M.Sc. Mathematics
Semester IV

COURSE CODECOURSE NAMECREDITS

JMMH401
Functional Analysis:
UNITI
Coordinate transformation, Covariant, Contravariant and mixed tensors, tensors of higher rank, symmetric and skew symmetric tensor, tensor algebra, Contraction, Inner Product, Quotient Law. Riemannianmetric tensor, Christoffel symbols, Transformation Laws of Christoffel symbols, Covariant derivatives of higher rank tensor, Riemannian curvature tensor.
UNITII
Differentiable curves and their parametric and implicit representations, Tangent vector, Principal normal, Binormal, curvature and torsion, SerretFrenet formulas, Fundamental theorem for space curve. Vector fields, Covariant differentiations, Connexion forms and structural equations in E3.
UNITIII
Curvilinear CoOrdinates on a Surface, First fundamental forms, Geodesic on surface, Geodesic coordinates.
UNITIV
Second fundamental forms, Tensor derivative, GaussWeingarten formulae, Integrability condition, Gauss & Mainardi Codazzi equations, Meusrier theorem, Geodesic curvature.
UNITV
Line of curvature, Asymptotic lines, Gauss and mean curvature, Minimal surfaces, Third fundamental forms.
Books:
 1. Introduction to Differential Geometry: Abraham Goetz; Addison Wesley Pub. Co.
 2. Differential Geometry: Nirmala Prakash; McGrawHill
 3. Elementary Differential Geometry: B.O. Neill; Academic Press.
 4. A course in tensors with Application to Riemannian Geometry: R.S. Mishra
 5. An introduction to Differential Geometry: T.J. Willmore
 6. Introduction to Riemannian Geometry and Tensor Calculus: Weitherburn.
04 
JMMH402
Graph Theory:
UNITI
Graph; Applications of Graph; Finite and Infinite Graphs; Null Graph; Incidence and Degree; Isolated Vertex; Pendant Vertex; Isomorphism; Sub graphs; Walks; Paths; Circuits; Connected Graphs, Disconnected Graphs and Components
UNITII
Euler’s Graph; Operations On Graphs; Hamiltonian Paths and Circuits; Shortest Path Algorithms; The Traveling Salesman Problem; Dijkastra’s Algorithm; Fleury’s Algorithm.
UNITIII
Trees; Properties of Trees; Pendant Vertices in a Tree; Distance and Centers in a Tree; Rooted and Binary Trees, On Counting Trees; Spanning Trees; Fundamental Circuits; Finding All Spanning Trees of a Graph; Spanning Trees in a Weighted Graph; CutSets; Some Properties of a CutSet; Fundamental Circuits and CutSets, Connectivity and Separability; Network Flows.
UNITIV
Combinatorial and Geometric Graphs; Planar Graphs; Kuratowski's Two Graphs; Different, Detection of Planarity; Geometric Dual; Combinatorial Dual; Thickness and Crossings; Vectors and Vector Space; Associated with a Graph.
UNITV
Matrix representation of graphs; Incidence matrix; Sub matrix of () AG; Circuit matrix,Fundamental circuit matrix and Rank of B;Cutset matrix; Path matrix; Adjacency Matrix; Adjacency Matrix;Chromatic Number; Chromatic Partitioning; Chromatic Polynomial; Matching Coverings, The Four Color Problem.
 1. Narsingh Deo; Graph Theory; PrenticeHall, Inc.
 2. Douglas B. West; Introduction to Graph Theory; Pearson Education Pvt. Ltd.
 3. Gary Chartrand; Chromatic Graph Theory; CRC Press.
 4. J.A. Bondy U.S.R. Murty; Graph Theory, Springer.
 5. Reinhard Diestel ; Graph Theory, Spring.
04 
JMMH403
Mathematical Statistics:
UNITI
Random variable and sample space, notion of probability, axioms of probability, empirical approach to probability, conditional probability, independent events, Bayes’ Theorem; probability distributions with discrete and continuous random variables, joint probability mass function, marginal distribution function, joint density function.
UNITII
Mathematical expectation, moment generating function; Chebyshev’s inequality, weak law of large numbers, Bernoullian trials; the Binomial, negative binomial, geometric, Poisson, normal, rectangular, exponential, Gaussian, beta and gamma distributions and their moment generating functions; fit of a given theoretical model to an empirical data.
UNITIII
Sampling and large sample tests, Introduction to testing of hypothesis, tests of significance for large samples, chisquare test, SQC, analysis of variance, T and F tests; Theory of estimation, characteristics of estimation, minimum variance unbiased estimator, method of maximum likelihood estimation.
UNITIV
Scatter diagram, linear and polynomial fitting by the method of least squares; linear correlation and linear regression, rank correlation, correlation of bivariate frequency distribution.
UNITV
Limit Theorems: Stochastic convergence, Bernoulli law of large numbers, Conv ergence of sequence of distribution functions, LevyCramer Theorems, deMoivre Theorem, Laplace Theorem, Poisson, Chebyshev, Khintchine Weak law of large numbers, Lindberg Theorem, Lyapunov Theroem, Borel Cantelli Lemma, Kolmogorov Inequality and Kolmogorov Strong Law of large numbers.
Books:
 1. Robert V. Hogg and Allen T. Craig, Introduction to Mathematical Statistics, Macmillan Publishing Co. Inc.
 2. Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, American Mathematical Society.
 3. Feller, W: Introduction to Probability and its Applications, Wiley Eastem Pvt. Ltd.
 4. K.L. Chung, A course in Probability, Academic Press.
 5. R. Durrett, Probability Theory and Examples, Duxbury Press.
04 
JMMH404
Number Theory:
UNITI
Algebraic numbers ,number fields,conjugates and discriminants,algebraic integers, integral bases,norms and traces,rings of integers, quadratic field and cyclometic fields
UNITII
Trivial factorizations, factorization in to irreducible, examples of nonunique factorization in to irreducible, prime factorization.
UNITIII
Euclidean domain and Euclidean quadratic fields, consequences of unique factorization,the RamanujanssNagell theorem, prime factorization of ideals, norm of an ideal, nonunique factorization in cyclometric fields
UNITIV
Lattices of dimension m, the quotient torus, Minkowiski theorem, two squares theorem, four squares theorem, the space L
UNITV
The classgroup, limitness of the class group, unique factorization of elements in an extension ring, factorization of a rational prime, Minkowiski constants, classnumber calculations.
Books:
 1. D.A Marcus, Number Fields,SpringerVerlag,New York Inc,1987.
 2. S. Lang, Algebraic Number Theory, Chapman and Hall, London, 1987
 3. K. Ireland and M Rosen, A classical Introduction to Modern Number Theory, SpringerVerlag.
04 
JMMH405
Minor Research Project and Seminar
04 
Total Credits20