M.Phil./Ph.D. (Applied Mathematics)

 

Minimum Eligibility

After 12 years of regular schooling, a 3/4 years Bachelor's degree and a Master’s degree leading to M.A./M.Sc./M.Tech. degree in Mathematics / Computer Science with at least 55% marks or an equivalent grade.

 

Or

 

After 12 years' of regular schooling, an integrated Bachelor's and Master's degrees in Mathematics / Computer Science provided that the total duration of education is of at least 5 years.

 

Degrees should be from an institution recognized by the Government of the respective SAARC countries.

 

Admission Procedure: Through an Entrance Test and an Interview.

 

Format of the Entrance Test Question Paper and Course:

The duration of the test will be three hours. The test will consist of 70 multiple choice questions of one mark each. Each question will have four options with only one correct option.There will be no negative marking.Calculators are not allowed. However Log Tables may be used.The test will be divided into the following two parts:

 

  • PART A : Undergraduate level knowledge of Mathematics – 30 Questions
 
  • PART B : Masters level knowledge of Mathematics – 40 Questions

 

Part A:

Real Analysis: Elementary set theory, real number system as a complete ordered field, Archimedean property, supremum, infimum, sequence and series, monotone sequences, convergence, limit superior, limit inferior, Bolzano Weierstrass theorem, Continuity, uniform continuity, differentiability, mean value theorems; partial derivatives and Leibnitz theorem, Sequences and series of functions, uniform convergence, power series, Riemann sums, Riemann integration, improper integrals, functions of several variables, multiple integrals, line, surface and volume integrals, theorems of Green, Stokes and Gauss, Fourier series.

Abstract Algebra: Groups and their elementary properties, order of group, subgroups, cyclic groups, cyclic subgroups, permutation groups, Lagrange's theorem, normal subgroups and quotient groups, homomorphism of groups, isomorphism and correspondence theorems; Rings, integral domains and fields, ring homomorphism and ideals; Vector space, vector subspace, linear independence of vectors, basis and dimension of a vector space.

Differential Equations: Ordinary differential. equations: First order and of higher degree, linear equations with constant coefficients, method of variation of parameters, equations reducible to linear equations with constant coefficients; Partial differential equations: Linear and quasi-linear first order partial differential equations, Lagrange and Charpits methods for solving first order partial differential equations, general solution of higher order linear partial differential equations with constant coefficients.

Numerical Analysis: Bisection, secant and Newton-Raphson methods for algebraic and transcendental equations, fixed point iteration, rate of convergence; Systems of linear equations: Gauss elimination and LU decomposition, Gauss Jacobe and Gauss Siedal methods, condition number; Numerical integration: trapezoidal and Simpsons rules, errors and their bounds and finite difference operators.

3Dimensional Geometry and Vector Calculus: Direction cosines and direction ratios, vector equation of a line, coplanar and skew lines, shortest distance between two lines, vector equation of a plane, angle between two planes, angle between a line and a plane, distance of a point from a plane; vector triple products, directional derivative, curl, divergence, gradient, Gauss- divergence theorem, Stoke's theorem and their applications.

Mechanics: Force vectors, equilibrium of a particle, force system resultants, equilibrium of a rigidbody, friction, kinematics of a particle, kinetics of a particle; Force and acceleration, work and energy, impulse and momentum, planar kinematics of a rigid body, planar kinetics of a rigid body: force and acceleration, work and energy, impulse and momentum; Common catenary, virtual work.

Linear Programming: Linear programming problem and its formulation, graphical method ofsolution, convex sets, feasible and infeasible solutions, optimal feasible solutions, simplex method, big-M and two phase methods, dual problem.

Probability and Statistics: Sample space, discrete and continuous random variables, cumulative distribution, mean and variance of random variable, expectation, moments generating functions, distribution: uniform, binomial, poisson, geometric, normal and exponential expectation of functions of two variables and conditional expectation.

 

Part B:

Linear Algebra: Algebra of matrices, rankof a matrices, systems of linear equations,eigen values and eigen vectors, minimal polynomial, Cayley-Hamilton Theroem, diagonalisation, Hermitian, Skew-Hermitian and unitary matrices; Matrix representation of linear transformations, change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms; Finite dimensional inner product spaces, orthonormal basis, Gram-Schmidt orthonormalization process, self-adjoint operators; Quadratic forms, reduction and classification of quadratic forms.

Complex Analysis: Analytic functions, Cauchy-Riemann equations, conformal mappings, bilineartransformations; complex integration, Cauchy's theorem, Liouville's theorem, maximum modulus principle, Taylor and Laurent's series, singularities, calculus of residues, Stereographic projection.

Analysis: Metric spaces, normed linear spaces, inner product spaces, Banach and Hilbert spaces,compactness, connectedness, completeness, Weierstrass approximation theorem; Functions of bounded variation, Lebesgue measure, measurable functions; Lebesgue integral, Fatou's lemma, monotone convergence theorem, dominated convergence theorem.

Abstract Algebra: Conjugacyclasses, finite abelian groups, solvable groups, subgroups and normal subgroups, Cauchy Theorem and p-groups, the Structure of Groups, Sylow's theorems and their applications; ideals, prime and maximal ideals, quotient rings, Euclidean domains, Principleideal domains and unique factorization domains; Polynomial rings and irreducibility criteria; finite fields, extensions of field.

Ordinary Differential Equations: First order ordinary differential equation (ODE), existence anduniqueness theorems and solutions of first order initial value problems, singular solution of first order ODEs, systems of linear first order ODEs; method of solution of dx/P=dy/Q=dz/R, orthogonal trajectories, solution of Pfaffian differential equations in three variables, linear second order ODEs with variable coefficients; Sturm-Liouville boundary value problems, theory of Green's function and solution of boundary value problems using Green's function, method of Laplace transforms for solving ODEs; Series solutions, Legendre and Bessel functions and their orthogonality.

Partial Differential Equations: Cauchy problem for first order PDEs, method of characteristics;second order linear equations in two variables and their classification; Method of separation of variables for Laplace, Heat and Wave equations; Cauchy, Dirichlet and Neumann problems; solutions of Laplace, wave and diffusion equations in two variables; Fourier series and Fourier transform and Laplace transform methods of solutions for the above equations; theory of Green's function and solution of above equations using Green's function.

Numerical Analysis: Numerical solution of algebraic and transcendental equations: methods forsystem of nonlinear equations, condition for convergence, rate of convergence, methods for complex roots; Iterative methods for the solution of systems of linear equations: Jacobi, Gauss-Seidel and SOR methods; eigen values of iteration matrix, determination of optimal relaxation parameter, matrix eigenvalue problems: Jacobi method, power method; interpolation: error of polynomial interpolation, Lagrange, Hermite, Newton and spline interpolations; numerical differentiation; numerical integration: quadrature formula, method based on undetermined parameters, Gauss-Legendre quadrature, composite integration, double integration; least square polynomial approximation, rational approximation; numerical solution of ordinary differential equations: initial value problems: Taylor series methods, Euler's forward and backward methods, Runge-Kutta methods.

Mechanics and Calculus of Variation: Virtual work, Generalized coordinates, Lagrange's equationsfor holonomic systems, Hamilton's canonical equations, canonical transformation, Hamilton's principle and principle of least action, two-dimensional motion of rigid bodies, Euler's dynamical equations for the motion of a rigid body about an axis; Variation of a functional, variation problems with fixed boundaries, Euler-Lagrange equation, necessary and sufficient conditions for extremum, variational methods for boundary value problems in ordinary and partial differential equations.

 
See the M.Phil./Ph.D (Applied Mathematics) Entrance Test paper for the year 2013

 

Interview: Candidates up to five times the number of seats will be short-listed for interview on the basis of their performance in the entrance test subject to a minimum cut off. Interview will carry a weightage30 marks. A minimum of 50% marks will have to be secured in both written test and interview in order to be eligible for the admission.

 

Candidates invited for interview will be given a travel subsidy (upper limit Indian Rs. 5000) towards actual travel cost by the shortest route as per instructions to be communicated to the selected candidates later. If candidates from outside India are unable to travel to New Delhi for the interview, they can seek permission for an interview through Skype.

 

A final merit list will be drawn by adding marks of the entrance test and the interview. Separate merit lists will be made for (a) candidates from all SAARC countries other than India, and (b) candidates from India. Equal number of candidates will be offered admission from these two lists, provided they secure overall qualifying marks of 50%. Up to 30% of the seats may be filled by candidates who have already secured JRF funding through a National competitive test in any of the SAARC countries.

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