M.Sc (Mathematics) IV semester syllabus

DEPARTMENT OF MATHEMATICS 

 M.Sc. Mathematics  

 Semester IV                                                                                                                             (401)

 Paper-I: Integral Equations & Calculus of Variation

 INTEGRAL EQUATIONS:

 Unit I 
Volterra Integral Equations: Basic concepts - Relationship between Linear differential equations and Volterra Integral equations - Re solvent Kernel of Volterra Integral equation. Differentiation of some re solvent kernels - Solution of Integral equation by Re solvent Kernel - The method of successive approximations - Convolution type equations - Solution of Integro-differential equations with the aid of the Laplace Transformation – Volterra integral equation of the first kind - Euler integrals - Abel’s problem - Abel’s integral equation and its generalisations.
Unit II
 Fredholm Integral Equations: Fredholm integral equations of the second kind – Fundamentals – The Method of Fredholm Determinants - Iterated Kernels constructing the Resolvent Kernel with the aid of Iterated Kernels - Integral equations with Degenerated Kernels. Hammerstein type equation - Characteristic numbers and Eigen functions and its properties. Green’s function: Construction of Green’s function for ordinary differential equations - Special case of Green’s function - Using Green’s function in the solution of boundary value problem.
 CALCULUS OF VARIATIONS:
Unit III 
Introduction - The Method of Variations in Problems with fixed Boundaries: Definitions of Functional – Variation and Its properties - Euler’s’ equation - Fundamental Lemma of Calculus of Variation-The problem of minimum surface of revolution - Minimum Energy Problem Brachistochrone Problem - Variational problems involving Several functions - Functional dependent on higher order derivatives - Euler Poisson equation.
Unit-IV
 Functional dependent on the functions of several independent variables - Euler’s equations in two dependent variables - Variational problems in parametric form - Applications of Calculus of Variation - Hamilton’s principle - Lagrange’s Equation, Hamilton’s equations.
 Text Books:
[1] M. KRASNOV, A. KISELEV, G. MAKARENKO, Problems and Exercises in Integral Equations (1971).
[2] S. Swarup, Integral Equations, (2008).
[3] L.ELSGOLTS, Differential Equations and The Calculus of Variations, MIR Publishers, MOSCOW.

M 402                                                                                                                                   Semester IV Paper-II: Elementary Operator Theory

 Unit I 
Spectral theory in finite dimensional normed spaces - Basic concepts of spectrum - Spectral properties of bounded linear operators - Further properties of re solvent and spectrum. (Sections 7.1, 7.2, 7.3 & 7.4 of [1]).
Unit II
 Compact linear operators on normed spaces - Further properties of compact linear operators - Spectral properties of compact linear operators on normed spaces - Operator equations involving compact linear operators. (Sections 8.1, 8.2, 8.3 and 8.5 of [1]).
 Unit III 
Spectral properties of bounded self adjoint linear operators - Further spectral properties of bounded linear operators - Positive operators - Square root of a positive operator. (Sections 9.1, 9.2, 9.3 and 9.4 of [1])
 Unit-IV
 Projection operators - Properties of projection operators - Spectral family - Spectral family of a bounded self adjoint linear operator. (Sections 9.5, 9.6, 9.7 and 9.8 of [1])
Text Book: Introductory Functional Analysis by E. Kreyszig , John Wiley and Sons, New York, 1978.
 References: [1] Elements of Functional Analysis by Brown and Page, D.V.N. Comp.
[2] Functional Analysis by B.V. Limaye, Wiley Eastern Limited, (2nd Edition).
[3] A Hilbert Space Problem Book by P.R.Halmos, D.Van Nostrand Company, Inc. 1967.

M 403                                                                                                                             Semester IV Paper-III: Analytic Number Theory 

Unit I 
Averages of arithmetical function: The big oh notation- Asymptotic equality of functions- Euler summation formula- Some asymptotic formulas- The average order of d(n)- The average order of the divisor functions σ(n)- The average order of φ(n)- An application to the distribution of lattice points visible trona the origin The average order of µ(n) and Λ(n)- The partial sums of Dirichlet product- Applications to µ(n) and Λ(n)- Another identity for the partial sums of a dirichlet product. (Sections 3.1 to 3.12 ).
Unit II
 Some elementary theorems on the distribution of prime numbers- Introduction chebyshev’s functions- ψ(x) and θ(x)- Relation connecting θ(n) and π(n)- Some equivalent forms of the prime number theorem- Inequalities for π(n) and pn. (Sections 4.1 to 4.5)
Unit III 
Shapiro’s Tauberian theorem- Applications of Shapiro's theorem An asymptotic formula for the partial sums P p≤x 1/p - The partial sums of the mobins function- Selberg Asymptotic formula. (Sections 4.6 to 4.11 except 4.10)
 Unit-IV
 Finite Abelian groups and their character: Construction of sub groups- Characters of finite abelian group The character group- The orthogonality relations for characters Dirichlet characters- Sums involving dirichelt characters the non vanishing of L(1, χ) for real non principal χ . (Sections 6.4 to 6.10)
 Text Book: Tom M. Apostol- An Introduction to Analytic Number Theory, Springer.

M 404(B)                                                                                                                       Semester IV Paper-IV: Graph Theory 

Unit I
 Basics of Graph Theory: Graphs, isomorphism, subgraphs, matrix representations, degree, operations on graphs, degree sequences. Connected graphs and shortest paths: Walks, trails, paths, connected graphs, distance, cut-vertices, cut-edges, blocks, connectivity, weighted graphs, shortest path algorithms.
 Unit II
 Trees: Characterizations, number of trees, minimum spanning trees. Special classes of graphs: Bipartite graphs, line graphs, chordal graphs. Eulerian graphs: Characterization, Fleury’s algorithm, chinese-postman-problem. Hamilton graphs: Necessary conditions and sufficient conditions
 Unit III 
Independent sets, coverings, matchings: Basic equations, matchings in bipartite graphs, perfect matchings, greedy and approximation algorithms. Vertex colorings: Chromatic number and cliques, greedy coloring algorithm, coloring of chordal graphs, Brook’s theorem. Edge colorings: Gupta-Vizing theorem, Class-1 graphs and class-2 graphs, equitable edge-coloring.
Unit-IV
 Planar graphs: Basic concepts, Eulers formula, polyhedrons and planar graphs, characterizations, planarity testing, 5-color-theorem. Directed graphs: Out-degree, in-degree, connectivity, orientation, Eulerian directed graphs, Hamilton directed graphs, tournaments
Text Books: [1] J.A. Bondy and U.S.R. Murty: Graph Theory with Applications (Freely downloadable from Bondy’s website; Google-Bondy).
[2] D.B. West: Introduction to Graph Theory, Prentice-Hall of India/Pearson, 2009 ( latest impression).
 References: [1] J.A. Bondy and U.S.R. Murty: Graph Theory, Springer, 2008.
[2] R.Diestel: Graph Theory, Springer( low price edition) 2000

M/AM 405(B)                                                                                                                    Semester IV Paper-V: Advanced Operations Research 

Unit I
 Characteristics of Game theory – Minimax(Maxmin) criterion and optimal strategy- Saddle points - Solution of Games with saddle points- Rectangular Games without saddle points - Minimax(Maxmin) principle for Mixed strategy Games - Equivalence of Rectangular Game and Linear programming problem - Solution of (m×n) Games by Simplex method-Arithmetic method for (2×2) Games - concept of Dominance - Graphical method for (3 × 3)Games without saddle point. Unit II
 Inventory Problems: Analytical structure of inventory Problem, ABC analysis, EOQ Problems with and without shortage, with (a) Production is instantaneous (b) finite constant rate (c) shortage permitted random models where the demand follows uniform distribution.
 Unit III
 Non - Linear programming-unconstrained problems of Maxima and Minima - constrained problems of Maxima and Minima - Constraints in the form of Equations – Lagrangian Method-Sufficient conditions for Max(Min) of Objective function with single equality constraint – With more than one equality constraints - Constraints in the form of Inequalities - Formulation of Non - Linear programming problems - General Nonlinear programming problem - Canonical form - Graphical Solution
 Unit-IV
 Quadratic programming - Kuhn-Tucker Conditions - Non-negative constraints, General quadratic programming problem - Wolfe’s modified simplex method-Beales’s Method - Simplex method for quadratic Programming.
 Text Books: [1] S.D. Sharma, Operations Research.
[2] Kanti Swarup, P. K. Gupta and Manmohan, Operations Research.
 [3] O.L. Mangasarian, Non-Linear Programming, McGraw Hill, New Delhi.