Teaching Aptitude

Teaching: Concept, Objectives, Levels of teaching (Memory, Understanding and Reflective), Characteristics and basic requirements.

Learner's characteristics: Characteristics of adolescent and adult learners(Academic, Social, Emotional and Cognitive), Individual differences.

Factors affecting teaching related to: Teacher, Learner, Support material,Instructional facilities, Learning environment and Institution.

Methods of teaching in Institutions of higher learning: Teacher centred vs.Learner centred methods; Off-line vs. On-line methods (Swayam,Swayamprabha, MOOCs etc.).

Teaching Support System: Traditional, Modern and ICT based.

Evaluation Systems: Elements and Types of evaluation, Evaluation in Choice Based Credit System in Higher education, Computer based testing, Innovations in evaluation systems.

Research Aptitude

Research: Meaning, Types, and Characteristics, Positivism and Post- positivistic approach to research.

Methods of Research: Experimental, Descriptive, Historical, Qualitative and Quantitative methods.

Steps of Research.

Thesis and Article writing: Format and styles of referencing.

Application of ICT in research.

Research ethics.

Comprehension

A passage of text be given. Questions be asked from the passage to be answered.

Communication

Communication: Meaning, types and characteristics of communication.

Effective communication: Verbal and Non-verbal, Inter-Cultural and group communications, Classroom communication.

Barriers to effective communication.

Mass-Media and Society.

Mathematical Reasoning and Aptitude

Types of reasoning.

Number series, Letter series, Codes and Relationships.

Mathematical Aptitude (Fraction, Time & Distance, Ratio, Proportion and Percentage, Profit and Loss, Interest and Discounting, Averages etc.).

Data Interpretation

Sources, acquisition and classification of Data.

Quantitative and Qualitative Data.

Graphical representation (Bar-chart, Histograms, Pie-chart, Table-chart and Line-chart) and mapping of Data.

Data Interpretation.

Data and Governance.

Logical Reasoning

Understanding the structure of arguments: argument forms, structure of categorical propositions, Mood and Figure, Formal and Informal fallacies, Uses of language, Connotations and denotations of terms, Classical square of opposition.

Evaluating and distinguishing deductive and inductive reasoning.

Analogies.

Venn diagram: Simple and multiple use for establishing validity of arguments.

Indian Logic: Means of knowledge.

Pramanas: Pratyaksha (Perception), Anumana (Inference), Upamana (Comparison), Shabda (Verbal testimony), Arthapatti (Implication) and Anupalabddhi (Non-apprehension).

Structure and kinds of Anumana (inference), Vyapti (invariable relation), Hetvabhasas (fallacies of inference).

Information and Communication Technology (ICT)

ICT: General abbreviations and terminology.

Basics of Internet, Intranet, E-mail, Audio and Video-conferencing.

Digital initiatives in higher education.

ICT and Governance.

Higher Education System

Institutions of higher learning and education in ancient India.

Evolution of higher learning and research in Post Independence India.

Oriental, Conventional and Non-conventional learning programmes in India.

Professional, Technical and Skill Based education.

Value education and environmental education.

Policies, Governance, and Administration.

People, Development and Environment

Development and environment: Millennium development and Sustainable development goals.

Human and environment interaction: Anthropogenic activities and their impacts on environment.

Environmental issues: Local, Regional and Global; Air pollution, Water pollution, Soil pollution, Noise pollution, Waste (solid, liquid, biomedical, hazardous, electronic), Climate change and its Socio-Economic and Political dimensions.

Impacts of pollutants on human health.

Natural and energy resources: Solar, Wind, Soil, Hydro, Geothermal, Biomass, Nuclear and Forests.

Natural hazards and disasters: Mitigation strategies.

Environmental Protection Act (1986), National Action Plan on Climate Change, International agreements/efforts -Montreal Protocol, Rio Summit, Convention on Biodiversity, Kyoto Protocol, Paris Agreement, International Solar Alliance.

1. Basic Concepts of Real and Complex Analysis

Sequences and series, Continuity, Uniform continuity, Differentiabilty, Mean Value Theorem, Sequences and series of functions, Uniform convergence, Riemann integral - definition and simple properties

Algebra of complex numbers, Analytic functions, Cauchy’s Theorem and integral formula, Power series, Taylor’s and Laurent’s series, Residues, Contour integration.

2. Basic Concepts of Linear Algebra

Space of n-vectors

Linear dependence

Basis, Linear transformation

Algebra of matrices

Rank of a matrix

Determinants

Linear equations

Quadratic forms

Characteristic roots and vectors.

3. Basic Concepts of Probability

Sample space, Discrete probability, Simple theorems on probability, Independence of events, Bayes Theorem, Discrete and continuous random variables, Binomial, Poisson and Normal distributions

Expectation and moments, Independence of random variables, Chebyshev’s inequality.

4. Linear Programming Basic Concepts

Convex sets, Linear Programming Problem (LPP)

Examples of LPP

Hyperplane, open and closed Half-spaces

Feasible, basic feasible and optimal solutions

Extreme point and graphical method.

5. Real Analysis

Finite, countable and uncountable sets, Bounded and unbounded sets, Archimedean property, ordered field, completeness of R, Extended real number system, limsup and liminf of a sequence, the epsilon-delta definition of continuty and convergence, the algebra of continuous functions, monotonic functions, types of discontinuties, infinite limits and limits at infinity, functions of bounded variation

elements of metric spaces.

6. Complex Analysis

Riemann Sphere and Stereographic projection

Lines, circles, crossratio

Mobius transformations, Analytic functions, Cauchy-Riemann equations, line integrals, Cauchy’s theorem, Morera’s theorem, Liouville’s theorem, integral formula, zero-sets of analytic functions, exponential, sine and cosine functions, Power siries representation, Classification of singularities, Conformal mapping.

7. Algebra

Group

subgroups

Normal subgroups

Quotient Groups

Homomorphisms

Cyclic Groups

Permutation Groups

Cayley’s Theorm

Rings

Ideals

Integral Domains

Fields

Polynomial Rings.

8. Linear Algebra

Vector spaces, subspaces, quotient spaces, Linear indepenence, Bases, Dimension

The algebra of linear Transformations, kernal, range, isomorphism, Matrix Representation of a linear transormation, change of bases, Linear functionals, dual space, projection, determinant function, eigenvalues and eigen vectorsCayley-Hamilton Theorem,Invariant Sub-spaces, Canonical Forms : diagonal form, Triangular form, Jordan Form

Inner product spaces.

9. Differential Equations

First order ODE, singular solutions initial value Problems of First Order ODE, General theory of homogeneous and non-homogeneous Linear ODE, Variation of Paraneters

Lagrange’s and Charpit’s methods of solving First order Partial Differental Equations

PDE’s of higher order with constant coefficients.

10. Data Analysis Basic Concepts

Graphical representation, measures of central tendency and dispersion

Bivariate data correlation and regression

Least squares-polynomial regression, Applications of normal distribution.

11. Probability

Axiomatic definition of probability

Random variables and distribution functions (univariate and multivariate); expectation and moments; independent events and independent random variables; Bayes theorem; marginal and conditional distribution in the multivariate case, covariance matrix and correlation coefficients (product moment, partial and multipal), regression

Moment generating functions, characteristic functions; probability inequalities (Tehebyshef, Markov, Jensen)

Convergence in probability and in distribution; weak law of large numbers and central limit theorem for independent indentically distributed random variables with finite variance.

12. Probability Distribution

Bemoulli, Binomial, Multinomial. Hypergeomatric, Poisson, Geometric and Negative binomial distributions, Uniform, exponential, Cauchy, Beta, Gamma, and normal (univariate and multivariate) distributions Transformations of random variables; sampling distributions

t, F and chi-square distributions as sampling distributions, as sampling distributions, Standard errors and large sample distributions

Distribution of order statistics and range.

13. Theory of Statistics

Methods of estimation : maximum likelihood method, method of moments, minimum chi-square method, least- squares method

Unbiasedness, efficiencey, consistency

Cramer-Rao inequality

Sufficient, Statistics

Rao-Blackwell theorem

Uniformly minimum variance unbiased estimators

Estimation by confidence intervals

Tests of hypotheses : Simple and composite hypotheses, two types of errors, critical region, randomized test, power function, most powerful and uniformly most powerful tests

Likelihood-ratio tests. Wald’s sequential probability ratio test.

14. Statistical Methods and Data Analysis

Tests for mean and variance in the normal distribution : one-population and two-population cases; related confidence intervals

Tests for product moment, partial and multiple correlation coefficients; comparison of k linear regressions

Fitting polynomial regression; related test Analysis of discrete date: chi-square test of goodness of fit, contingency tables

Analysis of variance : one-way and two-way classification (equal number of observations per cell)

Large sample tests through normal approximation

Nonparametric tests : sign test, Median test, Mann-Whitney test, Wilcoxon test for one and two-samples, rank correlation and test of independence.

15. Operational Research Modelling

Definition and scope of Operational Research

Different types of models

Replacement models and sequencing theory, Inventory problems and their analytical structure

Simple deterministic and stochastic models of inventory control

Basic characteristics of queueing system, different performance measures, steady state solution of Markovian queueing models: M/M/1, M/M/1 with limited waiting space M/M/C, M/M/C with limited waiting space.

16. Linear Programming

Linear Programming, Simplex method, Duality in linear programming

Transformation and assignment problems

Two person-zero sum games

Equivalence of rectangular game and linear programming.

17. Finite Population

Sampling Techniques and Estimation: Simple random sampling with and without replacement

Stratified sampling; allocation problem; systematic sampling Two stage sampling

Related estimation problems in the above cases.

18. Design of Experiments

Basic principles of experimental design

Randomisation structure and analysis of completely randomised, randomised blocks and Latin-square designs

Factorial experiments

Analysis of 2n factorial experiments in randomised blocks

Real Analysis

Riemann integrable functions; improper integrals, their convergence and uniform convergence

Eulidean space R", Bolzano-Weierstrass theorem, compact Subsets of R”, Heine-Borel T\theorem, Fourier series

Continuity of functions on R", Differentiability of F:R”-Rm

Properties of differential, partial and directional derivatives, continuously differentiable functions

Taylor’s series. Inverse function theorem, Implieit function theorem

Integral functions, line and surface integrals, Green’s theorem, Stoke’s theorem.

2. Complex Analysis

Cauchy’s theorem for convex regions

Power series representation of Analtic functions

Liouville’s theorem, Fundamental theorem of algebra Riemann’s theorem on removable singularities, maximum modulus principle

Schwarz Iemma, Open Mapping theorem, Casorattl-Weierstrass-theorem, Weierstrass’s theorem on uniform convergence on compact sets, Bilinear transformations, Multivalued Analytic Functions, Riemann Surfaces.

3. Algebra

Symmetric groups

Alternating groups

Simple groups

Rings

Maximal Ideals

Prime Ideals

Integral domains Euclidean domains

principal Ideal domains

Unique Factorisation domains

quotient fileds

Finite fields

Algebra of Linear Transformations

Reduction of matrices to Canonical Forms

Inner Product Spaces

Orthogonality

Quadratic Forms

Reduction of quadratic forms.

4. Advanced Analysis

Elements of Metric Spaces

Convergence

continuity

compactness

Connectedness

Weierstrass’s approximation Theorem

Completeness

Bare category theorem

Labesgue measure

Labesgue Integral

Differentiation and Integration.

5. Advanced Algebra

Conjugate elements and class equations of finite groups, Sylow theorems, solvable groups, Jordan Holder Theorem, Direct Products, Structure Theorem for finite abelian groups, Chain conditions on Rings

Characteristic of Field, Field extensions, Elements of Galois theory, solvability by Raducals, Ruler and compass construction.

6. Functional Analysis

Banach Spaces Hahn-Banach Theorem, Open mapping and closed Graph Theorems

Principal of Uniform boundedness, Boundedness and continuity of Linear Transformations, Dual Space, Embedding in the second dual, Hilbert Spaces, Projections

Orthonormal Basis, Riesz-representation theorem, Bessel’s Inequality, parsaval’s identity, self adjoined operators, Normal Operators.

7. Topology

Elements of Topological Spaces, Continuity, convergence, Homeomorphism, Compactness, Connectedness, Separation Axioms, First and Second Countability, Separability, Subspaces, Product Spaces, quotient spaces

Tychonoft’s Theorem, Urysohn’s Metrization theorem, Homotopy and Fundamental Group.

8. Discrete Mathematics

Partially ordered sets

Lattices

Cornplete Lattices

Distrbutive lattices

Complements

Boolean Algebra

Boolean Expressions

Application to switching circuits

Elements of Graph Theory

Eulerian and Hamiltonian graphs

planar Graphs

Directed Graphs

Trees

Permutations and Combinations

Pigeonhole principle

principle of Inclusion and Exclusion

Derangements.

9. Ordinary and partial Differential Equations

Existence and Uniqueness of solution dy/dx = f (x,y) Green’s function

sturm Liouville Boundary Value Problems

Cauchy Problems and Characteristics

Classification of Second Order PDE

Separation of Variables for heat equation

wave equation and Laplace equation

Special functions.

10. Number Theory

Divisibility; Linear diophantine equations

Congruences

Quadratic residues; Sums of two squares, Arithmatic functions Mu, Tau, and Signa (and ).

11. Machanics

Generalise coordinates

Lagranges equation

Hamilton’s cononical equations

Variational Principles-Hamilton’s pronciples and plrinciples of least action

Two dimensional motion of rigid bodies

Euler’s dynamical equations for the motion of rigid body

Motion of a rigid body about an axis

Motion about revolving axes.

12. Elasticity

Analysis of strain and stress, strain and stress tensors;

Geometrical representation

Compatibility conditions

Strain energy function

Constritutive relations

Elastic solids “Hookes law

Saint-Venant’s principle, Equations of equilibrium

Plane problem-Airy’s stress function vibrations of elastic, cylindrical and spherical media.

13. Fluid Mechanics

Equation of continuity in fluid motion

Euler’s equations of motion for perfect fluids

Two dimensional motion complex potential

Motion of sphere in perfect liquid and montion of liquid past a sphere; vorticity

Navier-Stokes’s equations for viscous flows- some exact solutions.

14. Differetial Geometry

Space curves-their curvature and torsion

Serret Frehat Formula

Fundamental theorem of space curves

Curves on sirfaces

First and second fundamental form

Gaussian curvatures

Principal directions and principal curvatures

Goedesics, Fundamental equations of surface theory.

15. Calculus of Variations

Linear functionals, minimal functional theorem, general variation of a functional, Euler-Lagrange equation

Variational methods of boundary value problems in ordinary and partial differential equations.

16. Linear Integral Equations

Linear Integral Equations of the first and second kind of Fredholm and Volterra type

solution by successive substitutions and successive approximations

Solution of equations with separable kernels

The Fredholm Alternative

Holbert-Schmidt theory for symmetric kernels.

17. Numerical analysis

Finite differences, Interpolation

Numerical solution of algebric equation

Iteration

Newton-Raphason Method

Solution on Linear system

Direct method

Gauss elminaiton method

Matrix-Inversion eigenvalue problems

Numerical differentiation and integration

Numerical solution of ordinary differential equation

iteration method, Picard’s method , Euler’s method and improved Euler’s method.

18. Integral Transform

Laplace transform

Transform of elementary functions, Transform of Derivatives, Inverse Transform, Convolution Theorem, Applications, Ordinary and Partial differential equations

Fourier transform

sine and cosine transform, Inverse Fourier Transform, Application to ordinary and partial differential equations.

19. Mathematical Programming

Revised simplex method, Dual simplex method, Sensitivity analysis and parametric linear programming

Kuhn-Tucker conditions of optimality

Quadratic programming; methods due to Beale, Wofle and Vandepanne, Duality in quadratic programmming, self duality, Integer programming.

20. Measure Theory

Measurable and measure spaces : Extension of measures,signed measures, Jordan-Hahn decomposition theorems

Integratiuon, monotone convergence theorem, Fatou’s lemma, dominaated convergence theorem

Absolute continuity, Radon Nikodym theorem, Product measures, Fubini's theorem.

21. Probability

Sequences of events and random variables: Zero-one laws of Borel and Kolmogorov

Almost sure convergence, convergence in mean square, Khintchine’s weak law of large numbers; Kolmogorov’s inequality, strong law of large numbers

Convergence of series of random variables, three-series criterion

Central limit theorems of Liapounov and Lindeberg-Feller

Conditional expectation, martingales.

22. Distribution Theory

Properties of distribution functions and characteristic functions; continuty theorem, inversion formula, Representation of distribution function as a mixture of discrete and continuous distribution functions; Convolutions, marginal and conditional distributions of bivariate discrete and continuous distributions

Relations between characteristic functions and moments; Moment inequalities of Holder and Minkowski.

23. Statistical Inference and Decision Theory

Statistical decision problem : non-randomized, mixed and randomized decision rules; risk function admissibility, Bayes rules, minimax rules, least favourable distributions, complete class and minimal complete class. Decision problem for finite parameter space. Convex loss function. Role of sufficiency

Admissible, Bayes and minimax estimators; illustrations

Unbiasedness

UMVU estimators

Families of distributons with monotone likelihood property, exponential family of distributions

Test of a simple hypothesis against a simple alternative from decision-theoretic viewpoint

Tests with Neyman structure

Uniformly most powerful unbiased tests

Locally most powerful tests

Inference on location and scale parameters; estimation and tests

Equivariant estimators

Invariance in hypothesis testing.

24. Large sample statistical methods

Various modes of convergence

Op and op, CLT, Sheffe’s theorem, Polya’s theorem and Slutsky’s theorem

Transformation and variance stabilizing formula

Asymptotic disribution of function of sample moments

Sample quantiles

Order statistics and their functions

Tests on correlations, coefficient of variation, skewness and kurtosis

Pearson Chi-square, contingency Chi-square and likelihood ratio statistics

U-statistics consistency of Tests

Asymptotic relative efficiency.

25. Multivariate Statistical Analysis

Singular and non-singular multivariate distributions

Characteristics functions

Multivariate normal distributions, margrinal and conditional distributions; distribution of linear forms, and quadratic forms, Cochran’s theorem

Inference on parameters of multivariate normal distributions, one-population and two population cases

wishart distribution

Hotellings T2, Mahalanobis D2 Discrimination analysis, Principal components, Canonical correlations, Cluster analysis.

26. Linear Models and Regression

Standard Gauss-Markov models; Estimability of parameters; best linear unbiased extimates(BLUE); Method of least squares and Gauss-Markovtheorem; Variance-covariance matrix of BLUES

Tests of linear hypothesis; One-way and two-way classifications

Fixed, random and mixed effects models (two-way classifications only); variance components, Bivariate and multiple linear regression; Polynomial regression; use of ortheogonal polynomials

Analysis of covarance

Linear and nonlinear regression outliers.

27. Sample Surveys

Sampling with varying probability of selection, Hurwitz-Thompson estimator; PPS sampling: Double sampling

Cluser sampling

Non-sampling errors: Interpentrating samples

Multiphase sampling

Ratio and regession methods of estimation.

28. Design of Experiments

Factorial experiments, confounding and fractional replication

Split and strip plot designs; Quasi-Latin square designs; Youden square

Design for study of response surfaces; first and second order designs

Incomplete block designs; Balanced, connectedness and orthogonality, BIBD with recovery of inter-block information PBIBD with 2 associate classes

Analysis of series of experiments, esimation of residual effects

Construction of orthogonal-Latin squares, BIB designs, and confounded factorial designs

Optimality criteria for experimental designs.

29. Time-Series Analysis

Discrete-parameter stochastic processes; strong and weak stationarity; autocovariance and autocorrelation

Moving average, autoregressive, autoregressive moving average and autoregressive intgegfrated moving average processes

Box-Jenkins models

Estimation of the parameters in ARIMA models; forecasting

Periodogram and correlogram analysis.

30. Stochastic Proceses

Markov chains with finite and countable state space, classification of states, limiting behaviour of n-step transition probabilities, stationary distribution; branching processes; Random walk; Gambler’s ruin

Markov processes in continuous time; Poisson processes, birth and death processes, Wiener process.

31. Demography and Vital Statistics

Measures of fertility and mortality, period and Cohort measures

Life tables and its applications; Methods of construction of abridged life tables

Application of stable population theory to estimate vital rates

Popultion projetions

Stochastic models of fertility and reproduction.

32. Industrial Statistics

Control charts for variables and attributes; Acceptance sampling by attribites; single, double and sequential sampling plans; OC and ASN functions, AOQL and ATI; Acceptance sampling by varieties

Tolerance limits Reliability analysis: Hazard function, distribution with DFR and IFR; Series and parallel systems

Life testing experiments.

33. Inventory and Queueing theory

Inventory (S,s) policy periodic review models with stochastic demand

Dynamic inventory models

Probabilistic re-order point, lot size inventory system with and without lead time

Distribution free analysis

Solution of inventory problem with unknown density function

Warehousing problem

Queues: Imbedded markov chain method to obtain steady state solution of M/G/1, G/M/1 and M/D/C, Network models

Machine maintenance models

Design and control of queueing systems.

34. Dynamic Programming and Marketing

Nature of dynamic programming, Deterministic processes, Non-sequential discrete optimisation-allocation problems, assortment problems

Sequential discrete optimisation long-term planning problems, multi stage production processes

Functional approximations

Marketing systems, application of dynamic programming to marketing problems

Introduction of new product, objective in setting market price and its policies, purchasing under Fluctuating prices, Advertising and promotional decisions, Brands swiching analysis, Distribution decisions.