Topics to be covered from January to March

Topics to be covered from April to June

CC-3:

Real Analysis

Unit-1:

Real Number System

AM

Intuitive idea of real numbers. Mathematical operations and usual order of real numbers revisited with their properties (closure, commutative, associative, identity, inverse, distributive). Idea of countable sets, uncountable sets and uncountability of R. Concept of bounded and unbounded sets in R. L.U.B. (supremum), G.L.B. (infimum) of a set and their properties. L.U.B. axiom or order completeness axiom. Archimedean

property of R. Density of rational (and Irrational) numbers in R.

Intervals. Neighbourhood of a point. Interior point. Open set. Union, intersection of open sets. Limit point and isolated point of a set. Bolzano-Weirstrass theorem for sets. Existence of limit point of every uncountable set as a consequence of Bolzano-Weirstrass theorem. Derived set. Closed set. Complement of open set and closed set. Union and intersection of closed sets as a consequence. No nonempty proper

subset of R is both open and closed. Dense set in R as a set having non-empty intersection with every open intervals. Q and R-Q are dense in R.

Unit-2: Sequences

SGD

• Real sequence. Bounded sequence. Convergence and non-convergence. Examples. Boundedness of convergent

sequence. Uniqueness of limit. Algebra of limits.

• Relation between the limit point of a set and the limit of a convergent sequence of distinct elements. Monotone sequences and their convergence. Sandwich rule. Nested interval theorem. Limit of some important sequences

, , , with and |l|<1, ,

,

(a > 0). Cauchy’s first and second limit theorems.

• Subsequence. Subsequential limits, lim sup as the L.U.B. and lim inf as the G.L.B of a set containing all the subsequential limits. Alternative definition of limsup and liminf of a sequence using inequality or as lim sup = and lim inf =

[Equivalence between these

definitions is assumed]. A bounded sequence is convergent if and only if lim sup = lim inf . Every

sequence has a monotone subsequence. Bolzano-Weirstrass theorem for sequence. Cauchy’s convergence criterion. Cauchy sequence.

Unit-3: Infinite Series

SG

Infinite series, convergence and non-convergence of infinite series, Cauchy criterion, tests for convergence :

comparison test, limit comparison test, ratio test, Cauchy’s n-th root test.

Kummer’s test and Gauss test

(statements only). Alternating series, Leibniz test. Absolute and conditional convergence.

Graphical Demonstration(Teaching Aid)

SG/AM/SGD

• Plotting of recursive sequences.

• Study the convergence of sequences through plotting.

• Verify Bolzano-Weierstrass theorem through plotting of sequences and hence identify convergent subsequences

from the plot.

• Study the convergence/divergence of infinite series by plotting their sequences of partial sum.

• Cauchy’s root test by plotting n-th roots.

• Ratio test by plotting the ratio of n-th and (n + 1)-th term.

CC-4:

Group Theory-I

Unit-1: Groups and Subgroups

MDG

&

MI

Symmetries of a square, definition of group, examples of groups including permutation groups, dihedral

groups and quaternion groups (through matrices), elementary properties of groups, examples of commutative

and non-commutative groups.

Subgroups and examples of subgroups, necessary and sufficient condition

for a nonempty subset of a group to be a subgroup. Normalizer, centralizer, center of a group, product of

two subgroups.

Unit-2:

Cyclic Groups, Permutation groups, etc.

MDG

Properties of cyclic groups, classification of subgroups of cyclic groups. Cycle notation for permutations, properties of permutations, even and odd permutations, alternating group.

Properties of cosets, order of

an element, order of a group. Lagrange’s theorem and consequences including Fermat’s Little theorem.

Unit-3:

Normal subgroups, Homomorphisms

SG

&

MDG

Normal subgroup and its properties. Quotient group. Group homomorphisms, properties of homomorphisms,

correspondence theorem and one one correspondence between the set of all normal subgroups of a group and the set of all congruences on that group,

.

Cayley’s theorem, properties of isomorphisms. First, Second and Third isomorphism theorems

GE-2

Unit-1: Differential Calculus-II

MDG

• Sequence of real numbers: Definition of bounds of a sequence and monotone sequence. Limit of a sequence. Statements of limit theorems. Concept of convergence and divergence of monotone sequences-applications of the theorems, in particular, definition of e. Statement of Cauchy’s general principle of convergence and

its application.

• Infinite series of constant terms; Convergence and Divergence (definitions). Cauchy’s principle as applied to infinite series (application only). Series of positive terms: Statements of comparison test. D.AIembert’s Ratio test. Cauchy’s nth root test and Raabe’s test Applications. Alternating series. Statement of Leibnitz

test and its applications.

• Real-Valued functions defined on an interval: Statement of Rolle’s Theorem and its geometrical interpretation.

Mean value theorems of Lagrange and Cauchy. Statements of Taylor’s and Maclaurin’s Theorems with Lagrange’s and Cauchy’s from of remainders. Taylor’s and Maclaurin’s Infinite series of functions like , sin x, cos x, , log(1 + x) with restrictions wherever necessary.

• Indeterminate Forms : L’Hospital’s Rule : Statement and Problems only.

• Application of the principle of Maxima and Minima for a function of single variable in geometrical, physical and to other problems.

• Maxima and minima of functions of not more than three variables Lagrange’s Method of undetermined

multiplier - Problems only.

Unit-2:

Differential Equation-II

SGD

Linear homogeneous equations with constant coefficients, Linear non-homogeneous equations, The method

of variation of parameters, The Cauchy-Euler equation, Simultaneous differential equations, Simple eigenvalue problem.

Order and degree of partial differential equations, Concept of linear and non-linear partial differential equations, Formation of first order partial differential equations, Linear partial differential equation of first order, Lagrange’s method, Charpit’s method.

Unit-3:

Vector Algebra

MI

Addition of Vectors, Multiplication of a Vector by a Scalar. Collinear and Coplanar Vectors. Scalar and

Vector products of two and three vectors.

Simple applications to problems of Geometry. Vector equation

of plane and straight line. Volume of Tetrahedron. Applications to problems of Mechanics (Work done and

Moment).

Unit-4:

Discrete

Mathematicsinate

SG

&

AM

• Integers: Principle of Mathematical Induction. Division algorithm. Representation of integer in an arbitrary

base. Prime Integers. Some properties of prime integers. Fundamental theorem of Arithmetic. Euclid’s Theorem. Linear Diophantine equations. [Statement of Principle of Mathematical Induction, Strong form of Mathematical induction. Applications in different problems. Proofs of division algorithm. Representation

of an integer uniquely in an arbitrary base, change of an integer from one base to another base. Computer operations with integers ˆa“ Divisor of an integer, g.c.d. of two positive integers, prime integer,

Proof of Fundamental theorem, Proof of Euclid’s Theorem. To show how to find all prime numbers less than or equal to a given positive integer. Problems related to prime number. Linear Diophantine equation ˆa“ when such an equation has solution, some applications.]

• Congruences: Congruence relation on integers, Basic properties of this relation. Linear congruences, Chinese Remainder Theorem. System of Linear congruences. [Definition of Congruence ˆa“ to show it is an equivalence relation, to prove the following : a º b ( mod m) implies

(i) (a + c) º(b + c) ( mod m)

(ii) ac º bc (mod m)

(iii) º (mod m), for any polynomial f(x) with integral coefficients f(a) º f(b) ( mod m) etc. Linear Congruence, to show how to solve these congruences, Chinese remainder theorem ˆa“ Statement and

proof and some applications. System of linear congruences, when solution exists ˆa“ some applications.]

• Boolean algebra : Boolean Algebra, Boolean functions, Logic gates, Minimization of circuits.

• Application of Congruences : Divisibility tests. Check-digit and an ISBN, in Universal product Code, in major credit cards. Error detecting capability. [Using Congruence, develop divisibility tests for integers based on their expansions with respect to different bases, if d divides (b − 1) then n = is divisible by d if and only if the sum of the digits is divisible by d etc. Show that congruence can be used to schedule Round-Robin tournaments. Check digits for different identification numbers ˆa“ International

standard book number, universal product code etc. Theorem regarding error detecting capability.]

• Congruence Classes : Congruence classes, addition and multiplication of congruence classes. Fermat’s little

theorem. Euler’s theorem. Wilson’s theorem. Some simple applications.

[Definition of Congruence Classes,

properties of Congruence classes, addition and multiplication, existence of inverse. Fermat’s little theorem.

Euler’s theorem. Wilson’s theorem - Statement, proof and some applications.]