Syllabus Dips Academy-Csir mathematics Part (2)

TIFR
The screening test is mainly based on mathematics covered in a reasonable B.Sc. course. The interview need not be confined to this. Algebra : Definitions and examples of groups (finite and infinite, commutative and non-commutative), cyclic groups, subgroups , homomorphisms, quotients. Definitions and examples of rings and fields. Basic facts about finite dimensional vector spaces, matrices, determinants, and ranks of linear transformations. Integers and their basic properties. Polynomials with real or complex coefficients in 1 variable. Analysis: Basic facts about real and complex numbers, convergence of sequences and series of real and complex numbers, continuity, differentiability and Riemann integration of real valued functions defined on an interval (finite or infinite), elementary functions (Polynomial functions, rational functions, exponential and log, trigonometric functions). Geometry / Topology: Elementary geometric properties of common shapes and figures in 2 and 3 dimensional Euclidean spaces (e.g. triangles, circles, discs, spheres, etc.), Plane analytic geometry (= coordinate geometry) and trigonometry. Definition and basic properties of metric spaces, examples of subsets of Euclidean spaces (of any dimension), connectedness, compactness. Convergence in metric spaces, continuity of functions between metric spaces. General : Pigeon-hole principle (box principle), induction, elementary properties of divisibility, elementary combinatorics (permutations and combanitions, binomial coefficients), elementary reasoning with graphs.

DRDO
Linear Algebra: Finite dimensional vector spaces. Linear transformations and their matrix representations, rank; systems of linear equations, eigenvalues and eigenvectors, minimal polynomial, Cayley-Hamilton theorem, diagonalization, Hermitian, Skew–Hermitian and unitary matrices. Finite dimensional inner product spaces, self – adjoint and Normal linear operators, spectral theorem, When the mind is tired, every single action requires great effort DIPS Academy /12
Quadratic forms. Complex Analysis : Analytic functions, conformal mappings, bilinear transformations, complex integration: Cauchy’s integral theorem and formula, Liouville’s theorem, maximum modulus principle, Taylor and Laurent’s series, residue theorem and applications for evaluating real integrals. Real Analysis: Sequences and series of functions, uniform convergence, power series, Fourier series, functions of several variables, maxima, minima, multiple integrals, line, surface and volume integrals, theorem of Green, Stokes and Gauss; metric spaces, completeness, Weierstrass approximation theorem, compactness. Lebesgue measure, measurable functions; Lebesgue integral, Fatou’s lemma, dominated convergence theorem. Ordinary Differential Equations : First order ordinary differential equations, existence and uniqueness theorems, systems of linear first order ordinary differential equations, linear ordinary differential of higher order with constant coefficients; linear second order ordinary differential equations with variable coefficients, method of Laplace transforms for solving ordinary differential equations, series solutions; Legendre and Bessel functions and their orthogonality, Sturm Liovullie system, Green’s functions. Algebra: Normal subgroups and homomorphisms theorems, automorphisms. Group actions, sylow’s theorems and their applications groups of order less than or equal to 20, Finite p-groups. Euclidean domains, Principal ideal domains and unique factorizations domains. Prime ideals and maximal ideals in commutative rings. Functional Analysis: Banach spaces, Hahn-Banach theorems , open mapping and closed graph theorems, principle of uniform boundedness; Hilbert spaces, orthonormal sets, Riesz representation theorem, self–adjoint, unitary and normal linear operators on Hilbert Spaces. Numerical Analysis: Numerical solution of algebraic and transcendental equations ; bisection, secant method, Newton-Raphson method, fixed point iteration, interpolation: existence and error of polynomial interpolation, Lagrange , Newton , Hermite (oscutatory) interpolations; numerical differentiation and integration. Trapezoidal and Simpson rules: Gaussian quadrature; (Gauss- Legendre and Gauss- Chebyshev), method of undetermined parameters, least square and orthonormal polynomial approximation; numerical solution of systems of linear equations; direct and iterative methods, (Jacobi, Gauss- Seidel and SOR) with convergence; matrix eigenvalue problems; Jacobi and Given’s methods, numerical solution of ordinary differential equations; initial value problems, Taylor series method , Runge-Kutta methods, predictor – corrector methods; convergence and stability. Partial Differential Equations: Linear and quasilinear first order partial differential equations, method of characteristics; second order linear equations in two variables and their classification; Cauchy Dirichilet and Neumann problems, Green’s functions; solutions of Laplace, wave and diffusion equations in two variables Fourier series and transform methods of solutions of the above equations and applications to physical problems. Mechanics: Forces in three dimensions, Poinsot central axis, virtual work, Lagrange’s equations for holonomic systems, theory of small oscillations, Hamiltonian equations. Topology: Basic concepts of topology , product topology, connectedness, compactness, countability and separation axioms, Urysohn’s Lemma, Tietze extension theorem, metrization theorems, Tychnoff theorem on compactness of product spaces. Probability and Statistics : Probability space, conditional probability , Baye’s theorem, independence , Random variables, joint and conditional distributions, standard probability distributions and their properties, expectation, conditional expectation, moments. Weak and strong law of large numbers, central limit theorem, Sampling distributions, UMVU estimators, sufficiency and consistency , maximum likelihood estimators. Testing of hypotheses, Neyman-Pearson tests, monotone likelihood ratio, likelihood ratio tests, standard parametric tests based on normal X2, t, F-distributions. Linear regression and test for linearity of regression. Interval estimation. Calculus of Variation and Integral Equaitons: Variational problems with fixed boundaries; sufficient conditions for extremum, linear integral equations of Fredholm and volterra type, their iterative solutions. Fredholm alternative.

ISI (JRF)
ISI Bangalore
General Topology: Topological spaces, Continuous functions, Connectednes, compactness, Separation Axioms. Product spaces. Complete metric spaces. Uniform continuity. Functional Analysis: Normed linear spaces, Banach spaces, Hilbert spaces, Compact operators. Knowledge of some standard examples like C[0, 1] LP[0, 1]. Continuous linear maps (linear operators). Hahn-Banach Theorem, Open mapping theorem, Open mapping theorem, closed graph theorem and the uniform boundedness principle. Real Analysis: Sequences and series, Continuity and differentiability of real valued functions of one and two real variables and applications, uniform convergence, Riemann integration. Give a lot of time to the improvement of yourself, then there is no time to criticise others DIPS Academy /13
Linear Algebra: Vector spaces, linear transformations, characteristic roots and characteristic vectors, systems of linear equations, inner product spaces, diagonalization of symmetric and Hermitian matrices, quadratic forms. Elementary number theory : Divisibility, congruence, standard arithmetic functions, permutations and combinations. Lebesgure integration: Lebesgue measure on the line, measurable functions, Lebesgue integral, convergence almost everywhere, monotone and dominated convergence theorems. Complex Analysis : Analytic functions, Cauchy’s theorem and Cauchy integral formula, maximum modulus principle, Laurent series, Singularities, Theory of residues, contour integration. Abstract Algebra : Groups, Symmetric and Alternating groups, Direct product and finite abelian groups, Sylow theorems; Rings, Polynomial rings, integral domains, Euclidean rings, fields, extension fields, roots of polynomials, finite fields. Ordinary differential Equations: First order ODE and their solutions, singular solutions, initial problems for first order ODE, general theory of homogeneous and nonhomogeneous linear differential equations.

I.I.Sc. Banglore
Real Analysis : Real valued functions of a real variable: Continuity and differentiability, sequences and series of real numbers and functions, uniform convergence, Riemann integration, fundamental theorem of integral calculus. Topology if Rn, Compactnes and connectedness. Complex Analysis: Continuity and differentiability, analytic functions, Cauchy’s theorem, Cauchy’s integral formula, Taylor and Maclaurin expansions, Laurent’s series, singularities, theory of residues and contour integral, conformal mappings. Linear Algebra: Vector Spaces: Linear independence, basis, dimension, linear transformations, matrices, systems of linear equations, rank and nullity, characteristic values and characteristic vectors, Cayley-Hamilton characteristic and minimal