Semester Courses of M A M Sc Mathematics

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Second Semester,Compulsory Papers,Paper I Fields and Modules. Paper II Differential Geometry of Manifolds,Paper III Partial Differential Equations. Paper IV Operations Research,Paper V Hydro Dynamics. Viva Voce and Project Work 50 marks, There will be a Viva Voce and Project Work examination of 50 marks in each semester based. on all the papers of respective semester Under the project the candidate will present a. dissertation in his her own handwriting The dissertation will consist of at least one. theorem article with proof and one problem with solution relevant definitions with examples. and or counter examples wherever necessary from each paper of Mathematics studied in. First and Second semesters,Distributions of marks are as follows.
1 Attendance To be verified by Head 05 marks a 91 100 05marks. in case of University and by Principal b 81 90 04marks. in case of college c 75 80 03marks,2 Presentation of at least 15 minutes 10 marks. to be evaluated by internal examiner, 2 Class test in each paper 10 marks Average of all five tests to be. 3 Dissertation 05 marks,4 Viva Voce 20 marks, For viva voce examination and evaluation of project work there will be a board of. examiners consisting of an external examiner and an internal examiner The dissertation will. be forwarded by the Head of Department at the university centre and by the Principal of the. college at the college centre,M A M Sc Semester I,Paper I Groups and Canonical Forms. Groups Conjugacy relation Normaliser of an element Class equation of a finite group. Center of a group Fundamental theorems on isomorphism of groups Automorphisms Inner. automorphism Maximal subgroups Composition series Jordan Holder theorem Solvable. groups Nilpotent groups Commutator subgroups External and iternal direct product of. groups Cauchy s theorem for finite group Sylow s theorem. 4 questions, Canonical forms Similarity of linear transformations Invariant subspaces Reduction to.
triangular forms Nilpotent transformations Index of nilpotency Invariants of a nilpotent. transformation The primary decomposition theorem Jordan blocks and Jordan forms. 2 questions,Books Recommended, 1 I N Herstein Topics in Algebra Wiley Eastern Ltd New Delhi. 2 P B Bhattacharya S K Jain and S R Nagpaul Basic Abstract Algebra Second. Edition Cambridge University Press Indian Edition, 3 Surjeet Singh and Qazi Zameeruddin Modern Algebra Vikas Publishing House Pvt. 4 K B Datta Matrix and Linear Algebra Prentice Hall of India Pvt Ltd New Delhi. 5 S Kumaresan Linear Algebra A Geometric Approach Prentice Hall of India. 6 A R Vasishtha A K Vasishtha Modern Algebra Krishna Prakashan Media P. Ltd Meerut, 7 H K Pathak Abstract Algebra Shiksha Sahitya Prakashan. Paper II Topology, Definition and examples of topological spaces Closed sets Closure Dense subsets. Neighbourhoods Interior exterior and boundary Accumulation points and derived sets. Bases and sub bases Subspaces and relative topology Alternate methods of defining a. topology in terms of Kuratowski Closure operator and Neighbourhood Systems. Continuous functions and homeomorphism First and second countable spaces Lindelof s. theorems Separable spaces Second Countability and Separability. 3 questions, Separation axioms T0 T1 T2 T3 T4 their characterizations and basic properties Urysohn.
Lemma Tietz extention theorem, Compact sets and their properties Finite intersection property Bolzano Weierstrass. property Continuous functions and compactness Sequential compactness countable. compactness and their comparison One point compactification Connected spaces. Connectedness on the real line Components Locally connected Spaces. 3 questions,Books Recommended, 1 George F Simmons Introduction to Topology and Modern Analysis Mc Graw Hill. Book Company, 2 J L Kelley General Topology Van Nostrand Reinhold Co New York. 3 K D Joshi Introduction to General Topology Wiley Eastern Ltd. 4 James R Munkres Topology Prentice Hall of India Pvt Ltd New Delhi. 5 Willard General Topology Addison Wesley Reading,Paper III Differential and Integral Equations. Solution of Differential Equations in ascending and descending power series. Hypergeometric Differential Equations Papperitz symbol Pochhammer symbol. Hypergeometric Function Solution of Gauss s Hypergeometric Differential Equation. Differentiation of Hypergeometric Functions, Legendre s Differential Equation Legendre s Functions Generating Function for x.
Laplace Definite Integrals for x Orthogonal Properties of Legendre s Polynomials. Recurrence Formulae Beltrami Result Christoffel s Expansion and Summation formulae. Rodrigue s Formula for x, Bessel s Differential Equation Bessel s Functions Generating Function for x. Differential Equations Reducible to Bessel s Differential Equations Orthogonality of. Bessel s Functions,3 questions, Integral Equation Linear Integral Equations Types of Linear Integral Equations Types of. Kernels Conversion of differential equations to integral equations 2 kernels and 2. Functions Eigen values and eigen functions Solution of Volterra Integral Equations by. Successive Approximations and Successive Substitution Methods. Fredholm Integral Equations of First and Second kinds Solution of Fredholm Integral. Equations by Successive Approximations and Successive Substitution Methods Neumann. Series Volterra solution of Fredholm Integral Equation of second kind Reduction of. Volterra Integral Equation into differential equation Reduction of Volterra Integral Equation. of first kind into Volterra Integral Equation of second kind. 3 questions,Books Recommended, 1 Differential and Integral Equations by B P Parashar. 2 Series Solution and Special Functions by V S Verma. 3 Fundamentals of Integral Equations by V S Verma,Paper IV Complex Analysis. Analytic continuation Uniqueness of analytic continuation Power series method of analytic. continuation Branches of many valued function Singularities of an analytic function. Riemann surfaces Gamma function Zeta Function Principle of reflection Hadamard s. multiplication theorem Functions with natural boundaries. 3 questions, Maximum modulus theorem Schwarz s lemma Vitali s convergence theorem Hadamard s.
three circles theorem Mean values of f z Borel Cartheodory theorem Pharagmen. Lindelof theorem, Conformal representation Linear bilinear transformations involving circles and half planes. Transformations w z2 w z 1 z 2 w log z w tan2 z 2 Simple function and its. properties The 1 4 theorem, Radius of convergence of the power series Analyticity of sum of power series Position of. the singularities,3 questions,Books Recommended, 1 E C Titchmarsh Theory of Functions Oxford University Press London. 2 Mark J Ablowitz and A S Fokas Complex Variables Introduction and Applications. Cambridge University Press South Asian Edition 1998. 3 R V Churchill J W Brown Complex Variables and Applications 5th Edition. McGraw Hill New York 1990, 4 Shanti Narayan Theory of Functions of a Complex Variable S Chand Co New. 5 H K Pathak Complex Analysis Shiksha Sahitya Prakashan. Paper V Real Analysis, Definition and Existence of Riemann Stieltjes integrals Properties of the integral.
integration and differentiation the fundamental theorem of calculus integration of vector. valued functions Rectifiable curves Rearrangements of terms of a series Riemann s. theorem Sequences and series of functions of real numbers Pointwise convergence and. uniform convergence Cauchy Criterion of uniform convergence Weierstrass M test Abel s. and Dirichlet s tests for uniform convergence Uniform convergence and continuity Uniform. convergence and integration Uniform convergence and differentiation Weierstrass. approximation theorem Power series uniqueness theorem for power series Abel s and. Tauber s theorem,4 questions, Functions of several variables linear transformations Derivatives in an open subset of R n. Chain rule Partial derivatives interchange of the order of differentiation derivative of. higher orders Taylor s theorem Inverse function theorem Implicit function theorem. Jacobians extremum problems with constraints Lagrange s multiplier method. 2 questions,Books Recommended, 1 Walter Rudin Principles of Mathematical Analysis 3rd edition McGraw Hill. Kogakusha 1976 International Student Edition, 2 H L Royden Real Analysis Macmillan Pub Co Inc New York 4th Edition 1993. 3 Richard Johnson Baugh Foundation of Mathematical Analysis. 4 H K Pathak Real Analysis Shiksha Sahitya Prakashan. M A M Sc Semester II,Paper I Fields and Modules, Field theory Extension fields Algebraic and transcendental extensions Splitting field. Separable and inseparable extensions Normal extension Perfect fields Finite fields. Antomorphisms of extensions Galois group Fundamental theorem of Galois theory. Construction with ruler and compass Solution of polynomial equations by radicals. Insolvability of the general equation of degree 5 by radicals. 4 questions, Modules Cyclic modules Simple modules Semi simple modules Schuler s lemma Free.
modules Noetherian and artinian modules Hilbert basis theorem. 2 questions,Books Recommended, 1 I N Herstein Topics in Algebra Wiley Eastern Ltd New Delhi. 2 P B Bhattacharya S K Jain and S R Nagpaul Basic Abstract Algebra Second. Edition Cambridge University Press Indian Edition, 3 Surjeet Singh and Qazi Zameeruddin Modern Algebra Vikas Publishing House Pvt. 4 K B Datta Matrix and Linear Algebra Prentice Hall of India Pvt Ltd New Delhi. 5 S Kumaresan Linear Algebra A Geometric Approach Prentice Hall of India. 6 A R Vasishtha A K Vasishtha Modern Algebra Krishna Prakashan Media P. Ltd Meerut, 7 H K Pathak Abstract Algebra Shiksha Sahitya Prakashan. Paper II Differential Geometry of Manifolds, Definition and examples of differentiable manifold Differentiable functions Differentiable. curves Tangent space Vector fields Lie bracket Invariant view point of connections. Covariant differentiation Torsion Curvature Parallelism Difference tensor of two. connections Lie derivative,3 questions, Riemannian Manifold Riemannian connection Riemannian curvature tensor and Ricci.
tensor Idenitities of Bianchi Sectional curvature, Exterior product of two vectors Exterior algebra of order r Exterior derivative Cartans s. structural equations Submanifolds Normals Induced connection Gauss formulae. Weingarten formulae Lines of curvature Mean curvature Equations of Gauss and Codazzi. 3 questions,Books Recommended, 1 B B Sinha An Introduction to Modern Differential Geometry Kalyani Publishers New. 2 N J Hickls Notes on Differential Geometry, 3 K Yano and M Kon Structure of Manifolds World Scientific Publishing Co Pvt. Paper III Partial Differential Equations, Partial Differential Equations of the First Order Origin of first order partial differential. equations Lagrange s solution of first order linear partial differential equation Non linear. partial differential equations of the first order Cauchy s method of characteristics Charpit s. method and Jacobi s method,3 questions, Partial Differential Equations of Second and Higher Orders Origin of second order.
partial differential equations Higher order partial differential equations with constant. coefficients Equations with variable coefficients Classification of second order partial. differential equations Canonical forms Solution of non linear second order partial. differential equations by Monge s method Method of separation of variables for solving. Laplace wave and diffusion equations,3 questions,Books Recommended. 1 V S Verma A Text Book of Partial Differential Equations. 2 A R Forsyth A Treatise on Differential Equations. 3 I N Sneddon Elements of Partial Differential Equations. Paper IV Operations Research, Origin and development of OR Objective nature definition and scope of OR Phases of. OR Methods, Network analysis Basic concepts and definition Network drawing and analysis Critical. path method Labelling method Methods based on time estimates to find critical path. Concept of slack and float Resource levelling and time cost trade off analysis Time cost. optimization procedure Project crashing PERT Requirements for application of PERT. technique Practical limitations in using PERT Diffferences in PERT and CPM Shortest path. problem Minimum spanning tree problem Maximum flow problem Minimum cost flow. 3 questions, Sequencing Problems Assumptions for seqnencing problem Processing n jobs on two. machines n jobs on three machines 2 jobs on m machines. Non Linear Programming Introduction and defintions Formulation of non Linear. programming problems General non linear programming problems Constrained. optimization with equality constraints Constrained optimization with inequality constaints. Saddle point problems Saddle points and NLPP, Goal Programming Introduction and definition Concept of goal programming Difference.
between linear programming approach and goal programming approach Goal programming. model formulation Methods of solution of goal programming problem Graphical method. and Simplex method,3 questions,Books Recommended, 1 H A Taha Operations Research An Introduction Macmillan Publishing Co Inc. 2 Kanti Swarup P K Gupta Man Mohan Operations Research Sultan Chand and. Sons New Delhi, 3 B S Goel S K Mittal Operations Research Pragati Prakashan Meerut. 1 Semester Courses of M A M Sc Mathematics The course of M A M Sc Mathematics will be spread in two years Previous amp Final There will be four semester examinations and a viva voce amp project work examination

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