M.A./M.Sc. (Mathematics) Third Semester Course
M301 Complex Analysis-I
Section – I
The algebra and the geometric representation of complex numbers. Limits and continuity. Analytic functions. Polynomials and rational functions. The exponential and the trignometric functions. The periodicity. The logarithm. Sets and elements. Arcs and closed curves, Analytic functions in region
Conformal mapping, length and area. The linear group, the cross ratio, symmetry, oriented circles, family of circles. The use of level curves, a survey of elementary mappings, elementary Riemann surfaces.
Section- II
Line integrals, rectifiable arcs, line integral as function of arcs, Cauchy’s theorem for a rectangle, Cauchy’s theorem in a disk. The index of a point with respect to a closed curve. The integral formula. Higher derivatives.
Sequences, Series, Uniform convergence, Power series and Abel’s limit theorem. Weierstrass’s theorem, the Taylor’s series and the Laurent series.
Removable singularities. Taylor’s theorem, zeros and poles. The local mapping and the maximum principle.
Section – III
Chains and cycles, simple connectivity, Homology, the general statement of Cauchy’s theorem. Proof of Cauchy’s theorem. Locally exact differentials and multiply connected regions. The residue theorem, the argument principle and evaluation of definite integral.
M302 Topology
Section - I
Elementary Set Theory
Partial ordered sets and lattices.
Metric Spaces
Open sets, closed sets, convergence, completeness, Baire’s category theorem, continuity
Topological Spaces
The definition and some examples, elementary concepts, Open bases and open subbases, weak topologies, the function algebras C (X, R) and C (X, C).
Section - II
Compactness
Compact spaces, products of spaces, Tychonoff’s theorem and locally compact spaces, compactness for metric spaces, Ascoli’s theorem.
Separation
T1-spaces and Hausdorff spaces, completely regular spaces and normal spaces, Urysohn’s lemma and Tietze’s extension theorem, the Urysohn imbedding theorem, the Stone-Cech compactification.
Section - III
Connectedness
Connected spaces, the components of a space, totally disconnected spaces, locally connected spaces.
Aproximation
The Weierstrass approximation theorem.
M303 Analytic Number Theory
Section – I
Divisibility Theory in the IntegersThe Division Algorithm, The Greatest Common Divisor, The Euclidean Algorithm, and The Diophantine Equation ax + by = c.
Primes and their Distribution
The Fundamental Theorem of Arithmetic. The Sieve of Eratosthenes and The Goldbach Conjecture.
The Theory of Congruences
Basic Properties of Congruence, Special Divisibility Tests and Linear Congruences
Section – II
Fermat’s TheoremFermat’s Factorization Method, The Little Theorem and Wilson’s Theorem
Number – Theoretic Functions
The Functions τ and σ, The Möbius Inversion Formula, The Greatest Integer Function and An Application to the Calendar.
Euler’s Generalization of Fermat’s Theorem
Euler’s Phi-Function, Euler’s Theorem and Some properties of the Phi-Function, An Application to Cryptography.
Section – III
Primitive Roots and IndicesThe Order of an Integer Modulo n, Primitive Roots for Primes, Composite Numbers Having Primitive Roots and The Theory of Indices.
The Quadratic Reciprocity Law
Euler’s Criterion, The Legendre Symbol and Its Properties, Quadratic Reciprocity and Quadratic Congruences with Composite Moduli.
M304 Operations Research-II
Section – I
Queueing TheoryQueueing systems, Queueing problem, Transient and steady states, Probability Distributions in Queueing systems. Poisson process (pure birth process), Properties of possions arrivals, Exponential process, Markovian property, Pure death process, Service time distribution, Erlang service time distribution, Solution of Queueing Models.
Dynamic Programming
Decision Tree and Bellman’s principle of optimality, Concept of dynamic programming, minimum path problem, Mathematical formulation of Multistage Model, Backward & Forward Recursive approach, Application in linear programming.
Section – II
Non-Linear Programming Problems (NLPP):Formulation of a NLPP, General non-linear NLPP, Constrained optimization with equality constraint, Necessary and sufficient condition for a general NLPP (with one constraint), with m(<n) constrained optimization with inequality constraints (Kuhn – Tucker conditions), Saddle point problem, saddle point and NLPP, Graphical solution of NLPP, Verification of Kuhn – Tucker conditions, Kuhn – Tucker conditions with Non-negative constraints.
Section – III
Quadratic programmingQuadratic programming; Wolfe’s Modified Simplex method, Beale’s Method.
Separable Programming
Separable Programming, Piecewise linear approximation, Separable programming algorithm.
Simulation
Definition, Types of simulation, Event type simulation, Generation of random numbers, Monte – Carlo Simulation.
M305 Mathematical Statistics
Section – I
Distributions of Random VariablesThe probability set Function, random variables, The probability density Function, the distribution function, Certain probability Models, Mathematical Expectation, Some special Mathematical expectations, Chebyshev’s Inequality, conditional probability, Marginal and conditional distributions, the correlation coefficient, Stochastic Independence.
Section – II
Some Special DistributionsThe Binomial, trinomial, and Multinomial Distributions, the Poisson Distribution, The Gamma and Chi-square Distributions, the normal distribution, and the bivariate normal distribution.
Sampling theory, Transformations of variables of the Discrete type, Transformations of the variables of the continuous type. The t and F distributions.