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M .Sc (Mathematics) Semester -1

Math 601S: Real Analysis

UNIT-I

24 July to 15 Sepetember

(i) Basic Topology : Finite, countable and uncountable sets. Metric spaces, compact sets. Perfect sets.

Connected sets. (ii) Sequences and series : Convergent sequences (in metric spaces). Subsequences.

Cauchy sequences. Upper and lower limits of a sequence of real numbers. Riemann’s Theorem on

Rearrangements of series of real and complex numbers. (iii) Continuity : Limits of functions (in metric

spaces). Continuous functions. Continuity and compactness. Continuity and connectedness. Monotonic

functions.

UNIT- II

16 september to dispersal of classes

(iv) The Riemann-Stieltjes integral: Definition and existence of the Riemann-Stieltjes integral. Properties

of the integral. Integration of vector-valued functions. Rectifiable curves. (v) Sequences and series of

functions: Problem of interchange of limit processes for sequences of functions. Uniform convergence.

Uniform convergence and continuity. Uniform convergence and integration. Uniform convergence and

differentiation. Equicontinuous families of functions, The Stone-Weierstrass theorem

Math 602S: Algebra

UNIT-1

24 July to 15 September

Review of basic concepts of groups with emphasis on exercises. Permutation groups, Even and odd

permutations, Conjugacy classes of permutations, Alternating groups, Simplicity of An, n > 4. Cayley's

Theorem, Direct products, Fundamental Theorem for finite abelian groups, Sylow theorems and their

applications, Finite Simple groups

UNIT-II


Survey of some finite groups, Groups of order p2 , pq (p and q primes). Solvable groups, Normal and

subnormal series, composition series, the theorems of Schreier and Jordan Holder. Review of basic

concepts of rings with emphasis on exercises. Polynomial rings, formal power series rings, matrix rings,

the ring of Guassian Integers.

Math 603S: Differential Equations

UNIT-I

24 July to 15 september

Differential Equations Existence and uniqueness of solution of first order equations. Boundary value

problems and StrumLiouville theory. ODE in more than 2-variables.

UNIT-II

16 August to dispersal of classes

Partial differential equations of first order. Partial differential equations of higher order with constant

coefficients. Partial differential equations of second order and their classification.

Math 604S: Complex Analysis

UNIT-I

24 August to 15 september

Complex plane, geometric representation of complex numbers, joint equation of circle and straight line,

stereographic projection and the spherical representation of the extended complex plane. Topology on

the complex plane, connected and simply connected sets. Complex valued functions and their

continuity. Curves, connectivity through polygonal lines.Analytic functions, Cauchy-Riemann equations,

Harmonic functions and Harmonic conjugates.Power series, exponential and trigonometric functions,

arg z, log z, az and their continuous branches.

UNIT-II


Complex Integration, line integral, Cauchy’s theorem for a rectangle, Cauchy’s theorem in a disc, index

of a point with respect to a closed curve, Cauchy’s integral formula, Higher derivatives, Morrera’s

theorem, Liouville’s theorem, fundamental theorem of Algebra. The general form of Cauchy’s theorem.

Math 605S: Number Theory

UNIT-I

24 July To 15 September

Divisibility, Greatest common divisor, Euclidean Algorithm, The Fundamental Theorem of arithmetic,

congruences, Special divisibility tests, Chinese remainder theorem, Fermat’s little theorem, Wilson’s

theorem, residue classes and reduced residue classes, Euler’s theorem, An Application to cryptography,

Arithmetic functions φ (n), d(n), σ(n), µ(n), Mobius inversion Formula, the greatest integer function,

perfect numbers, Mersenne primes and Fermat numbers

UNIT-II

16 September To dispersal of classes

Primitive roots and indices. Quadratic residues, Legendre symbol, Quadratic reciprocity law, Jacobi

symbol, Binary quadratic forms and their reduction, sums of two and four squares, positive definite

binary quadratic forms, Diophantine equations ax + by = c , x 2 +y2 =z 2 , x 4 +y4 =z 2 .

Third Semester

Math-617S: Field Theory

UNIT I


Fields, examples, characteristic of a field, subfield and prime field of a field, field extension, the degree

of a field extension, algebraic extentions and transcendental extension, Adjunction of roots, splitting

fields, finite fields, existence of algebraic closure, algebraically closed fields. Separable, normal and

purely inseparable extensions. Perfect fields, primitive elements. Langrange’s theorem on primitive

elements.

UNIT II

16 September to dispersal of classes

Galois extensions, the fundamental theorem of Galois theory, Cyclotomic extensions, and Cyclic

extensions, Applications of cyclotomic extensions and Galois theory to the constructability of regular

polygons, Solvability of polynomials by radicals.

Math 618S: Topology

UNIT – I


Topological Spaces, bases for a topology, the order topology, the product topology on X × Y , the

subspace topology, closed sets and limit points, continuous functions, the product topology, the metric

topology, the quotient topology. Connected spaces, connected subspaces of the real line, components

and local connectedness.

UNIT-II


Compact spaces, compact space of the real line, limit point compactness, local compactness, nets.The

countability axioms, the separation axioms, Normal spaces, the Urysohn Lemma, the Urysohn

Metrization Theorem, the Tietze Extension Theorem, the Tychonoff Theorem

Math 661S: Probability and Mathematical Statistics

UNIT – I


Nature of Data and methods of compilation: Measurement scales, Attribute and variable, Discrete and

continuous variables. Collection, Compilation and Tabulation of data. Representation of data:

Histogram, Frequency Polygon, Frequency Curve, Ogives. Measures of central tendency: Mean, Median,

Mode, Geometric Mean, Harmonic Mean and their properties. Measuring variability of data: Range,

Quartile deviation, Deciles and Percentiles. Standard deviation, Central and non-central moments,

Sample and Population variance. Skewness and Kurtosis, Box and Whisker plot. 19 Correlation &

Regression Analysis: Scatter diagram. Karl Pearson’s and Spearman’s rank correlation coefficient. Linear

Regression and its properties. Theory of attributes, independence and association.

UNIT – II


Probability: Intuitive concept of Probability, Combinatorial problems, conditional probability and

independence, Bayes’ theorem and its applications. Random Variables and Distributions: Discrete and

Continuous random variables. Probability mass function and Probability density function. Cumulative

distribution function. Expectation of single and two dimensional random variables. Properties of random

variables. Moment generating function and probability generating functions. Distributions: Bernoulli

distribution. Binomial distribution. Poisson distribution, Negative Binomial and Hypergeometric

distributions. Uniform, Normal distribution. Normal approximation to Binomial and Poisson

distributions. Beta, Gamma, Chi-square and Bivariate normal distributions. Sampling distribution of

mean and variance (normal population). Chebyshev’s inequality, weak law of large numbers, Central

limit theorems.

Math 672S: Computational Techniques

UNIT-I

24 July 15 September

Programmer’s model of a computer, Types of computers, General awareness of Computer Hardware –

CPU, Input, Output and peripherals, Software and Programming languages, General awareness of MS –

Word. Programming in FORTRAN 77: Character set, constants, variables, Arithmetic expressions, Format

specification, READ, WRITE statements, unformatted I/O Statements, Unconditional GO TO, Computed

GO TO, Arithmatic and Logical IF statements, IF-THEN-ELSE, Nested IF-THEN-ELSE, ELSE-IF-THEN, DO

loops, Nested DO loops, CONTINUE Statement, Data statement, Double Precision, Logical Data, Complex

Data, WHILE Structure, Arrays-One and multidimensional, Subscripted Variables, Implied DO loops,

Sorting Problem, Function Subprograms and Subroutine subprograms, COMMON, EQUIVALENCE, Simple

programs.

UNIT-II


Solution of non-linear equations: Functional iteration, Bisection, Secant, Regula-Falsi, Newton-Raphson

and Bairstow’s methods, Rate of convergence of these methods, Solution of linear system of equations:

Gauss elimination, Gauss Seidal and Triangularization methods, Condition of convergence of these

methods. Interpolation: Finite difference operators, Newton interpolation, Gauss Forward and backward

interpolation formulae, Newton’s divided difference formula, Lagrange’s Formula, Inverse interpolation,

Hermite interpolation.

Math-678S: Linear Programming

UNIT-I


Linear Programming and examples, Convex Sets, Hyperplane, Open and Closed half-spaces, Feasible,

Basic Feasible and Optimal Solutions, Extreme Point & graphical methods. Simplex method, Charnes-M

method, Two phase method, Determination of Optimal solutions, unrestricted variables, Duality theory,

Dual linear Programming Problems, fundamental properties of dual Problems, Complementary

slackness, Unbounded solution in Primal. Dual Simplex Algorithm, Sensitivity analysis.

UNIT-II


Parametric Programming, Revised Simplex method, Transportation Problems, Balanced and unbalanced

Transportation problems, U-V method, Paradox in Transportation problem, Assignment problems,

Integer Programming problems: Pure and Mixed Integer Programming problems, 0-1 programming

problem, Gomary’s algorithm, Branch & Bound Technique. Travelling salesman problem

M .Sc Chemistry(Semester -1)

Mathematics For Chemists

UNIT 1


Vectors (15 Hrs.) Vector, dot, cross and triple products etc. The gradient, divergence and curl. Vector

calculus, Gauss theorem, divergence theorem etc. Matrix Algebra Addition and multiplication; inverse,

adjoint and transpose of matrices, special matrices (Symmetric, skew-symmetric, Hermitian, unit,

diagonal, unitary, etc.) and their properties. Matrix equation: Homogeneous, non-homogenous linear

and conditions for the solution, linear dependence and independence. Introduction to vector spaces,

matrix eigen values and eigen vectors, diagonalization, determinants (examples from Huckel theory).

Elementary Differential Equations Variables-separable and exact, first-order differential equations,

homogenous, exact and linear equations. Applications to chemical kinetics, secular equilibria, quantum

chemistry, etc. Solutions of differential equations by the power series method, Fourier series, spherical

harmonics, second order differential equations and their solutions.

UNIT 2


Differential Calculus (15 Hrs.) Functions, continuity and differentiability, rules for differentiation,

applications of differential calculus including maxima and minima (examples related to maximally

populated rotational energy levels, Bohr’s radius and most probable velocity from Maxwell’s distribution

etc), exact and inexact differentials with their applications to thermodynamic properties. Integral

calculus, basic rules for integration, integration by parts, partial fraction and substitution. Reduction

formulae, applications of integral calculus. Functions of several variables, partial 9 differentiation, co-

ordinate transformations (e.g. Cartesian to spherical polar), curve sketching. Permutation And

Probability Permutations and combinations, probability and probability theorems, probability curves,

average, root mean square and most probable errors, examples from the kinetic theory of gases etc.,

curve fitting (including least squares fit etc.) with a general polynomial fit

B.A/ B.Sc Semester -1

Plane Geometry

Unit-I

24 Juy To 15 September

Transformation of axes in two dimensions: Shifting of origin, rotation of axes, invariants. Pair of Straight

Lines : Joint equation of pair of straight lines and angle between them, Condition of parallelism and

perpendicularity, Joint equation of the angle bisectors, Joint equation of lines joining origin to the

intersection of a line and a curve. Circle : General equation of circle, Circle through intersection of two

lines, tangents, normals, chord of contact, pole and polar, pair of tangents from a point, equation of

chord in terms of mid-point, angle of intersection and orthogonality, power of a point w.r.t. circle,

radical axis, co-axial family of circles, limiting points.

Unit-II


Conic : General equation of a conic, tangents, normals, chord of contact, pole and polar, pair of tangents

from a point, equation of chord in terms of mid-point, diameter. Conjugate diameters of ellipse and

hyperbola, special properties of parabola, ellipse and hyperbola, conjugate hyperbola, asymptotes of

hyperbola, rectangular hyperbola. Indentification of conic in general second degree equations

Calculus

Unit-I

24 Juy to 15 September

Properties of real numbers : Order property of real numbers, bounds, l.u.b. and g.l.b. order

completeness property of real numbers, archimedian property of real numbers. Limits: ε -δ definition of

the limit of a function, basic properties of limits, infinite limits, indeterminate forms. Continuity:

Continuous functions, types of discontinuities, continuity of composite functions, continuity of f x( ) ,

sign of a function in a neighborhood of a point of continuity, intermediate value theorem, maximum and

minimum value theorem.

Unit-II

16 September To dispersal of Classes

Mean value theorems: Rolle’s Theorem, Lagrange’s mean value theorem, Cauchy’s mean value theorem,

their geometric interpretation and applications, Taylor’s theorem, Maclaurin’s theorem with various

form of remainders and their applications. Hyperbolic, inverse hyperbolic functions of a real variable and

their derivatives, successive differentiations, Leibnitz’s theorem.

Trignometry And Matrices

Unit-I


D’Moivre’s theorem, application of D’Moivre’s theorem including primitive nth root of unity. Expansions

of sin nθ , cos nθ , sinn θ , cosn θ (n∈N). The exponential, logarithmic, direct and inverse circular and

hyperbolic functions of a complex variable. Summation of series including Gregory Series.

Unit-II


Hermitian and skew-hermitian matrices, linear dependence of row and column vectors, row rank,

column rank and rank of a matrix and their equivalence. Theorems on consistency of a system of linear

equations (both homogeneous and non-homogeneous). Eigen-values, eigen-vectors and characteristic

equation of a matrix, Cayley-Hamilton theorem and its use in finding inverse of a matrix.

Diagonalization.

B.A./B.Sc Semester-3

Advanced Calculus

Unit-I


Limit and continuity of functions of two and three variables. Partial differentiation. Change of variables.

Partial derivation and differentiability of real-valued functions of two and three variables. Schwarz and

Young’s theorem. Statements of Inverse and implicit function theorems and applications. Vector

differentiation, Gradient, Divergence and Curl with their properties and applications.

Unit-II


Euler’s theorem on homogeneous functions. Taylor’s theorem for functions of two and three variables.

Jacobians. Envelopes. Evolutes. Maxima, minima and saddle points of functions of two and three

variables. Lagrange’s multiplier method

Differential Equations

Unit-I


Exact differential equations. First order and higher degree equations solvable for x, y, p. Clairaut’s form.

Singular solution as an envelope of general solutions. Geometrical meaning of a differential equation.

Orthogonal trajectories. Linear differential equations with constant coefficients.

Unit-II


Linear differential equations with variable coefficients- Cauchy and Legendre Equations. Linear

differential equations of second order- transformation of the equation by changing the dependent

variable/the independent variable, methods of variation of parameters and reduction of order.

Simultaneous Differential Equations

Statics

Unit-I


Basic notions. Composition and resolution of concurrent forces – Parallelogram law of forces,

Components of a force in given directions, Resolved parts of a force, Resultant of any number of

coplanar concurrent forces, Equilibrium conditions for coplanar concurrent forces, equilbrium of a body

resting on a smooth inclined plane. Equilibrium of three forces acting at a point – Triangle law of forces,

theorem,

Unit –II


Moments and Couples – Moment of a force ab theorems on moment of a couple,

Equivalent couples, Varignon’s theorem, generalized theorem of moments, resultant of a force and a

couple, resolution of a force into a force and a couple, reduction of a sy coplanar forces to a force and a

couple. Equilibrium conditions for any number of coplanar non forces. Friction: Definition and nature of

friction, laws of friction, equilibrium of a particle on a rough plane, Problems on ladders, rods, spheres

and circles.

B.A/B.Sc Semester -V

Analysis

Unit-I


Countable and uncountable sets. Riemann integral, Integrability of continuous and monotonic functions,

Properties of integrable functions, The fundamental theorem of integral calculus, Mean value theorems

of integral calculus. Beta and Gamma functions.

Unit-II


Improper integrals and their convergence, Comparison tests, Absolute and conditional convergence,

Abel’s and Dirichlet’s tests, Frullani’s integral. Integral as a function of a parameter. Continuity,

derivability and integrability of an integral of a function of a parameter.

Modern Algebra

Unit-I


Groups, Subgroups, Lagrange’s Theorem, Normal subgroups and Quotient Groups, Homomorphisms,

Isomorphism Theorems, Conjugate elements, Class equation, Permutation Groups, Alternating groups,

Simplicity of n A , n ≥ 5 (without proof).

Unit-II


Rings, Integral domains, Subrings and Ideals, Characteristic of a ring, Quotient Rings, Prime and Maximal

Ideals, Homomorphisms, Isomorphism Theorems, Polynomial rings.

Probability Theory

SECTION A


Review of notion of Probability, conditional Probability and independence, Bayes’ Theorem. Random

Variables : Concept, probability density function, cumulative distribution function, discrete and

continuous random variables, expectations, mean, variance, moment generating function, skewness and

kurtosis. Discrete Random Variables : Bernoulli random variable, binomial random variable, negative

binomial random variable, geometric random variable, Poisson random variable.

SECTION B


Continuous Random Variables : Uniform random variable, exponential random variable, Beta random

variable, Gamma random variable, Chi-square random variable, normal random variable. Bivariate

Random Variables : Joint distribution, joint and conditional distributions, Conditional Expectations,

Independent random variables, the correlation coefficient, Bivariate normal distribution

B C A Semester -1

Fundamental Of Mathematical Statistics

SECTION-A


Basic Statistics: Types of Statistics, Different Statistical Techniques, Steps in Statistical Investigation, Uses

and Limitations of statistics, Collection of Data: Sources of collecting primary and Secondary Data,

Limitations of Secondary Data, Criteria of evaluating secondary data, Organization of data, Graphs of

Grouped Frequency Distribution, Tabulation of Data, Parts of Table Measures of Central Tendency: Kinds

of measures of central tendency (statistical averages or averages): Arithmetic Mean: Simple Arithmetic

Mean, Methods of calculating Simple Arithmetic Mean, Arithmetic Mean in case of Individual Series,

Discrete series and continuous series, Weighted Arithmetic Mean, Combined Arithmetic Mean.

Geometric Mean: Simple Geometric Mean , Methods of calculating Simple Geometric Mean, Geometric

Mean in case of Individual Series, Discrete series and continuous series, Weighted Geometric Mean,

Combined Geometric Mean. Harmonic Mean: Simple Harmonic Mean ,Methods of calculating Simple

Harmonic Mean, Harmonic Mean in case of Individual, Discrete series and continuous series, Weighted

Harmonic Mean, Combined Harmonic Mean.

SECTION-B


Median: Methods of Calculating Median in case of Individual, Discrete series and continuous series

Partition Value: Quartile, Quintiles, Hexiles, Septiles, Octiles, Deciles, Percentiles Mode: Methods of

Calculating Mode in case of Individual Series, Discrete series and continuous series Range: Computation

of Range, Inter Quartile Range, Computation of Inter Quartile Range, Percentile Range and Computation

of Percentile Range. Mean Deviation, Computation of Mean Deviation, Standard Deviation, Calculation

of Standard Deviation, Variance, Calculation of Standard Deviation for individual Series, Discrete Series

and Continuous Series, Coefficient of Standard Deviation and coefficient of variation, Combined

Standard Deviation, Correcting incorrect Standard Deviation

SECTION-C


Correlation Analysis : Correlation Analysis: Definition, Types of Correlation: Positive, Negative, Simple,

Multiple, Partial, Total, Linear and Non-Linear. Need of Correlation Analysis, Correlation and Causation,

Techniques for Measuring Correlation: Scatter Diagram Method, Graphic Method, Karl Pearson’s

Coefficient of Correlation: Correcting incorrect coefficient of correlation, calculating Karl Pearson’s

coefficient of correlation in case of grouped series, Probable Error, Coefficient of Determination,

Spearman’s coefficient of Correlation (Rank correlation): Calculation of Correct Coefficient of rank

correlation, Difference between Rank Coefficient and Karl Pearson’s coefficient of coefficient,

Coefficient of concurrent deviation.

SECTION-D


Regression Analysis (Linear Regression): Definition, Difference between Correlation and Regression,

Types of Regression Analysis: Simple, Multiple, Partial, Total, Linear and Non-Linear, Objectives of

Regression Analysis, Methods of obtaining regression analysis: Regression Lines, Regression Equations.

Methods of obtaining regression equations: Normal Equations and Regression Coefficient, Properties of

Regression Coefficient, Standard Error of Estimate, Regression Coefficient in case of Grouped Data, Uses

of Regression Analysis and Limitations of Regression Analysis.

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