DOWNLOAD PG TRB MATHS Syllabus PDF – 2019

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DOWNLOAD PG TRB  MATHS Syllabus PDF – 2019

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                Syllabus: MATHS (Subject Code:
P03)

Unit-I – Algebra

Groups –
Examples – Cyclic Groups- Permulation Groups – Lagrange’s theorem- Cosets – Normal
groups – Homomorphism – Theorems – Cayley’s theorem – Cauchy’s Theorem – Sylow’s
theorem – Finitely Generated Abelian Groups – Rings- Euclidian Rings-
Polynomial Rings- U.F.D. – Quotient – Fields of integral domains- Ideals-
Maximal ideals – Vector Spaces – Linear independence and Bases – Dual spaces –
Inner product spaces – Linear transformation – rank – Characteristic roots of matrices
– Cayley Hamilton Theorem – Canonical form under equivalence – Fields –
Characteristics of a field – Algebraic extensions – Roots of Polynomials –
Splitting fields – Simple extensions – Elements of Galois theory- Finite
fields.

Unit-II – Real Analysis

Cardinal
numbers – Countable and uncountable cordinals – Cantor’s diagonal process –
Properties of real numbers – Order – Completeness of R-Lub property in R-Cauchy
sequence – Maximum and minimum limits of sequences – Topology of R.Heine Borel
– Bolzano Weierstrass – Compact if and only if closed and bounded – Connected
subset of R-Lindelof’s covering theorem – Continuous functions in relation to
compact subsets and connected subsets- Uniformly continuous function –
Derivatives – Left and right derivatives – Mean value theorem – Rolle’s
theorem- Taylor’s theorem- L’ Hospital’s Rule – Riemann integral – Fundamental
theorem of Calculus –Lebesgue measure and Lebesque integral on R’Lchesque
integral of Bounded Measurable function – other sets of finite measure –
Comparison of Riemann and Lebesque integrals – Monotone convergence theorem –
Repeated integrals.

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Unit-III – Fourier series and Fourier Integrals

Integration
of Fourier series – Fejer’s theorem on (C.1) summability at a point –
Fejer’s-Lebsque theorem on (C.1) summability almost everywhere – Riesz-Fisher
theorem – Bessel’s inequality and Parseval’s theorem – Properties of Fourier
co-efficients – Fourier transform in L (-D, D) – Fourier Integral theorem –
Convolution theorem for Fourier transforms and Poisson summation formula.

Unit-IV – Differential Geometry

Curves in
spaces – Serret-Frenet formulas – Locus of centers of curvature – Spherical
curvature – Intrinsic equation – Helices – Spherical indicatrix surfaces –
Envelope – Edge of regression – Developable surfaces associated to a curve –
first and second fundamental forms – lines of curvature – Meusnieu’s theorem –
Gaussian curvature – Euler’s theorem – Duplin’s Indicatrix – Surface of
revolution conjugate systems – Asymptritic lines – Isolmetric lines –
Geodesics.

Unit-V – Operations Research

Linear
programming – Simplex Computational procedure – Geometric interpretation of the
simplex procedure – The revised simplex method – Duality problems – Degeneracy
procedure – Peturbation techniques – integer programming – Transportation
problem – Non-linear programming – The convex programming problem – Dyamic
programming – Approximation in function space, successive approximations – Game
theory – The maximum and minimum principle – Fundamental theory of games – queuing
theory / single server and multi server models (M/G/I), (G/M/I), (G/G1/I)
models, Erlang service distributions cost Model and optimization – Mathematical
theory of inventory control – Feed back control in inventory management –
Optional inventory policies in deterministic models – Storage models – Damtype
models – Dams with discrete input and continuous output – Replacement theory –
Deterministic Stochostic cases – Models for unbounded horizons and uncertain
case – Markovian decision models in replacement theory – Reliability – Failure
rates – System reliability – Reliability of growth models – Network analysis –
Directed net work – Max flowmin cut theorem – CPM-PERT – Probabilistic
condition and decisional network analysis.

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Unit-VI – Functional Analysis

Banach
Spaces – Definition and example – continuous linear transformations – Banach
theorem – Natural embedding of X in X – Open mapping and closed graph theorem –
Properties of conjugate of an operator – Hilbert spaces – Orthonormal bases –
Conjugate space H – Adjoint of an operator – Projectionsl as a Hilbert space –
lp space – Holders and Minkowski inequalities – Matrices – Basic operations of matrices
– Determinant of a matrix – Determinant and spectrum of an operator – Spectral
theorem for operators on a finite dimensional Hilbert space – Regular and
singular elements in a Banach Algebra – Topological divisor of zero – Spectrum
of an element in a Branch algebra – the formula for the spectral radius radical
and semi simplicity.

Unit-VII – Complex Analysis

Introduction
to the concept of analytic function – limits and continuity – analytic
functions – Polynomials and rational functions elementary theory of power
series – Maclaurin’s series – uniform convergence power series and Abel’s limit
theorem – Analytic functions as mapping – conformality arcs and closed curves –
Analytical functions in regions – Conformal mapping – Linear transformations –
the linear group, the cross ratio and symmetry – Complex integration – Fundamental
theorems – line integrals – rectifiable arcs – line integrals as functions of
arcs – Cauchy’s theorem for a rectangle, Cauchy’s theorem in a Circular disc,
Cauchy’s integal formula – The index of a point with respect to a closed curve,
the integral formula – higher derivatives – Local properties of Analytic
functions and removable singularitiesTaylor’s theorem – Zeros and Poles – the
local mapping and the maximum modulus Principle.

Unit-VIII – Differential Equations

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Linear
differential equation – constant co-efficients – Existence of solutions –
Wrongskian – independence of solutions – Initial value problems for second
order equations – Integration in series – Bessel’s equation – Legendre and
Hermite Polynomials – elementary properties – Total differential equations –
first order partial differential equation – Charpits method.

Unit-IX – Statistics – I

Statistical
Method – Concepts of Statistical population and random sample – Collections and
presentation of data – Measures of location and dispersion – Moments and
shepherd correction – cumulate – Measures of skewness and Kurtosis – Curve
fitting by least squares – Regression – Correlation and correlation ratio –
rank correlation – Partial correlation – Multiple correlation coefficient –
Probability Discrete – sample space, events – their union – intersection etc. –
Probability classical relative frequency and axiomatic approaches – Probability
in continuous probability space – conditional probability and independence –
Basic laws of probability of combination of events – Baye’s theorem –
probability functions – Probability density functions – Distribution function –
Mathematical Expectations – Marginal and conditional distribution – Conditional
expectations.

Unit-X – Statistics-II

Probability distributions – Binomial, Poisson, Normal, Gama, Beta, Cauchy, Multinomial Hypergeometric, Negative Binomial – Chehychev’s lemma (weak) law of large numbers – Central limittheorem for independent identical variates, Standard Errors – sampling distributions of t, F and Chi square – and their uses in tests of significance – Large sample tests for mean and proportions – Sample surveys – Sampling frame – sampling with equal probability with or without replacement – stratified sampling – Brief study of two stage systematic and cluster sampling methods – regression and ratio estimates – Design of experiments, principles of experimentation – Analysis of variance – Completely randomized block and latin square designs.

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