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Syllabus
RT
Ring Theory (Syllabus)
RING THEORY
UNIT-I
Definition and examples of rings, properties of rings, subrings, integral domains and fields, characteristic of a ring, Ideals, ideal generated by a subset of a ring, factor rings, operations on ideals.
UNIT-II
Prime and maximal ideals. Ring homomorphisms, properties of ring homomorphisms, Isomorphism theorems I, II and III, field of quotients.
UNIT-III
Polynomial rings over commutative rings, division algorithm and consequences, principal ideal domains, factorization of polynomials, reducibility tests, irreducibility tests, Eisenstein criterion, Unique factorization in Z[x].
UNIT-IV
Divisibility in integral domains, irreducibles, primes, unique factorization domains, Euclidean domains.
BOOKS RECOMMENDED :
1. Joseph A. Gallian, Contemporary Abstract Algebra (4th Edition), Narosa Publishing House, New Delhi.
2. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002.
BOOK FOR REFERENCES:
1. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011.
2. Joseph 1. Rotman, An Introduction to the Theory of Groups, 4th Ed., Springer Verlag, 1995.
3. I. N. Herstein, Topics in Algebra, Wiley Eastern Limited, India, 1975.
TMS
Topology of Metric Spaces (Syllabus)
TOPOLOGY OF METRIC SPACES
UNIT-I
Metric spaces, sequences in metric spaces, Cauchy sequences, complete metric spaces, open and closed balls, neighborhood, open set, interior of a set, limit point of a set, closed set, diameter of a set, Cantor's theorem.
UNIT-II
Subspaces, Countability Axioms and Separability, Baire’s Category theorem.
UNIT-III
Continuity: Continuous mappings, Extension theorems, Real and Complex valued Continuous functions, Uniform continuity, Homeomorphism, Equivalent metrics and isometry, uniform convergence of sequences of functions.
UNIT-IV
Contraction mappings and applications, connectedness, Local connectedness, Bounded sets and compactness, other characterization of compactness, continuous functions on compact spaces.
NMSC
Numerical Methods and Scientific Computing (Syllabus)
NUMERICAL METHODS AND SCIENTIFIC COMPUTING
UNIT-I
Rate of convergence, Algorithms, Errors: Relative, Absolute, Round off, Truncation. Approximations in Scientific computing, Error propagation and amplification, conditioning, stability and accuracy, computer arithmetic mathematical software and libraries, visualisation, Numerical solution of non-linear equations: Bisection method, Regula- Falsi method, Secant method, Newton- Raphson method, Fixed-point Iteration method.
UNIT-II
Rate of convergence of the above methods. System of linear algebraic equations: Gaussian Elimination and Gauss Jordan methods. Gauss Jacobi method, Gauss Seidel method and their convergence analysis. Computing eigen-values and eigenvectors.
UNIT-III
Polynomial interpolation: Existence uniqueness of interpolating polynomials. Lagrange and Newtons divided difference interpolation, Error in interpolation, Central difference & averaging operators, Gauss-forward and backward difference interpolation. Hermite and Spline interpolation, piecewise polynomial interpolation.
UNIT-IV
Numerical Integration: Some simple quadrature rules, Newton-Cotes rules, Trapezoidal rule, Simpsons rule, Simpsons 3/8th rule, Numerical differentiation and integration, Chebyshev differentiation and FFT, Richard-son extrapolation.