M.Sc. Mathematics

Semester IV

  • COURSE CODE
    COURSE NAME
    CREDITS
  • JMMH-401

    Functional Analysis:

    UNIT-I

    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.

    UNIT-II

    Differentiable curves and their parametric and implicit representations, Tangent vector, Principal normal, Binormal, curvature and torsion, Serret-Frenet formulas, Fundamental theorem for space curve. Vector fields, Covariant differentiations, Connexion forms and structural equations in E3.

    UNIT-III

    Curvilinear Co-Ordinates on a Surface, First fundamental forms, Geodesic on surface, Geodesic co-ordinates.

    UNIT-IV

    Second fundamental forms, Tensor derivative, Gauss-Weingarten formulae, Integrability condition, Gauss & Mainardi Codazzi equations, Meusrier theorem, Geodesic curvature.

    UNIT-V

    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; McGraw-Hill
    • 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
  • JMMH-402

    Graph Theory:

    UNIT-I

    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

    UNIT-II

    Euler’s Graph; Operations On Graphs; Hamiltonian Paths and Circuits; Shortest Path Algorithms; The Traveling Salesman Problem; Dijkastra’s Algorithm; Fleury’s Algorithm.

    UNIT-III

    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; Cut-Sets; Some Properties of a Cut-Set; Fundamental Circuits and Cut-Sets, Connectivity and Separability; Network Flows.

    UNIT-IV

    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.

    UNIT-V

    Matrix representation of graphs; Incidence matrix; Sub matrix of () AG; Circuit matrix,Fundamental circuit matrix and Rank of B;Cut-set matrix; Path matrix; Adjacency Matrix; Adjacency Matrix;Chromatic Number; Chromatic Partitioning; Chromatic Polynomial; Matching Coverings, The Four Color Problem.

    • 1. Narsingh Deo; Graph Theory; Prentice-Hall, 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
  • JMMH-403

    Mathematical Statistics:

    UNIT-I

    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.

    UNIT-II

    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.

    UNIT-III

    Sampling and large sample tests, Introduction to testing of hypothesis, tests of significance for large samples, chi-square test, SQC, analysis of variance, T and F tests; Theory of estimation, characteristics of estimation, minimum variance unbiased estimator, method of maximum likelihood estimation.

    UNIT-IV

    Scatter diagram, linear and polynomial fitting by the method of least squares; linear correlation and linear regression, rank correlation, correlation of bivariate frequency distribution.

    UNIT-V

    Limit Theorems: Stochastic convergence, Bernoulli law of large numbers, Conv ergence of sequence of distribution functions, Levy-Cramer Theorems, de-Moivre 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
  • JMMH-404

    Number Theory:

    UNIT-I

    Algebraic numbers ,number fields,conjugates and discriminants,algebraic integers, integral bases,norms and traces,rings of integers, quadratic field and cyclometic fields

    UNIT-II

    Trivial factorizations, factorization in to irreducible, examples of non-unique factorization in to irreducible, prime factorization.

    UNIT-III

    Euclidean domain and Euclidean quadratic fields, consequences of unique factorization,the Ramanujan-ssNagell theorem, prime factorization of ideals, norm of an ideal, non-unique factorization in cyclometric fields

    UNIT-IV

    Lattices of dimension m, the quotient torus, Minkowiski theorem, two squares theorem, four squares theorem, the space L

    UNIT-V

    The class-group, limitness of the class group, unique factorization of elements in an extension ring, factorization of a rational prime, Minkowiski constants, class-number calculations.

    Books:

    • 1. D.A Marcus, Number Fields,Springer-Verlag,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, Springer-Verlag.
    04
  • JMMH-405

    Minor Research Project and Seminar
    04
  • Total Credits
     
    20