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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