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Topics covered for ARAMBH : Mathematical Science Batch for CSIR-UGC NET June 2022

Mathematical Sciences

Common Subject

Linear Algebra

Matrices

Diagonalizability & Canonical Forms

Linear Equations

Linear Transformations

Polynomials

Determinants

Elementary Canonical Forms

Rational and Jordan Forms

Inner Product Spaces

Operators on Inner Product Spaces

Bilinear Forms

Numerical Analysis

Soln. of Algebraic & Trans Eqn

Interpolation and Approximation

Numerical Differentiation

Initial Value Problems for ODE

Boundary Value Problems-ODE & PDE

Linear System of Algebraic Eqn

Numerical Integration

Complex Analysis

Preliminaries

Bilinear Transformation & Mapping

Elementary Functions

Analytic Functions

Power Series

Complex Integration

Applications of Cauchy's Theorem

Laurent Series & Residue Theorem

Harmonic Functions

Riemann Mapping Theorem

Entire and Meromorphic Functions

Analytic Continuation

Max. & Min. Modulus Principle & Schwarz Lemma

Fundamental Concepts of Complex Analysis

Stereographic Projection & Point Set Topology

Limit, Continuity & Differentiability

Singularites of Analytic Functions

Important Theorems & Applications

Taylor and Laurent Expansion

Special Functions:The Exponential

Argument and Rouche’s Theorem

Calculus of Residues

Conformal Mapping

Descriptive Statistics

Introduction Meaning and Scope

Frequency Distributions

Measures of Central Tendency

Measures of Dispersion

Skewness and Kurtosis

Theory of Probability

Random Variables

Mathematical Expectation

Discrete Distributions

Continuous Distributions

Curve Fitting

Correlation and Regression

Theory of Attributes

Sampling and Large Sample Tests

Chi-Square Distribution

t,F and z distributions

Theory of Estimation

Statistical Inference-II

Probability

Special Univariate Distributions

Joint & Conditional Distributions

Inference - Parameter Estimation

Maximum Likelihood Estimation

Hypothesis Testing

Elements of Bayesian Inference

Markov Chains

Miscellaneous

Ordinary Differential Equation

Differential Equations

Equations of First Order & Degree

Trajectories

Sol. of 1st Order & Higher Degree

LDE with Constant Coefficients

Homogeneous Linear Equations

Method of Variation of Parameters

Ordinary Simultaneous DE

Exact Differential Equations

Equations of Special Forms

Linear Equations of Second Order

Picard’s Method & Existence

Simultaneous Equations

Partial Differential Equations

Riccati’s Equation

Chebyshev Polynomials

Beta and Gamma Functions

Power Series

Integration in Series

Legendre Polynomials

Legendre Functions of Second Kind

Bessel Functions

Orthogonal Sets of Functions

Introduction and Preliminaries

First Order First Degree Differential Equation

General Theory of Linear Differential Equation of Higher Order

LDE Solution with Const. & Var. Coefficients

Uniqueness and Existence

System of Differential Equations

Boundary Value Problems

Equtn of the 1st Order but Not of The 1st Degree

Partial Differential Equation

Origin of PDE

Linear PDE of Order One

Non-linear PDE of Order One

Linear PDE with Constant Coeff.

PDE Reducible to Equations

2nd Order Variable Coeff. PDE

PDE into Cononial/Normal Form

Monge’s Method

Classification & Formation of PDE

Heat, Wave & Laplace Equation

Classification of Second Order PDE

Linear Integral Equations

Fredholm and Volterra Type

Sol. with Separable Kernels

Characteristic Numbers

Eigen Functions

Resolvent Kernel

Volterra’s Integral Equation

Fredholm Integral Equation

Hilbert Schmidt Theory

Integral Equation

Calculus of Variations

Variation of a Functional

Euler-Lagrange Equation

Sufficient Conditions for Extrema

Boundary Value Problems in PDE

Miscellaneous

Real Analysis

Countability

Point Set Topology

Sequence

Series

Function and its Property

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