ENGINEERING MATHEMATICS
Mathematical Logic: Propositional Logic;
First Order Logic.
Probability: Conditional Probability; Mean,
Median, Mode and Standard Deviation; Random
Variables; Distributions; uniform, normal,
exponential, Poisson, Binomial.
Set Theory & Algebra: Sets; Relations;
Functions; Groups; Partial Orders; Lattice;
Boolean Algebra.
Combinatorics: Permutations; Combinations;
Counting; Summation; generating functions;
recurrence relations; asymptotics.
Graph Theory: Connectivity; spanning trees;
Cut vertices & edges; covering;
matching; independent sets; Colouring;
Planarity; Isomorphism.
Linear Algebra: Algebra of matrices,
determinants, systems of linear equations,
Eigen values and Eigen vectors.
Numerical Methods: LU decomposition for
systems of linear equations; numerical
solutions of non linear algebraic equations
by Secant, Bisection and Newton-Raphson
Methods; Numerical integration by
trapezoidal and Simpson's rules.
Calculus: Limit, Continuity &
differentiability, Mean value Theorems,
Theorems of integral calculus, evaluation of
definite & improper integrals, Partial
derivatives, Total derivatives, maxima &
minima.
FORMAL LANGUAGES AND AUTOMATA
Regular Languages: finite automata, regular
expressions, regular grammar.
Context free languages: push down automata,
context free grammars
COMPUTER HARDWARE
Digital Logic: Logic functions,
minimization, design and synthesis of
combinatorial and sequential circuits,
number representation and computer
arithmetic (fixed and floating point)
Computer organization: Machine instructions
and addressing modes, ALU and data path,
hardwired and microprogrammed control,
memory interface, I/O interface (interrupt
and DMA mode), serial communication
interface, instruction pipelining, cache,
main and secondary storage
SOFTWARE SYSTEMS
Data structures and Algorithms: the notion
of abstract data types, stack, queue, list,
set, string, tree, binary search tree, heap,
graph, tree and graph traversals, connected
components, spanning trees, shortest paths,
hashing, sorting, searching, design
techniques (greedy, dynamic, divide and
conquer), asymptotic analysis (best, worst,
average cases) of time and space, upper and
lower bounds, intractability
Programming Methodology: C programming,
program control (iteration, recursion,
functions), scope, binding, parameter
passing, elementary concepts of object
oriented programming
Operating Systems (in the context of Unix):
classical concepts (concurrency,
synchronization, deadlock), processes,
threads and interprocess communication, CPU
scheduling, memory management, file systems,
I/O systems, protection and security
Information Systems and Software
Engineering: information gathering,
requirement and feasibility analysis, data
flow diagrams, process specifications,
input/output design, process life cycle,
planning and managing the project, design,
coding, testing, implementation,
maintenance.
Databases: relational model, database
design, integrity constraints, normal forms,
query languages (SQL), file structures
(sequential, indexed), b-trees, transaction
and concurrency control
Data Communication: data encoding and
transmission, data link control,
multiplexing, packet switching, LAN
architecture, LAN systems (Ethernet, token
ring), Network devices: switches, gateways,
routers
Networks: ISO/OSI stack, sliding window
protocols, routing protocols, TCP/UDP,
application layer protocols & systems
(http, smtp, dns, ftp), network security
Web technologies: three tier web based
architecture; JSP, ASP, J2EE, .NET systems;
html, XML
MA -
MATHEMATICS
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, diagonalisation, Hermitian, Skew-Hermitian
and unitary matrices. Finite dimensional
inner product spaces, self-adjoint and
Normal linear operators, spectral theorem,
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, theorems 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 equations 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 Liouville system,
Greeen'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(osculatory)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, Dirichlet 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, Tychonoff
theorem on compactness of product spaces.
Probability and Statistics: Probability
space, conditional probability, Bayes'
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.
Linear Programming: Linear programming
problem and its formulation, convex sets
their properties, graphical method, basic
feasible solution, simplex method, big-M and
two phase methods, infeasible and unbounded
LPP's, alternate optima. Dual problem and
duality theorems, dual simplex method and
its application in post optimality analysis,
interpretation of dual variables. Balanced
and unbalanced transportation problems,
unimodular property and u-v method for
solving transportation problems. Hungarian
method for solving assignment problems.
Calculus of Variations and Integral
Equations: Variational problems with fixed
boundaries; sufficient conditions for
extremum, Linear integral equations of
Fredholm and Volterra type, their iterative
solutions. Fredholm alternative.
|