite Automata. Acceptors. Non-deterministic Finite Automata and equivalence of its power to that of Deterministic Finite Automata. Moore and Mealy Machines.
Turing Machine and Partial Recursive Functions.
Grammars and Languages-Structure Grammars. Rewriting Rulers. Derivations. Sentential Forms. Language generated by a Grammar. Regular, Context-Free, and Context Sensitive Grammars and Languages. Regular sets, Regular Expressions and the Pumping Lemma. Kleene’s Theorem.
Notions of Syntax Analysis. Polish Notations. Conversion of Infix Expressions to Polish Notation.
References
J.P.Tremblay & R.Manohar, Discrete Mathematical Structures with Applications to Computer Science, McGraw-Hill Book Co.,1999.
J.L. Gersting, Mathematical Structures for Computer Science, (3rd edition), Computer Science Press, New York. Seymour Lepschutz, Finite Mathematics (International edition 1983), McGraw-Hill Book Company, New York.
S. Witala, Discrete Mathematics – A Unified Approach, McGraw-Hill Book Co. J.E.Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages & Computation, Narosa Publishing House.
C. L. Liu, Elements of Discrete Mathematics, McGraw-Hill Book Co.
N. Deo, Graph Theory with Applications to Engineering and Computer Sciences, Prentice Hall of India.