AMG-3 Real Analysis and Lebesgue Integration
MG-3 Group Theory and Ring Theory
MS-3 Set Theory, Logic and Metric Spaces
MS-4 Ordinary and Partial Di erential equations
FMG-3 C-programming
1
AMG-3: Real Analysis and Lebesgue Integration
First Term: Real Analysis
1. Sequences of real numbers : De nition of sequence and subsequence, Limit of
a sequence, convergent sequences, Limit superior and Limit inferior, Cauchy
sequences. [10 Lectures]
2. Series of Real numbers : Convergence and divergence of series of real numbers,
alternating series, Conditional and absolute convergence of series, test
of absolute convergence (Ratio test and Root test), series whose terms form a
non-increasing sequence. [10 Lectures]
3. Riemann integral : Sets of measure zero, De nition and existence of a Riemann
integral, properties of Riemann integral, Fundamental theorem of integral calculus,
Mean value theorems of integral calculus. [14 Lectures]
4. Sequence and series of functions : Pointwise and uniform convergence, sequence
of functions, consequences of uniform convergence, convergence and uniform
convergence of series of functions, integration and di erentiation of series of
functions. [14 Lectures]
Text Books:
1. R.R. Goldberg - Methods of Real Analysis (Oxford and IBH Publications
(1970)).
Ch. 2 Art. 2.1, 2.9, 2.10.
Ch. 3 to 3.3, 3.4A, 3.4B, 3.6F, 3.6G, 3.7.
Ch. 7 Art. 7.1 to 7.4, 7.8 to 7.10.
Ch. 9 Art. 9.1 to 9.5
Reference Books:
1. D. Somasundaram, B. Choudhary - A rst course in Mathematical Analysis,
Narosa Publishing House, 1997.
2. Robert, G. Bartle, Donald Sherbert - Introduction to real analysis, Third edition,
John Wiley and Sons.
3. Shantinarayan and Mittal - A course of Mathematical Analysis, Revised edition,
S. Chand and Co.(2002).
4. S.C. Malik and Savita Arora - Mathematical Analysis , New Age International
Publications,Third Edition,(2008).
2
Second term: Lebesgue Integration
1. Measurable Sets [12 Lectures]
(i) Length of open sets and closed sets.
(ii) Inner and outer measure.
(iii) Measurable sets.
(iv) Properties of measurable sets.
2. Measurable Functions [12 Lectures]
3. The Lebesgue integrals [16 Lectures]
(i) De nition and example of the Lebesgue integrals for bounded functions.
(ii) Properties of Lebesgue integrals for bounded measurable functions.
(iii) The Lebesgue integral for unbounded functions.
(iv) Some fundamental theorems.
4. Fourier Series [8 Lectures]
(i) De nition and examples of Fourier Series.
(ii) Formulation of convergence problems.
Text-Book:
Richard R. Goldberg, Methods of Real Analysis, Oxford and IBH Publishing Co.
Pvt. Ltd. (1970).
(Chapter No. 11, 11.1 to 11.8, 12.1, 12.2. Theorem No. 11.1B and 11.1C, 11.8D
Statements only).
Reference Books:
1. Tom Apostol, Advanced Calculus, 2nd Edition, Prentice Hall of India, (1994).
2. D. Somasundaram and B. Choudhari, A rst course in Mathematical Analysis,
Narosa Publishing House, (1997).
3. R.G. Bartle and D.R. Scherbert, Introduction to real analysis 2nd Edition,
John Wiley, (1992).
4. Inder K. Rana, Measure and Integration
3
MG-3: Group Theory and Ring Theory
First term: Group Theory
Groups [12 Lectures]
1. Groups : de nition and examples.
2. Abelian group, nite group, in nite group.
3. Properties of groups.
4. Order of an element - de nition, examples, properties.
5. Examples of groups including Z;Q;R;C; Klein 4-group, Group of quaternions,
S1(= the unit circle in C);GLn(R); SLn(R);On(=the group of n n real orthogonal
matrices), Bn(= the group of n n nonsingular upper triangular
matrices), and groups of symmetries of plane gures such as D4 and S3.
Subgroups [10 Lectures]
1. Subgroups : de nition, necessary and su cient conditions, examples on nding
subgroups of nite groups, union and intersection of subgroups.
2. Subgroup generated by a subset of the group.
3. Cyclic groups : de nition, examples of cyclic groups such as Z and the group
n of the n-th roots of unity, properties :
(a) Every cyclic group is abelian.
(b) If G = (a); then G = (aô€€€1):
(c) Every subgroup of a cyclic group is cyclic.
(d) Let G be a cyclic group of order n. Let G = (a): The element as 2 G
generates a cyclic group of order n
gcd(n; s) :
(e) Let G = (a) and o(G) = n: Then (am) = G if and only if (m; n) = 1:
4. Cosets : de nition and properties.
5. Lagrange's theorem and corollaries.
Permutation Groups [6 Lectures]
1. De nition of Sn and detail discussion of the group S3:
2. Cycles and transpositions, even and odd permutations.
3. Order of permutation.
4
4. Properties : (i) o(Sn) = n! (ii) An is a subgroup of Sn:
5. Discussion of the group A4 including converse of Lagrange's theorem does not
hold in A4:
Normal Subgroups [8 Lectures]
1. De nition.
2. Properties with examples:
(a) If G is an abelian group, then every subgroup of G is a normal subgroup.
(b) N is a normal subgroup of G if and only if gNgô€€€1 = N for every g 2 G:
(c) The subgroup N of G is a normal subgroup of G if and only if every left
coset of M in G is a right coset of N in G.
(d) A subgroup N of G is a normal subgroup of G if and only if the product
of two right cosets of N in G is again a right coset of N in G,.
(e) If H is a subgroup of index 2 in G then H is a normal subgroup of G.
(f) If H is the only subgroup of G of a xed nite order then H is a normal
subgroup of G.
3. Quotient groups and examples.
Homomorphism and Isomorphism [12 Lectures]
1. Homomorphism.
2. Isomorphism : de nition, examples, establish isomorphism of two nite groups.
3. Fundamental Theorem of homomorphisms of groups.
4. The group Z=nZ of residue classes (mod n). Characterization of cyclic groups
(as being isomorphic to Z or Z=nZ for some n 2 N):
5. Cayley's Theorem for nite groups.
6. Classi cation of groups of order 5:
7. Cauchy's theorem for Abelian Groups.
Text book:
I.N. Herstein, Topics in Algebra, Wiley, 1990.
Reference Books :
1. M. Artin, Algebra, Prentice Hall of India, New Delhi, 1994.
5
2. P.B. Bhattacharya, S.K. Jain and S.R. Nagpal, Basic Abstract Algebra, Second
Ed., Foundation Books, New Delhi, 1995.
3. J.B. Fraleigh, A. First Course in Abstract Algebra, Third Ed., Narosa, New
Delhi, 1990.
4. N.S. Gopalakrishnan, University Algebra, Second Ed., New Age International,
New Delhi, 1986.
5. D.A.R. Wallace, Groups, Rings and Fields, Springer-Verlag, London, 1998.
6. I.N. Herstein, Abstract Algebra.
7. I. H. Sheth, Abstract Algebra, Second Revised Edition, 2009, PHL,India.
Second term: Ring Theory
1. De nition and properties of Ring, Subring. [5 Lectures]
2. Integral Domains: Zero divisors, Cancellation Law, Field, Characteristics of
Ring. [5 Lectures]
3. Ideals and Factor Rings: Existence of Factor Ring, Prime Ideals, Maximal
Ideals. : [6 Lectures]
4. Homomorphism of Rings: Properties of Ring Homomorphism, Kernel, First
isomorphism Theorem for Ring, Prime Fields. The eld of Quotients. [8 Lec-
tures]
5. Polynomial Ring: De nition. The division Algorithm, Principle Ideal Domain.
[6 Lectures]
6. Factorization of Polynomial: Reducibility and Irreducibility Tests, Eisenstein
criterion. Ideals in F[x]: Unique Factorization in Z[x]. [8 Lectures]
7. Divisibility in Integral Domain: Associates, Irreducible and Primes, Unique
Factorization Domains, Ascending chain Condition for PID, PID implies UFD,
Euclidean Domains. ED Implies PID,
D is UFD implies D[x] is UFD. [10 Lectures]
Text Book:
Joseph, A. Gallian, Contemporary Abstract Algebra,(4th Edition), Narosa Publishing
House.
Chapter Numbers : 12,13,14,15,16,17 and 18.
Reference Books:
1. J.B. Fraleigh, First course in Abstract Algebra (4rd Edition). Narosa Publishing
House.
6
2. I.N. Herstein. Abstract Algebra, (3rd Edition), Prentitice Hall of India, 1996.
3. N.S. Gopalkrishnan, University of Algebra, Wiley Eastern 1986.
4. C. Musili, Rings and Modules, Narosa Publishing House, 1992.
7
MS-3: Set Theory, Logic and Metric Spaces
First term: Set Theory and Logic
Sets and Relations : Cantor's concept of a set, Intuitive set theory, Inclusion,
Operations for sets, Algebra of sets, Equivalence relations, Functions, Composition
and Inversion of Functions, Operations for collections of sets, Ordering relations,
Power sets, Numerical Equivalence of sets. [8 Lectures]
Natural Number sequence :
Induction and Recursion, Cardinal numbers and Cardinality, Cardinal arithmetic,
Countable and Uncountable sets, Schroeder-Bernstein Theorem (without proof),
Paradoxes of Intuitive set theory, Russell's Paradox. [12 Lectures]
Logic :
Statement calculus (Sentential connectivities, Truth tables, Validity, Consequence,
Applications), Predicate Calculus (Symbolizing every day language, Formulation,
Validity, Consequence). [4 Lectures]
Basic Logic :
(Revision) Introduction, proposition, truth table, negation, conjunction and disjunction,
Implications, biconditional propositions, converse, contra positive and inverse
propositions and precedence of logical operators. [6 Lectures]
Propositional equivalence :
Logical equivalences, Predicates and quanti ers : Introduction, Quanti ers, Binding
variables and Negations. [6 Lectures]
Methods of Proof:
Rules of inference, valid arguments, methods of proving theorems; direct proof, proof
by contradiction, proof by cases, proofs by equivalence, existence proofs, Uniqueness
proofs and counter examples. [12 Lectures]
Text Books:
1. Set Theory and Logic, Robert R. Stoll, Errasia publishers, New Delhi. Sections
1.1 to 1.10, 2.3, 2.4, 2.5
2. Discrete Mathematics and its Applications, K.H. Rosen, Tata McGraw, New
Delhi. Chapter 4
Reference Books :
1. Symbolic Logic, I.M. Copi, Fifth Edition, Prentice Hall of India, 1995.
2. Naive Set Theory, P.R. Halmos, 1974.
8
Second Term: Metric Spaces
1. Chapter 1 : Basic Notions. [8 Lectures]
2. Chapter 2: Convergence. [8 Lectures]
3. Chapter 3 : Continuity. [8 Lectures]
4. Chapter 4 : Compactness. [10 Lectures]
5. Chapter 5 : Connectedness. [6 Lectures]
6. Chapter 6 : Complete Metric Spaces. [8 Lectures]
Text Book:
Topology of Metric Spaces by S. Kumaresan, Narosa Publishing House, 2005.
Sections : 1.1, 1.2 (except the Sections 1.2.51 to 1.2.65), 2.1, 2.2, 2.3, 2.4, 2.5 and
2.7, 3.1, 3.2 (up to 3.2.32 only), 3.3, 3.4,3.5.(Uniform Continuity to be dropped), 4.1,
4.2, (Proposition 4.2.13 without proof) and 4.3 (Theorem 4.3.24 without proof), 5.1
and 6.1 (Theorems 6.1.1, 6.1.3, 6.1.11, without proofs).
Note: All the problems which are based on normed linear spaces and matrices be
dropped.
Reference books :
1. Real Analysis, Carothers, Cambridge University Press, 2000.
2. Methods of Real Analysis, R.R. Goldberg, Oxford and IBH Publishing Company.
3. Metric Spaces, E.T. Copson, University Press, Cambridge, 2nd edition, Mumbai,
1978.
4. Introduction to Topology and Modern Analysis, G.F. Simmons. McGraw Hill
International Book Company, International Student Edition.
9
MS-4: Ordinary and Partial Di erential equations
First term: Ordinary Di erential Equations
1. What is a Di erential Equation?: [14 Lectures]
Introductory Remarks, the nature of solutions, separable equations, rst-order
linear equations, exact equations, orthogonal trajectories and families of curves,
homogeneous equations, integrating factors,
reduction of order:(1) dependent variable missing, (2) independent variable
missing, electrical circuits.
2. Second-Order Linear Equations: [12 Lectures]
Second-order linear equations with constant coe cients, the method of undetermined
coe cients, the method of variation of parameters, the use of a
known solution to nd another, vibrations and oscillations : (1) undamped
simple harmonic motion (2) damped vibrations (3) forced vibrations.
3. Power Series Solutions and Special Functions: [12 Lectures]
Introduction and review of power series, series solutions of rst-order di erential
equations, second-order linear equations, ordinary points, regular singular
points, more on regular singular points.
4. System of First-Order Equations: [10 Lectures]
Introductory remarks, linear systems, homogeneous linear systems with constant
coe cients.
Text Book: Di erential Equations by George F. Simmons, Steven G. Krantz, Tata
McGraw-Hill.
Reference Book:
1. W.R. Derrick and S.I. Grossman, A First Course in Di erential Equations with
Applications. CBS Publishers and distributors, Delhi-110 032. Third Edition.
2. Rainville, Bedient: Di erential Equations
10
Second Term: Partial Di erential Equations
1. Ordinary Di erential Equations in More Than Two Variables
(a) Surface and Curves in Three Dimensions [20 Lectures]
(b) Simultaneous Di erential Equations of the First Order and the First Degree
in Three Variables.
(c) Methods of solution of dx
P = dy
Q = dz
R .
(d) Orhogonal Trajectories of a System of curves on a Surface.
(e) Pfa an Di erential Forms and Equations.
(f) Solution of Pfa an Di erential Equations in Three Variables.
First Order Partial Di erential Equations : [28 Lectures]
(a) Curves and surfaces.
(b) Genesis of First Order Partial Di erential Equations.
(c) Classi cation of Integrals.
(d) Linear Equations of the First Order.
(e) Pfa an Di erential Equations.
(f) Compatible Systems.
(g) Charpit's Method.
(h) Jacobi's Method.
(i) Integral Surfaces through a given curve.
(j) Quasi-Linear Equations.
Text Book:
1. Ian Sneddon, Element of Partial Di erential Equations, McGraw-Hill Book
Company, McGraw-Hill Book Company. Chapter 1 x1 to x6.
2. T. Amaranath, An Elementary Course in Partial Di erential Equations, Narosa
Publishing, House 2nd Edition, 2003 (Reprint, 2006). Chapter 1 x1 to x10.
Reference Book:
1. Frank Ayres Jr., Di erential Equations, McGraw-Hill Book Company, SI Edition
(International Edition, 1972)
2. Ravi P. Agarwal and Donal O'Regan, Ordinary and Partial Di erential Equations,
Springer, First Edition (2009).
3. W.E. Williams, Partial Di erential Equations, Clarendon Press, Oxford,(1980).
11
FMG 3: C Programming
First Term
1. Introductory Concepts: Introduction to computer. Computer Characteristics.
Types of Programming Languages. Introduction to C. [2 Lectures]
2. C Fundamentals: The character set. Identi er and keywords. Data types.
Constants. Variables and arrays. Declarations. Expressions. Statements. Symbolic
constants. [4 Lectures]
3. Operators and Expressions: Arithmetic operators. Unary operators. Relational
and Logical operators. Assignment operators. Conditional Operator. Library
functions. [6 Lectures]
4. Data Input and Outputs: Preliminaries. Single character input-getchar()
function. Single character output-putchar() function. Writing output data-printf
function. Formatted input-output. Get and put functions. [8 Lectures]
5. Preparing and Running a Program: Planning and writing a C Program.
Compiling and Executing the Program. [2 Lectures]
6. Control Statements: Preliminaries. The while statement. The do-while statement.
The for statement. Nested loops. The if-else statement. The switch statement.
The break statement. The continue statement. The comma operator. [8 Lectures]
7. Functions: A brief overview. De ning a function. Accessing a function. Passing
arguments to a function. Specifying argument data types. Function prototypes.
Recursion. [8 Lectures]
8. Arrays: De ning an array. Processing an array. Passing arrays to a function.
Multidimensional arrays. Arrays and strings. [10 Lectures]
Text Book: Programming with C. By Byron S. Gottfried. Schaum's Outline series.
Chapters:1,2,3,4,5,6,7,9.
Reference Book: The C Programming Language. By Brian W. Kernighan, Dennis
M. Ritchie.
Second term: C programming
1. Program Structures: Storage classes. Automatic variables. External variables.
Static variables. [4 Lectures]
2. Pointers: Fundamentals. Pointer declarations. Passing pointer to a function.
Pointer and one dimensional arrays. Dynamic memory allocation. Operations on
pointers. Pointers and multidimensional arrays. Array of pointers. Pointer to function.
Passing functions to other functions. More about pointer declarations.
. [12 Lectures]
3. Structures and Unions: De ning a structure. Processing a structure. Userde
ned data types (typedef). Structures and pointers. Passing structure to a function.
Self-referential structures. Unions. [12 Lectures]
4. Data Files: Opening and closing a data le. Creating a data le. Processing a
data le. Unformatted data les. [10 Lectures]
12
5. Low-Level Programming: Bitwise operators. Register variables. Enumerations.
Macros. Command line arguments. The C processor. [10 Lectures]
Book: Programming with C. By Byron S. Gottfried. Schaum's Outline series. Chapters:
8,10,11,12,13,14.
Reference Book: The C Programming Language. By Brian W. Kernighan, Dennis
M. Ritchie.
13
MG-3 Group Theory and Ring Theory
MS-3 Set Theory, Logic and Metric Spaces
MS-4 Ordinary and Partial Di erential equations
FMG-3 C-programming
1
AMG-3: Real Analysis and Lebesgue Integration
First Term: Real Analysis
1. Sequences of real numbers : De nition of sequence and subsequence, Limit of
a sequence, convergent sequences, Limit superior and Limit inferior, Cauchy
sequences. [10 Lectures]
2. Series of Real numbers : Convergence and divergence of series of real numbers,
alternating series, Conditional and absolute convergence of series, test
of absolute convergence (Ratio test and Root test), series whose terms form a
non-increasing sequence. [10 Lectures]
3. Riemann integral : Sets of measure zero, De nition and existence of a Riemann
integral, properties of Riemann integral, Fundamental theorem of integral calculus,
Mean value theorems of integral calculus. [14 Lectures]
4. Sequence and series of functions : Pointwise and uniform convergence, sequence
of functions, consequences of uniform convergence, convergence and uniform
convergence of series of functions, integration and di erentiation of series of
functions. [14 Lectures]
Text Books:
1. R.R. Goldberg - Methods of Real Analysis (Oxford and IBH Publications
(1970)).
Ch. 2 Art. 2.1, 2.9, 2.10.
Ch. 3 to 3.3, 3.4A, 3.4B, 3.6F, 3.6G, 3.7.
Ch. 7 Art. 7.1 to 7.4, 7.8 to 7.10.
Ch. 9 Art. 9.1 to 9.5
Reference Books:
1. D. Somasundaram, B. Choudhary - A rst course in Mathematical Analysis,
Narosa Publishing House, 1997.
2. Robert, G. Bartle, Donald Sherbert - Introduction to real analysis, Third edition,
John Wiley and Sons.
3. Shantinarayan and Mittal - A course of Mathematical Analysis, Revised edition,
S. Chand and Co.(2002).
4. S.C. Malik and Savita Arora - Mathematical Analysis , New Age International
Publications,Third Edition,(2008).
2
Second term: Lebesgue Integration
1. Measurable Sets [12 Lectures]
(i) Length of open sets and closed sets.
(ii) Inner and outer measure.
(iii) Measurable sets.
(iv) Properties of measurable sets.
2. Measurable Functions [12 Lectures]
3. The Lebesgue integrals [16 Lectures]
(i) De nition and example of the Lebesgue integrals for bounded functions.
(ii) Properties of Lebesgue integrals for bounded measurable functions.
(iii) The Lebesgue integral for unbounded functions.
(iv) Some fundamental theorems.
4. Fourier Series [8 Lectures]
(i) De nition and examples of Fourier Series.
(ii) Formulation of convergence problems.
Text-Book:
Richard R. Goldberg, Methods of Real Analysis, Oxford and IBH Publishing Co.
Pvt. Ltd. (1970).
(Chapter No. 11, 11.1 to 11.8, 12.1, 12.2. Theorem No. 11.1B and 11.1C, 11.8D
Statements only).
Reference Books:
1. Tom Apostol, Advanced Calculus, 2nd Edition, Prentice Hall of India, (1994).
2. D. Somasundaram and B. Choudhari, A rst course in Mathematical Analysis,
Narosa Publishing House, (1997).
3. R.G. Bartle and D.R. Scherbert, Introduction to real analysis 2nd Edition,
John Wiley, (1992).
4. Inder K. Rana, Measure and Integration
3
MG-3: Group Theory and Ring Theory
First term: Group Theory
Groups [12 Lectures]
1. Groups : de nition and examples.
2. Abelian group, nite group, in nite group.
3. Properties of groups.
4. Order of an element - de nition, examples, properties.
5. Examples of groups including Z;Q;R;C; Klein 4-group, Group of quaternions,
S1(= the unit circle in C);GLn(R); SLn(R);On(=the group of n n real orthogonal
matrices), Bn(= the group of n n nonsingular upper triangular
matrices), and groups of symmetries of plane gures such as D4 and S3.
Subgroups [10 Lectures]
1. Subgroups : de nition, necessary and su cient conditions, examples on nding
subgroups of nite groups, union and intersection of subgroups.
2. Subgroup generated by a subset of the group.
3. Cyclic groups : de nition, examples of cyclic groups such as Z and the group
n of the n-th roots of unity, properties :
(a) Every cyclic group is abelian.
(b) If G = (a); then G = (aô€€€1):
(c) Every subgroup of a cyclic group is cyclic.
(d) Let G be a cyclic group of order n. Let G = (a): The element as 2 G
generates a cyclic group of order n
gcd(n; s) :
(e) Let G = (a) and o(G) = n: Then (am) = G if and only if (m; n) = 1:
4. Cosets : de nition and properties.
5. Lagrange's theorem and corollaries.
Permutation Groups [6 Lectures]
1. De nition of Sn and detail discussion of the group S3:
2. Cycles and transpositions, even and odd permutations.
3. Order of permutation.
4
4. Properties : (i) o(Sn) = n! (ii) An is a subgroup of Sn:
5. Discussion of the group A4 including converse of Lagrange's theorem does not
hold in A4:
Normal Subgroups [8 Lectures]
1. De nition.
2. Properties with examples:
(a) If G is an abelian group, then every subgroup of G is a normal subgroup.
(b) N is a normal subgroup of G if and only if gNgô€€€1 = N for every g 2 G:
(c) The subgroup N of G is a normal subgroup of G if and only if every left
coset of M in G is a right coset of N in G.
(d) A subgroup N of G is a normal subgroup of G if and only if the product
of two right cosets of N in G is again a right coset of N in G,.
(e) If H is a subgroup of index 2 in G then H is a normal subgroup of G.
(f) If H is the only subgroup of G of a xed nite order then H is a normal
subgroup of G.
3. Quotient groups and examples.
Homomorphism and Isomorphism [12 Lectures]
1. Homomorphism.
2. Isomorphism : de nition, examples, establish isomorphism of two nite groups.
3. Fundamental Theorem of homomorphisms of groups.
4. The group Z=nZ of residue classes (mod n). Characterization of cyclic groups
(as being isomorphic to Z or Z=nZ for some n 2 N):
5. Cayley's Theorem for nite groups.
6. Classi cation of groups of order 5:
7. Cauchy's theorem for Abelian Groups.
Text book:
I.N. Herstein, Topics in Algebra, Wiley, 1990.
Reference Books :
1. M. Artin, Algebra, Prentice Hall of India, New Delhi, 1994.
5
2. P.B. Bhattacharya, S.K. Jain and S.R. Nagpal, Basic Abstract Algebra, Second
Ed., Foundation Books, New Delhi, 1995.
3. J.B. Fraleigh, A. First Course in Abstract Algebra, Third Ed., Narosa, New
Delhi, 1990.
4. N.S. Gopalakrishnan, University Algebra, Second Ed., New Age International,
New Delhi, 1986.
5. D.A.R. Wallace, Groups, Rings and Fields, Springer-Verlag, London, 1998.
6. I.N. Herstein, Abstract Algebra.
7. I. H. Sheth, Abstract Algebra, Second Revised Edition, 2009, PHL,India.
Second term: Ring Theory
1. De nition and properties of Ring, Subring. [5 Lectures]
2. Integral Domains: Zero divisors, Cancellation Law, Field, Characteristics of
Ring. [5 Lectures]
3. Ideals and Factor Rings: Existence of Factor Ring, Prime Ideals, Maximal
Ideals. : [6 Lectures]
4. Homomorphism of Rings: Properties of Ring Homomorphism, Kernel, First
isomorphism Theorem for Ring, Prime Fields. The eld of Quotients. [8 Lec-
tures]
5. Polynomial Ring: De nition. The division Algorithm, Principle Ideal Domain.
[6 Lectures]
6. Factorization of Polynomial: Reducibility and Irreducibility Tests, Eisenstein
criterion. Ideals in F[x]: Unique Factorization in Z[x]. [8 Lectures]
7. Divisibility in Integral Domain: Associates, Irreducible and Primes, Unique
Factorization Domains, Ascending chain Condition for PID, PID implies UFD,
Euclidean Domains. ED Implies PID,
D is UFD implies D[x] is UFD. [10 Lectures]
Text Book:
Joseph, A. Gallian, Contemporary Abstract Algebra,(4th Edition), Narosa Publishing
House.
Chapter Numbers : 12,13,14,15,16,17 and 18.
Reference Books:
1. J.B. Fraleigh, First course in Abstract Algebra (4rd Edition). Narosa Publishing
House.
6
2. I.N. Herstein. Abstract Algebra, (3rd Edition), Prentitice Hall of India, 1996.
3. N.S. Gopalkrishnan, University of Algebra, Wiley Eastern 1986.
4. C. Musili, Rings and Modules, Narosa Publishing House, 1992.
7
MS-3: Set Theory, Logic and Metric Spaces
First term: Set Theory and Logic
Sets and Relations : Cantor's concept of a set, Intuitive set theory, Inclusion,
Operations for sets, Algebra of sets, Equivalence relations, Functions, Composition
and Inversion of Functions, Operations for collections of sets, Ordering relations,
Power sets, Numerical Equivalence of sets. [8 Lectures]
Natural Number sequence :
Induction and Recursion, Cardinal numbers and Cardinality, Cardinal arithmetic,
Countable and Uncountable sets, Schroeder-Bernstein Theorem (without proof),
Paradoxes of Intuitive set theory, Russell's Paradox. [12 Lectures]
Logic :
Statement calculus (Sentential connectivities, Truth tables, Validity, Consequence,
Applications), Predicate Calculus (Symbolizing every day language, Formulation,
Validity, Consequence). [4 Lectures]
Basic Logic :
(Revision) Introduction, proposition, truth table, negation, conjunction and disjunction,
Implications, biconditional propositions, converse, contra positive and inverse
propositions and precedence of logical operators. [6 Lectures]
Propositional equivalence :
Logical equivalences, Predicates and quanti ers : Introduction, Quanti ers, Binding
variables and Negations. [6 Lectures]
Methods of Proof:
Rules of inference, valid arguments, methods of proving theorems; direct proof, proof
by contradiction, proof by cases, proofs by equivalence, existence proofs, Uniqueness
proofs and counter examples. [12 Lectures]
Text Books:
1. Set Theory and Logic, Robert R. Stoll, Errasia publishers, New Delhi. Sections
1.1 to 1.10, 2.3, 2.4, 2.5
2. Discrete Mathematics and its Applications, K.H. Rosen, Tata McGraw, New
Delhi. Chapter 4
Reference Books :
1. Symbolic Logic, I.M. Copi, Fifth Edition, Prentice Hall of India, 1995.
2. Naive Set Theory, P.R. Halmos, 1974.
8
Second Term: Metric Spaces
1. Chapter 1 : Basic Notions. [8 Lectures]
2. Chapter 2: Convergence. [8 Lectures]
3. Chapter 3 : Continuity. [8 Lectures]
4. Chapter 4 : Compactness. [10 Lectures]
5. Chapter 5 : Connectedness. [6 Lectures]
6. Chapter 6 : Complete Metric Spaces. [8 Lectures]
Text Book:
Topology of Metric Spaces by S. Kumaresan, Narosa Publishing House, 2005.
Sections : 1.1, 1.2 (except the Sections 1.2.51 to 1.2.65), 2.1, 2.2, 2.3, 2.4, 2.5 and
2.7, 3.1, 3.2 (up to 3.2.32 only), 3.3, 3.4,3.5.(Uniform Continuity to be dropped), 4.1,
4.2, (Proposition 4.2.13 without proof) and 4.3 (Theorem 4.3.24 without proof), 5.1
and 6.1 (Theorems 6.1.1, 6.1.3, 6.1.11, without proofs).
Note: All the problems which are based on normed linear spaces and matrices be
dropped.
Reference books :
1. Real Analysis, Carothers, Cambridge University Press, 2000.
2. Methods of Real Analysis, R.R. Goldberg, Oxford and IBH Publishing Company.
3. Metric Spaces, E.T. Copson, University Press, Cambridge, 2nd edition, Mumbai,
1978.
4. Introduction to Topology and Modern Analysis, G.F. Simmons. McGraw Hill
International Book Company, International Student Edition.
9
MS-4: Ordinary and Partial Di erential equations
First term: Ordinary Di erential Equations
1. What is a Di erential Equation?: [14 Lectures]
Introductory Remarks, the nature of solutions, separable equations, rst-order
linear equations, exact equations, orthogonal trajectories and families of curves,
homogeneous equations, integrating factors,
reduction of order:(1) dependent variable missing, (2) independent variable
missing, electrical circuits.
2. Second-Order Linear Equations: [12 Lectures]
Second-order linear equations with constant coe cients, the method of undetermined
coe cients, the method of variation of parameters, the use of a
known solution to nd another, vibrations and oscillations : (1) undamped
simple harmonic motion (2) damped vibrations (3) forced vibrations.
3. Power Series Solutions and Special Functions: [12 Lectures]
Introduction and review of power series, series solutions of rst-order di erential
equations, second-order linear equations, ordinary points, regular singular
points, more on regular singular points.
4. System of First-Order Equations: [10 Lectures]
Introductory remarks, linear systems, homogeneous linear systems with constant
coe cients.
Text Book: Di erential Equations by George F. Simmons, Steven G. Krantz, Tata
McGraw-Hill.
Reference Book:
1. W.R. Derrick and S.I. Grossman, A First Course in Di erential Equations with
Applications. CBS Publishers and distributors, Delhi-110 032. Third Edition.
2. Rainville, Bedient: Di erential Equations
10
Second Term: Partial Di erential Equations
1. Ordinary Di erential Equations in More Than Two Variables
(a) Surface and Curves in Three Dimensions [20 Lectures]
(b) Simultaneous Di erential Equations of the First Order and the First Degree
in Three Variables.
(c) Methods of solution of dx
P = dy
Q = dz
R .
(d) Orhogonal Trajectories of a System of curves on a Surface.
(e) Pfa an Di erential Forms and Equations.
(f) Solution of Pfa an Di erential Equations in Three Variables.
First Order Partial Di erential Equations : [28 Lectures]
(a) Curves and surfaces.
(b) Genesis of First Order Partial Di erential Equations.
(c) Classi cation of Integrals.
(d) Linear Equations of the First Order.
(e) Pfa an Di erential Equations.
(f) Compatible Systems.
(g) Charpit's Method.
(h) Jacobi's Method.
(i) Integral Surfaces through a given curve.
(j) Quasi-Linear Equations.
Text Book:
1. Ian Sneddon, Element of Partial Di erential Equations, McGraw-Hill Book
Company, McGraw-Hill Book Company. Chapter 1 x1 to x6.
2. T. Amaranath, An Elementary Course in Partial Di erential Equations, Narosa
Publishing, House 2nd Edition, 2003 (Reprint, 2006). Chapter 1 x1 to x10.
Reference Book:
1. Frank Ayres Jr., Di erential Equations, McGraw-Hill Book Company, SI Edition
(International Edition, 1972)
2. Ravi P. Agarwal and Donal O'Regan, Ordinary and Partial Di erential Equations,
Springer, First Edition (2009).
3. W.E. Williams, Partial Di erential Equations, Clarendon Press, Oxford,(1980).
11
FMG 3: C Programming
First Term
1. Introductory Concepts: Introduction to computer. Computer Characteristics.
Types of Programming Languages. Introduction to C. [2 Lectures]
2. C Fundamentals: The character set. Identi er and keywords. Data types.
Constants. Variables and arrays. Declarations. Expressions. Statements. Symbolic
constants. [4 Lectures]
3. Operators and Expressions: Arithmetic operators. Unary operators. Relational
and Logical operators. Assignment operators. Conditional Operator. Library
functions. [6 Lectures]
4. Data Input and Outputs: Preliminaries. Single character input-getchar()
function. Single character output-putchar() function. Writing output data-printf
function. Formatted input-output. Get and put functions. [8 Lectures]
5. Preparing and Running a Program: Planning and writing a C Program.
Compiling and Executing the Program. [2 Lectures]
6. Control Statements: Preliminaries. The while statement. The do-while statement.
The for statement. Nested loops. The if-else statement. The switch statement.
The break statement. The continue statement. The comma operator. [8 Lectures]
7. Functions: A brief overview. De ning a function. Accessing a function. Passing
arguments to a function. Specifying argument data types. Function prototypes.
Recursion. [8 Lectures]
8. Arrays: De ning an array. Processing an array. Passing arrays to a function.
Multidimensional arrays. Arrays and strings. [10 Lectures]
Text Book: Programming with C. By Byron S. Gottfried. Schaum's Outline series.
Chapters:1,2,3,4,5,6,7,9.
Reference Book: The C Programming Language. By Brian W. Kernighan, Dennis
M. Ritchie.
Second term: C programming
1. Program Structures: Storage classes. Automatic variables. External variables.
Static variables. [4 Lectures]
2. Pointers: Fundamentals. Pointer declarations. Passing pointer to a function.
Pointer and one dimensional arrays. Dynamic memory allocation. Operations on
pointers. Pointers and multidimensional arrays. Array of pointers. Pointer to function.
Passing functions to other functions. More about pointer declarations.
. [12 Lectures]
3. Structures and Unions: De ning a structure. Processing a structure. Userde
ned data types (typedef). Structures and pointers. Passing structure to a function.
Self-referential structures. Unions. [12 Lectures]
4. Data Files: Opening and closing a data le. Creating a data le. Processing a
data le. Unformatted data les. [10 Lectures]
12
5. Low-Level Programming: Bitwise operators. Register variables. Enumerations.
Macros. Command line arguments. The C processor. [10 Lectures]
Book: Programming with C. By Byron S. Gottfried. Schaum's Outline series. Chapters:
8,10,11,12,13,14.
Reference Book: The C Programming Language. By Brian W. Kernighan, Dennis
M. Ritchie.
13
