M.Sc. Mathematics
Semester I

COURSE CODECOURSE NAMECREDITS

JMMH101
Real Analysis:
UNITI
Definition and existence of Reimann Stieltjes integral; Properties of the integral; Integration and differentiation; Fundamental theorem of calculus; Integration of vectorvalued functions.
UNITII
Sequences and series of functions; Point wise and uniform convergence; Cauchy criterion for uniform convergence; Uniform convergence and continuity; Uniform convergence and RiemannStieltjes integration; Uniform convergence and differentiation; Weierstrass approximation theorem.
UNITIII
Power series; Algebra of power series; Uniqueness theorem for power series; Abel’s and Tauber’s theorems.
UNITIV
Functions of several variables; Linear transformation; Derivatives in an open subset of Rn; Chain rule; Partial derivatives; Interchange of the order of differentiation; Derivatives of higher orders; Taylor’s theorem.
UNITV
Inverse function theorem and Implicit function theorem (without proof); Jacobians; Extremum problems with constraints; Lagrange’s multiplier method; Differentiation of integrals.
Books:
 1. Walter Rudin, Principles of Mathematical Analysis, McGrawHill.
 2. T. M. Apostol, Mathematical Analysis, Narosa Publishing.
 3. J. White, Real Analysis, An Introduction, AddisonWesley Publishing.
04 
JMMH102
Partial Differential Equations:
UNITI
Examples of PDE. Classification, Transport Equation: Initial value Problem, Nonhomogeneous Equation. Laplace's Equation: Fundamental Solution, Mean Value Formulas, Properties of Harmonic Functions, Energy Methods.
UNITII
Heat Equation: Fundamental Solution; Mean Value Formula, Properties of Solutions, Energy Methods. Wave Equation: Solution by Spherical Means. Nonhomogeneous Equations, Energy Methods.
UNITIII
Nonlinear First Order PDEComplete Integrals, Envelopes, Characteristics; Hamilton –Jacobi Equations (Calculus of Variations, Hamilton's ODE, Legendre Transform, HopfLax Formula, Weak Solutions, Uniqueness), Conservation Laws ( RankineHugoniot condition, LaxOleinik formula, Weak Solutions, Uniqueness).
UNITIV
Representation of SolutionsSeparation of Variables, Similarity Solutions (Plane and Traveling Waves, Solitons, Similarity Linder Scaling), Fourier and Laplace Transform, HopfCole Transform, Hodograph and Legendre Transforms. Potential Functions.
UNITV
Deriving Difference Equations, Elliptic Equations: Solution of Laplace’s equation, Leibmann,s method, relaxation method, solution of Poisson’s equation, Parabolic equation: solution of heat equation, BenderSchmidt method. The CrankNicholson method, Hyperbolic equations: solution of hyperbolic equation.
 1. L.C. Evans, Partial Differential equations, Graduate Studies in Mathematics. Volume 19, AMS, 1998.
 2. P. Prasad and R. Ravindran, Partial Differential equations, New Age International Pub. F. John, Partial Differential equations, Springer Verlag.
 3. W. E. William, Partial Differential equations, Clarendon pressoxford.
 4. E. T. Copson, Partial differential equations, Cambridge university press.
 5. I.N. Sneddon, Elements of partial differential equations, McGraw Hill book company
04 
JMMH103
Operation Research:
UNITI
Introduction: Definition and scope of O.R., Different O.R. models, General methods for solving O.R. models, Main characterization and phases of O.R., Linear programming and Simplex method with simple problems, Twophase and BigM methods.
UNITII
Inventory Management: Inventory control, Types of inventories, Cost associated with inventories, Factors affecting inventory control, Single item deterministic problems with and without shortages, Inventory control with price breaks, Inventory control for one period without setup cost with uncertain demands (News paper boy type problem).
UNITIII
Sequencing Theory: Introduction, Processing with njobs and two machines, njobs and three machines, njobs and m machines, Concept of jobs blocks; Nonlinear Programming: Convex sets and convex functions, Quadratic programming, Wolfe’s complementary pivot method and Beale’s methods.
UNITIV
Queuing Theory: Introduction, Characteristics of queuing systems, Poisson process and Exponential distribution; Classification of queues, Transient and steady states; Poisson queues (M/M/1, M/M/C).
UNITV
NonPoisson Queuing systems: (M/Ek/1) queuing systems; Replacement Problems: Replacement of items that deteriorate gradually and value of money does not change with time; Replacement of items that fail suddenly, Individual and group replacement policies.
Books:
 1. S.D. Sharma Operation Research, Kedar Nath Ram Nath.
 2. H.A. Taha, Operation Research An introduction, Macmillan Publishing Company.
 3. P.K. Gupta, K. Swarup & Man Mohan, Operation Research, Sultan Chand & Co.
 4. R.L. Ackoff and N.W. Sasieni, Fundamental of Operations Research, John Willy.
04 
JMMH104
Lattices and Algebra:
UNITI
Partially ordered set, Least upper bound, Greatest lower bound, Lattice, Sublattice, Their Characterizations, Ideals in a lattice, Properties of ideals.
UNITII
Interval, Homomorphism, Isomorphism and its characterization, Zero and all elements in a lattice, Complete lattice, Modular and distributive lattices, Characterization of a modular lattice, Isomorphic, Similar and projective intervals.
UNITIII
Refinement of a chain, Schreier’s refinement theorem, JordanHolder theorem. A.C.C, and D.C.C., Fundamental dimensionality relation for modular lattice, Decomposition theory for lattices with A.C.C., Independent (join) elements in a lattice & their properties, Complemented modular lattices, Points, Properties of complemented modular lattices with chain condition.
UNITIV
Distributive and complemented lattices, (Boolean Algebra), Boolean rings, Conversion of a Boolean algebras into Boolean rings and viceversa. Algebras, Different types of algebras (Quarternions, Caley), Endomorphism, Derivation of a ring and algebras.
UNITV
Lie ring, Lie ring endomorphism of an additive abelian group, Inner derivations, Inner derivation for associative rings and Lie rings, Homomorphism of a ring onto the Lie ring of inner derivations.
Books:
 1. N. Jacobson, Lectures in Abstract Algebra
 2. Medha & Medha, Lattice Theory
 3. Grazer, General Lattice Theory
 4. S. Birkhoff, Lattice Theory
 5. Kuroosh, General Algebra
04 
JMMH105
Modern Algebra:
UNITI
Groups–Properties, Examples; subgroups, cyclic groups, homomorphism of groups and Lagrange’s theorem; permutation groups, permutations as products of cycles, even and odd permutations, normal subgroups, quotient groups, isomorphism theorems, correspondence theorem.
UNITII
Group action; Cayley's theorem, group of symmetries, dihedral groups and their elementary properties; orbit decomposition; counting formula; class equation, consequences for pgroups; Sylow’s theorems.
UNITIII
Applications of Sylow’s theorems, conjugacy classes in Sn and An, simplicity of An. Direct product; structure theorem for finite abelian groups; invariants of a finite Abelian group.
UNITIV
Basic properties and examples of ring, domain, division ring and field; direct products of rings, characteristic of a domain, field of fractions of an integral domain; ring homomorphisms (alwaysunitary); ideals, factor rings, prime and maximal ideals, principal ideal domain; Euclidean domain, unique factorization domain.
UNITV
A brief review of polynomial rings over a field; reducible and irreducible polynomials, Gauss’ theorem for reducibility of f(x) ∈ Z[x]; Eisenstein’s criterion for irreducibility of f(x) ∈ Z[x] over Q, roots of polynomials; finite fields of orders 4, 8, 9 and 27 using irreducible polynomials over Z2and Z3.
Books:
 1. I. N. Herstein, Topics in Algebra, Wiley Eastern Ltd.
 2. M. Artin, Algebra, PrenticeHall of India.
 3. N. Jacobson, Basic Algebra, Hindustan Publishing Corporation.
 4. Maclane and Birkhoff, Algebra, Macmillan Company.
 5. S. Lang Addision, Linear Algebra, Wesley.
 6. Hofmann and Kunz, Linear Algebra, Prentice Hall.
04 
Total Credits20