Maharashtra State Eligibility Test for Lectureship
UNIVERSITY OF PUNE
Ganeshkhind, Pune-411007
SYLLABUS AND SAMPLE QUESTIONS-(Mathematical Sciences)
Subject Subject
Code No.
30 Mathematical Sciences
SET (Mathematical Sciences) / 2
set booklets \ mathematical science_SET syllabus (03-09)
[30] : MATHEMATICAL SCIENCES
SYLLABUS
PAPER II
General Information : Units 1, 2, 3 and 4 are compulsory for all candidates. Candidates with
Mathematics background may omit units 10-14 and units 17,18. Candidates with Statistics background
may omit units 6,7,9,15 and 16. Adequate alternatives would be given for candidates with O.R.
background.
1.
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.
transformation, Algebra of matrices, Rank of a matrix, Determinants, Linear equations, Quadratic
forms, Characteristic roots and vectors.
3.
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.
Examples of LPP. Hyperplane, open and closed Half-spaces. Feasible, basic feasible and
optimal solutions. Extreme point and graphical method.
5.
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.
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.
Groups, Permutation Groups, Cayley’s Theorm, Rings, Ideals, Integral Domains, Fields,
Polynomial Rings.
8.
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.
Basic Concepts of Real and Complex Analysis : Sequences and series, Continuity, UniformBasic Concepts of Linear Algebra : Space of n-vectors, Linear dependence, Basis, LinearBasic Concepts of Probability : Sample space, Discrete probability, Simple theorems onLinear Programming Basic Concepts : Convex sets, Linear Programming Problem (LPP).Real Analysis : Finite, countable and uncountable sets, Bounded and unbounded sets,Complex Analysis : Riemann Sphere and Stereographic projection. Lines, circles, crossratio.Algebra : Group, subgroups, Normal subgroups, Quotient Groups, Homomorphisms, CyclicLinear Algebra : Vector spaces, subspaces, quotient spaces, Linear indepenence, Bases,SET (Mathematical Sciences) / 3
set booklets \ mathematical science_SET syllabus (03-09)
9.
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.
dispersion. Bivariate data correlation and regression. Least squares-polynomial regression,
Applications of normal distribution.
11.
(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.
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.
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.
: 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.
Differential Equations : First order ODE, singular solutions initial value Problems of FirstData Analysis Basic Concepts : Graphical representation, measures of central tendency andProbability : Axiomatic definition of probability. Random variables and distribution functionsProbability Distribution : Bemoulli, Binomial, Multinomial. Hypergeomatric, Poisson,Theory of Statistics : Methods of estimation : maximum likelihood method, method ofStatistical Methods and Data Analysis : Tests for mean and variance in the normal distributionSET (Mathematical Sciences) / 4
set booklets \ mathematical science_SET syllabus (03-09)
15.
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.
Transformation and assignment problems. Two person-zero sum games. Equivalence of
rectangular game and linear programming.
17.
without replacement. Stratified sampling; allocation problem; systematic sampling Two stage
sampling. Related estimation problems in the above cases.
18.
and analysis of completely randomised, randomised blocks and Latin-square designs. Factorial
experiments. Analysis of 2n factorial experiments in randomised blocks.
Operational Research Modelling : Definition and scope of Operational Research. DifferentLinear Programming : Linear Programming, Simplex method, Duality in linear programming.Finite Population : Sampling Techniques and Estimation: Simple random sampling with andDesign of Experiments : Basic principles of experimental design. Randomisation structureSYLLABUS
PAPER III
1.
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”-R
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.
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.
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.
Connectedness, Weierstrass’s approximation Theorem, Completeness, Bare category theorem,
Labesgue measure, Labesgue Integral, Differentiation and Integration.
Real Analysis : Riemann integrable functions; improper integrals, their convergence andm. Properties of differential, partialComplex Analysis : Cauchy’s theorem for convex regions. Power series representation ofAlgebra : Symmetric groups, Alternating groups, Simple groups, Rings, Maximal Ideals,Advanced Analysis : Elements of Metric Spaces, Convergence, continuity, compactness,SET (Mathematical Sciences) / 5
set booklets \ mathematical science_SET syllabus (03-09)
5.
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.
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.
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.
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.
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.
Sums of two squares, Arithmatic functions Mu, Tau, and Signa (and ).
11.
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.
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.
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 flowssome
exact solutions.
Advanced Algebra : Conjugate elements and class equations of finite groups, Sylow theorems,Functional Analysis : Banach Spaces Hahn-Banach Theorem, Open mapping and closedTopology : Elements of Topological Spaces, Continuity, convergence, Homeomorphism,Discrete Mathematics : Partially ordered sets, Lattices, Cornplete Lattices, Distrbutive lattices,Ordinary and partial Differential Equations : Existence and Uniqueness of solutionNumber Theory : Divisibility; Linear diophantine equations. Congruences. Quadratic residues;Machanics : Generalise coordinates; Lagranges equation; Hamilton’s cononical equations;Elasticity : Analysis of strain and stress, strain and stress tensors; Geometrical representation;Fluid Mechanics : Equation of continuity in fluid motion; Euler’s equations of motion forSET (Mathematical Sciences) / 6
set booklets \ mathematical science_SET syllabus (03-09)
14.
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.
of a functional, Euler-Lagrange equation; Variational methods of boundary value problems in
ordinary and partial differential equations.
16.
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.
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.
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.
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.
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.
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.
Differetial Geometry : Space curves-their curvature and torsion; Serret Frehat Formula;Calculus of Variations : Linear functionals, minimal functional theorem, general variationLinear Integral Equations : Linear Integral Equations of the first and second kind of FredholmNumerical analysis : Finite differences, Interpolation; Numerical solution of algebric equation;Integral Transform : Laplace transform; Transform of elementary functions, Transform ofMathematical Programming : Revised simplex method, Dual simplex method, SensitivityMeasure Theory : Measurable and measure spaces : Extension of measures,signed measures,Probability : Sequences of events and random variables: Zero-one laws of Borel andSET (Mathematical Sciences) / 7
set booklets \ mathematical science_SET syllabus (03-09)
22.
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.
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.
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.
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.
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.
PPS sampling: Double sampling. Cluser sampling. Non-sampling errors: Interpentrating samples.
Multiphase sampling. Ratio and regession methods of estimation.
Distribution Theory : Properties of distribution functions and characteristic functions; continutyStatistical Inference and Decision Theory : Statistical decision problem : non-randomized,Large sample statistical methods : Various modes of convergence. Op and op, CLT, Sheffe’sMultivariate Statistical Analysis : Singular and non-singular multivariate distributions.Linear Models and Regression : Standard Gauss-Markov models; Estimability of parameters;Sample Surveys : Sampling with varying probability of selection, Hurwitz-Thompson estimator;SET (Mathematical Sciences) / 8
set booklets \ mathematical science_SET syllabus (03-09)
28.
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.
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.
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.
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.
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.
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.
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.
————Design of Experiments : Factorial experiments, confounding and fractional replication.Time-Series Analysis : Discrete-parameter stochastic processes; strong and weak stationarity;Stochastic Proceses : Markov chains with finite and countable state space, classification ofDemography and Vital Statistics : Measures of fertility and mortality, period and CohortIndustrial Statistics : Control charts for variables and attributes; Acceptance sampling byInventory and Queueing theory : Inventory (S,s) policy periodic review models with stochasticDynamic Programming and Marketing : Nature of dynamic programming, Deterministic
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