M.Sc. (Mathematics) – Non-Semester
(To be offered under Distance and Continuing Education)
SCHEME OF EXAMINATIONS
First year
S.No. Paper
1.1 Advanced Abstract Algebra
1.2 Real Analysis
1.3 Differential Equations
1.4 Fuzzy Mathematics and Statistics
1.5 Graph Theory and Combinatorics
Hours Marks Passing Min.
3
100
50
Second Year
1.1 Programming in C and Numerical Methods
1.2 Measure theory and Complex Analysis
1.3 Topology and Functional Analaysis
1.4 Operations Research
1.5 Mechanics
1.1 Advanced Abstract Algebra
Unit I : Groups – A counting principle – Normal subgroups and quotient groups –
homomorphism – isomorphism – Cayley’s theorem – permutation groups.
Unit II : Another counting principle – Sylow’s Theorems
Unit III : Rings – homomorphism – Ideals and quotient rings – Field of quotients of
an integral domain – Euclidean domain – Polynomial rings.
Unit IV : Vector spaces – Linear transformation and bases – Algebra of linear
transformations – Characteristic roots – triangular form.
Unit V: Extension fields – roots of polynomials – more about roots.
Text : I.N. Herstein Topics in Algebra (Second Edition)
Chapter 2, Section 2.1 to 2.12
Chapter 3, section 3. 1 to 3.9
Chapter 4, sections 4.1 , 4.2
Chapter 5, sections 5.1., 5.3., 5.5
Chapter 6, sections 6.1 to 6.4
1.2 Real Analysis
Unit I : Basic topology – convergent sequences – subsequences – upper and lower limits
– some special sequences –
Unit II : Series – Series of non-negative terms – The number e – The root and ratio tests –
Power series – suromation by parts – Absolute convergence – Addition and
multiplication of series – Rearrangements.
Unit III : Continuity – Differentiation.
Unit IV : The Riemann – Steiltjes integral – Sequences and series of functions –
Discussion of the main problem – Uniform convergence – Uniform convergence and
continuity – Uniform convergence and intergration.
Unit V: Uniform Convergence and differentiation – Equicontinuity – Equicontinuous
family of functions – Stone Weierstrass’ theorem – some special functions.
Text : Rudin – Principles of Mathematical Analysis (Tata McGrows Hill) Third Edition,
Chapters 2 to 8.
1.3 Differential Equations
Unit I : Second order linear equations – The general solution of a homogeneous
equation – Use of a known solution to find another – The method of variation of
parameters – Power series solution – Series solution of a first order equation.
Unit II : Second order linear equations – Ordinary points – regular singular points –
Legendre polynomials .
Unit III : Bessel functions and Gamma functions – Linear systems – Homogeneous
linear systems with constant coefficients – The method of successive approximation –
Piccard’s theorem.
Unit IV : Partial Differential Equations – Cauchy’s problem for first order equations –
Linear equations of first order – Nonlinear partial differential equations of first order
– Cauchy’s method of characteristics – Compatible system of first order equations.
Unit V: Charpit’s method – special types of first order equations – Solutions
satisfying given conditions – Jacobi’s method – Linear Partial Differential Equations
with constant coefficients – Equation with variable coefficients.
Tests: G.F. Simmons, Differential Equations (Torta McGrow Hill )sections 14,15,
16,19 26-29, 32-35, 37,38,55 and 56.
I.N, Senddon, Elements of Partial Differential Equations, (Mc-Grow Hill) Chapter 2
sections 1-4, 7-13, Chapter 3 sections 1,4 and 5.
1.4 Fuzzy Mathematics and Statistics
Unit I : The concept of Fuzziness – Some Algebra of fuzzy sets.
Unit II: Fuzzy quantities- Logical aspects of fuzzy sets.
Unit III: Distribution of random variables.
Unit IV: Conditional Probability and stochastic independence – Some special
distributions
Unit V: Distributions of Functions of random variables – Limiting Distributions.
Texts: H.T. Nguyen and E.A. Walker, A first course in Fuzzy Logic (Second Edition)
CRC Chapters 1 to 4.
R.V. Hagg and A.T. Craig, Introduction to Mathematical Statistics (fourth Edition)
Macmillan, Chapters 1 to 3, Chapter 4 (except section 4.6) and Chapter 5.
1.5 Graph Theory and Combinatorics
Unit I : Graphs and Subgraphs – Trees – Connectivity.
Unit II : Euler Tours and Hamilton Cycles – Matchings – Edge Colorings.
Unit III : Independent sets amd cliques – Verter Colorings.
Unit IV: Generating Functions – Recurrence relations.
Unit V: The principle of Inclusion and Exclusion – Polya’s theory of counting.
Texts: J.A. Bondy and U.S. R. Murty, Graph theory with Applications (Macmillan),
Chapter 1, (sections 1 to 7), chapters 2,3,4,5 and 6. (Exclusing sections dealing with
Applications), Chapters 7 (Sections 1 to 3) , Chapter 8. C.L. Liu, introduction to
combinatorics (Mc. Grow Hill) chapters 2 to 5.
1.1 Programming in C and Numerical Methods
Unit I : Overview of C – Constants, Variables and data types – Operators and
expressions – managing input and output operations – Decision making branching
and looping.
Unit II: Arrays – Handling of character strings – User defined functions –
Structures and Unions.
Unit III: Pointers – File management in C.
Unit IV : Interpolation – Lagrange’s interpolation formula – Numerical solution
of ordinary differential equations – Taylor series method – Piccard’s method –
Euler’s method.
Unit V: Runge – Kuttar Fourth order method – Predictor – Corrector methods –
Milne’s method.
Texts: E. Balagurusamy, Programming in Ansi C (Tata Mc. Graw Hill) Chapters
1 to 12. M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical methods for
scientific and engineering computation. (Second Edition) Wiley Eastern, Sections
4.1, 4.2, 6.1, 6.2 and 6.3.
1.2 Measure Theory and complex Analysis
Unit I : Lebesgue Measure – Lebesgue Integral
Unit II: Measure and Integration – Measure and Outer Measure.
Unit III: Complex numbers –Analytic functions – Elementary Theory of Power
series .
Unit IV: Cauchy’s theorem – Cauchy’s integral formula – singularities.
Unit V: Taylor’s theorem – Maximum principle – The calculus of Residues .
Texts: Royden – Real Analysis Third Edition(PHI) Chapters 3,4,11 and 12.
Ahlfors – Complex analysis (Tata – McGraw Hill) Second Edition, Chapter 1,
Chapter 2, sections 1 and 2. Chapter 4 sections 1,2,3 and 5.
1.3 Topology and Functional analysis
Unit I : Topological Spaces – Compactness.
Unit II: Separation – connectedness
Unit III: Banach Spaces
Unit IV: Hilbert spaces – Finite dimensional spectral Theory
Unit V: General Preliminaries on Banach Algebras – The structure of
commutative Banach Algebra.
Text: G.F. Simmons, introduction to Topology and Modern Analysis (Mc. Graw
Hill)
Chapters 3,4,5,6,9, 10, 11, 12 and 13.
1.4 Operations Research
Unit I : Linear Programming – Simplex method – Transportation and its variation
Unit II: Network Models – CPM – PERT
Unit III : Integer Programming
Unit IV: Inventory models – Dicision Analysis and Games
Unit V: Queueing Models.
Text: Taha – Operations Research – An Introduction (sixth Edition) PHI,
Chapters 2,3,5,6,9,11,14 and 17.
1.5 Mechanics
Unit I: Kinematics – Kinetic Energy and Angular Momentum – Methods of
Dynamics in space.
Unit II: The simple pendulum – The spherical pendulum – Motion of a rigid body.
Unit III: The equations of Lagrange and Hamilton – Hamiltonian methods.
Unit IV: Real fluids and ideal fluids – Velocity – Stream lines – Steady and
unsteady flows – velocity potential.
Unit V: Vorticity vector – Equation of continuity – Euler’s equation of motion –
Bernoulli’s equation – some three dimensional flows.
Texts: J.L. Synge and B.A Griffith, Principles of Mechanics (Mc. Graw Hill)
Chapters 11,12 and 13 *sections 2 and 3), Chapters 14,15, and 16 (Section 1) F.
Chorlton, Text book of Fluid Dynamics (CBs Publishers) Chapter 2, sections 2.1.
to 2.7, Chapter 3, sections 3.1 and chapter 4.
1.6 Operations Research
Unit I: Linear Programming – Simplex method – Transporation model and its
variation
Unit II: Networks Models – CPM – PERT
Unit III : Integer Programming
Unit IV: Inventory models – Decision Analysis and Games
B. Sc Mathematics Main (Non- Semester)
(to be offered under DD& CE)
Scheme of Examination
I Year ( 3 papers)
Paper I Calculus
Paper II Classical Algebra
Paper III Analytical Geometry 3D and Vector Calculus
II Year (3 papers)
Paper IV Modern algebra
Paper V Statistics
Paper VI operations Research
III Year (5 papers)
Paper VII Analysis
Paper VIII Mechanics
Paper IX Astronomy
Paper X Numerical Analysis
Paper XI Discrete Mathematics
I Year – Paper I – Calculus
Unit I : Curvature – radius of curvature – Cartesian and polar – centre of curvature –
Involute and evolute – Asymptotes in Cartesian co-ordinates – Multiple points – double
points.
Unit II:
Evaluation of double and triple integrals – jocobians, change of variables.
Unit III:
First order differential: equations of higher degree – solvable for p,x and y – Clairaut’s
form/ linear differential equations of second order – particular integrals for functions of
the form, Xn, eax, eax (f(x). Second order differential equations with variable
coefficients.
Unit IV: Laplace transform-Inverse transform – Properties – Solving differential
equations. Simultaneous equations of first order using Laplace transform.
Unit V: Partial differential equations of first order – formation – different kinds of
solution – four standard forms – Lagranges method.
Books:
1. Calculus 1,2 & 3, T.K. Manickavachagom pillai & others.
2. Calculus 1&2, S. Arumugam and Isaac.
I year – Paper II – Classical Algebra
Unit I :
Theory of Equations: Every equation f(x) =0 of nth degree has ‘n’ roots. Symmetric
functions of the roots in terms of the coefficients – sum of the rth powers of the roots – Newton’s
theorem – Descartes rule of sign – Rolle’s theorem.
Unit II
Reciprocal Equations – Transformation of equations – solution of cubic and biquadratic equation
– Cardon’s land Ferrari’s methods – Approximate solution of numerical equations- Netwon’s
and Horner’s methods.
Unit III
Sequences and series: Sequences – limits, bounded, monotonic, convergent, oscillatory and
divergent sequence – algebra of limits – subsequences – Cauchy sequences in R and Cauchy’s
general principle of convergence.
Unit IV
Series – convergence, divergence – geometric, harmonic, exponential, binomial and logratithmic
series – Cauchy’s general principle of convergence – comparison test – tests of convergence of
positive termed series – Kummer’s test, ratio test, Raabe’s test, Cauchy’s root test, Cauchy’s
condensation test.
Summation of series using exponential, binomial and logarithmic series.
Books for reference:
1.
2.
3.
4.
Sequences and series, S. Arumugam & Others
Algebra – Vol. I, T.K. Manickavachagom pillai & Others
Real Analysis – Vol.I, K. Chandrasekara Rao & K.S. Narayanan
Infinite series, Bromwich.
I Year – Paper III – Analytical Geometry 3D and Vector Calculus
Unit I : Rectangular Cartesian Coordinates in space – Distance formula – Direction ratio and
cosines – Angle between lines – simple problems.
Plane – different forms of equation – angle between two planes – perpendicular distance from a
point on a plane – projection of a line or a point on a plane.
Unit II: Lines – symmetrical form – plane and a straight line – The perpendicular from a point on
a line – Coplanar lines – shortest distance between two skew lines and its equation.
Sphere – Different forms of equations- plane section – the circle and its radius and centre –
tangent plane – condition for tangency – touching spheres – common tanget plane – point of
orthogonality of intersection of two spheres.
Unit III
Vector differentiation – Gradient, Divergence and Curl operators – solenoidal and irrotational
fields- formulas involving the laplace operator.
Unit IV: Double and triple integrals – Jacobian – change of variables – Vector integration –
single scalar variables – line, surface and volume integrals.
Unit V: Gauss’s Stoke’s and Green’s theorems – statements and verification only.
Books for reference:
1.
2.
3.
4.
Analytical Geometry of 3D-Part II, Manickavachagom Pillai
Analytical Geometry of 3D & Vector Calculus – P. Duraipandian & Others
Analytical Geometry of 3D & Vector Calculus – S. Arumugam & Others
Vector Analysis, K. Viswanathan.
II year – PaperIV – Modern Algebra
Unit I :
Sets – functions – relations – partitions – composition of functions – groups – subgroups –cyclic
groups.
Unit II:
Normal subgroups – cosets – lagrage’s theorem – Quotient groups – Homomorphism – Kernel –
Cayley’s theorem – Fundamental theorem of homomorphism.
Unit III:
Rings – types – subring – ordered integral domain – ideals – Quotient rings – P.I.D. –
Homomorphism of rings – fundamental theorem of homomorphism – Euclidean rings.
Unit IV: Definition and example of vector speaces – subspaces – sum and direct sum of
subspaces – linear span, linear dependence, independence and their basic properties – Basis –
finite dimensional vector spaces – dimension of sums of subspaces – Quotient space and its
dimension.
Unit V:
Linear transformation and their representation as matrices – Algebra of linear transformations –
dual spaces – Eigen values & eigen vectors of a linear transformation – inner product spaces –
Schwartz inequality – orthogonal sets and basis – Gram Schmidt orthogonalization process.
Reference books:
1.Modern Algebra, S. Arumugam and Issac
2. Modern Algebra, Vasistha
3. Topics in Algebra, I.N. Herstein, Vikas Publishers
II year – Paper V – Statistics
Unit I :
Correlation – Karl Pearson’s coefficient of Correlation, Lines of Regression – Regression
coefficients – Rank correlation.
Unit II:
Probability – Definition – application of addition and multiplication, theorems – conditional,
Probability – Mathematical Expectations – Moment generating function – special distributions,
(Binomial distribution, Poisson distribution, Normal distribution – properties).
Unit –III:
Association of attributes – Coefficient of association – consistency – time series – Definition –
Components of a time series – Seasonal and cyclic variations.
Unit –IV
Sampling – definition – large samples. Small samples – Population with one sample and
population with two samples – students – t-test-applications – chi – square test and goodness of
fit – applications.
Unit – V
Index numbers – Types of Index Numbers – Tests – Unit test, Commodity reversal test, time
reversal test, factor reaversal test – Chain index numbers – cost of living index- Interpolation –
finite differences operators -------------- - Newton’s forward, backward interpolation formulae,
Lagrange’s formula.
Books:
1.
2.
3.
4.
Statistics: S. Arumugam & others
Statistics: D.C. Saucheti & Kapoor
Statistics: Mangaladas & Others
Statistics: T. Sankaranarayana & Others.
II Year – Paper VI- Operations Research
Unit: 1
Linear programming problem – Mathematical formulation – Graphical method of solution simplex method – The big M method (Charnes method of penalties) – Two phase simplex
method – Duality – Dual simplex method – integer programming.
Unit – II
Transportation problem – mathematical formulation – North –west corner rule – Vogel’s
approximation method (unit penalty method ) – method of matrix minima – optimality test –
maximization – Assignment problem – mathematical formulation – method of solution –
maximization of the effective matrix.
Unit III:
Sequencing problem – introduction – n jobs and two machines – n jobs and three machines – two
jobs and n machines – graphical method – inventory models: types of inventory modals:
Deterministic: 1) Uniform rate of demand, infinite rate of production and no shortage – 2)
Uniform rate of demand, finite rate of replenishment and no shortage – 3) Uniform rate of
demand, instantaneous production with shortages – 4) Uniform rate of demand, instantaneous
production with shortage and fixed time.
Unit IV:
Probabilistic Models: Newspaper boy problem – discrete and continuous type cases- Inventory
models with one price break.
Queueing Theory: General concept and definitions – classification of queues – Poisson process,
properties of poisson process – models: 1) (M/M/1) : (∞/FCFS), 2)(M/M/1): (N/FCFS),
3)(µ/M/S): (∞/FCFS).
Unit V: Network Analysis: Drawing network diagram – Critical path method – labelling method
– concept of slack and floats on network – PERT – Algorithm for PERT – Differences in PERT
and CPM.
Resource Analysis in Network Scheduling : Project cost – Crashing cost – Time-cost
optimization algorithm – Resource allocation and scheduling.
Books for Reference:
1.
2.
3.
4.
5.
Operations Research : Kantiswarup, P.K. Gupta and Man Mohan.
Operations Research : P.K. Gupta, D.S. Hira.
Operations Research : V.K. Kapoor
Operations Research : S.D. Sharma
Operations Research : Mangaladoss.
III Year – Paper VII – Analysis
Unit I:
Metric spaces – open sets – Interior of a set – closed sets – closure – completeness – Cantor’s
intersections theorem – Baire – Category Theorem.
Unit II:
Continuity of functions – Continuity of compositions of functions – Equivalent conditions for
continuity – Algebra of continuous functions – hemeomorphism – uniform continuity –
discontinuities connectednon – connected subsets of R – Connectedness and continuity –
continuous image of a connected set is connected – intermediate value theorem.
Unit III:
Compactness – open cover – compact metric spaces – Herni Borel theorem. Compactness and
continuity – continuous image of compact metric space is compact – Continuous function on a
compact metric space in uniformly continuous – Equivalent forms of compactness – Every
compact metric space is totally bounded – Bolano – Weierstrass property – sequentially compact
metric space.
Unit IV:
Algebra of complex numbers – circles and straight lines – regions in the complex plane –
Analytic functions Cauchy – Rienann equations – Harmonic functions – Bilinear transformation
translation, rotation, inversion – Cross – ratio- Fixed points – Special bilinear transformations.
Unit V: Complex Integration – Cauchy’s integral theorem – Its extension – Cauchy’s integral
formula – Morera’s theorem – Liouville’s theorem – fundamental theorem of algebra – Taylor’s
series – Laurent’s series – Singularities. Residues – Residue Theorem – Evaluation of definite
integrals of the following types.
∫02π F (Cos x, sin x) dx
2 ∫-∞∞
f(x)
g(x)
dx
Books for reference:
1. Modern Analysis – Arumugam and Issac.
2. Real Analysis – Vol. III – K. Chandrasekhara Rao and K.S. Narayanan, S. Viswanathan
Publisher.
3. Complex Analysis – Narayanan & Manicavachagam Pillai
4. Complex Analysis – S. Arumugam & Issac.
5. Complex Analysis – P. Durai Pandian
6. Complex Analysis – Karunakaran, Narosa Publishers.
III Year – Paper VIII – Mechanics
Unit I:
Forces acting at a point – parallelogram of forces – triangle of forces – Lami’s theorem, Parallel
forces and moments – Couples – Equilibrium of three forces acting on a rigid body – Coplanar
forces – Reduction of any number of Coplanar forces theorems. General conditions of quilibrium
of a system of Coplanar forces.
Unit II:
Friction – Laws of friction – Equilibrium of a particle (i) on a rough inclined plane. (ii) under a
force parallel to the plane (iii) under any force – Equilibrium of strings – Equation of the
common catenary – Tension at any point – Geometrical properties of common catenary –
uniform chain under the action of gravity – Suspension bridge.
Unit III:
Dynamics – Projectiles – Equation of path, Range etc – Range on an inclined plane – Motion on
an inclined plane. Impulsive forces – Collision of elastic bodies – Laws of impact – direct and
oblique impact – Impact on a fixed plane.
Unit IV:
Simple harmonic motion in a straight line – Geometrical representation – Composition of SHM’s
of the same period in the same line and along two perpendicular directions – Particles suspended
by spring – S.H.M. on a curve – Simple pendulum – Simple Equivalent pendulum – The seconds
pendulum.
Unit V:
Motion under the action of Central forces – velocity and acceleration in polar coordinates –
Differential equation of central orbit – Pedal equation of central orbit – Apses – Apsidal
distances – Inverse square law.
Books for Reference:
1.
2.
3.
4.
Statics and Dynamics: S. Narayanan
Statics and Dynamics : M.K. Cenkataraman
Statics: Manickavachagom pillai
Dynamics: Duraipandian.
III Year – Paper IX – Astronomy
Unit I:
Spherical Trigonometry (only formulae) celestial sphere – four systems of coordinates – Diurnal
motion – Zones of the earth – Perpetual day and night – Terrestrial longitude and latitude –
International date line.
Unit II:
Dip of horizon – effects – Twilight – shortest twilight.
Unit III:
Refraction – Tangent formula – Cassini’s formula – Effects – Horizontal refraction – Geocentric
parallax.
Unit IV:
Kepler’s laws – verification – Newton’s deductions – Anomalies – Planets – Inferior and
superior planet – Bode’s law – Elongation – Sidereal period – Synodic period – Phase of the
planet – Stationary positions of a planet.
Unit V :
Moon – Phase – sidereal and synodic period – elongation – Metonic cycle – golden number –
Eclipses – Lunar and solar eclipses – conditions – Synodic period of the nodes – Ecliptic limits –
Maximum and minimum number of eclipses near a node and in a year – Saros – Lunar and solar
eclipses compared.
Books:
1. Astronomy : S. Kumaravelu & Susheela Kumaravelu.
2. Astronomy: G.V. Ramachandran
3. Astronomy: K. Subramanian and L.V. Subramanian
III Year – Paper X – Numerical Analysis
Unit I :
Finite differences – difference table – operators E,∆ and - Relations between these operatous –
Factorial notation – Expressing a given polynomial in factorial notation – Difference equation –
Linear difference equations – Homogeneans linear difference equation with constant
coefficients.
Unit II
Interpolation using finite differences – Newton – Gregory formula for forward interpolation –
Divided differences – Properties – Newton’s formula for unequal intervals - Lagrange’s formula
– Relation between ordinary differences and divided differences – inverse interpolation.
Unit III
Numerical differentiation and integration – General Quadratue formula for equidistant ordinates
– Trapezoidal Rule – Simpson’s one third rule – Simpson’s three eight rule – Waddle’s rule –
Cote’s method.
Unit IV:
Numerical solution of ordinary differential equations of first and second orders – Piccards
method. Eulers method and modified Euleis method – Taylor’s series method – Milne’s method
– Runge – Kutta method of order 2 and 4 – Solution of algebraic and transcendent equations.
Finding the initial approximate value of the root – Iteration method – Newton Raphson’s
method.
Unit V:
Simultaneous linear algebraic equations – Different methods of obtaining the solution – The
elimination method by Gauss – Jordan method – Grouts’ method – Method of factorization .
Books:
Calculus of finite differences and Numerical Analysis, P.P. Gupta & G.S. Malik, Krishna
Prakasham Mardin, Mecrutt.
III Year – Paper XI – Discrete Mathematics
Unit I:
Definition and examples of graphs – degrees – subgraphs – ismorphims – Ramsey numbers –
independent sets and coverings – intersection graphs and line graphs – matrices – operations in
graphs – degree sequences, graphic sequences.
Unit II:
Walks – trails and paths – connectedness and components – blocks – connectivity – Eulerian
graphs – Hamiltonian graphs – trees – characterization of trees – centre of a tree.
Unit III:
Planas graph and their properties – characterization of planas graphs – thickness – crossing and
outerplanarity – Chromatic number – chromatic index – five colour theorm – four colour
problem – chromatic polynomials – Directed graphs and basic properties – paths and connections
in digraphs – digraphs and matrices – tournaments.
Unit IV:
Permutations – ordered selections – unordered selections – further remarks on binomial theorem
– Pairings within a set – pairings between sets, - an optimal assignment problem.
Unit V:
Recurrence relations – Fibonacci type relations – Using generating functions – miscellaneous
methods – The inclusion exclusion principle and rook polynomials.
Text Books:
1. Invitation to graph theory, S. Arumugam and S. Ramachandran, Scitech Publications.
2. A first course in combinational mathematics, Ian Anderson (Oxford applied Math. Series)