THE Ph.D. PROGRAMME
QUALIFYING EXAMINATION REQUIREMENT
Subject to the final approval from PGPC, the following shall
be applicable for all Ph.D. students joining the Department of Mathematics in
Autumn 2007 or thereafter.
1. Each student in the Ph.D. programme in the Department of
Mathematics is required to take a Qualifying Examination to be administered by
the DPGC, Department of Mathematics.
2. The qualifying examination shall consist of written
examinations on any two of the following six topics: (i) Algebra, (ii) Analysis,
(iii) Geometry and Topology, (iv) Differential Equations, (v) Probability and
(vi) Statistics.
3. The qualifying examination shall be conducted in the week
before the beginning of the Autumn semester as well as the Spring semester. The
results of the qualifying examination shall be made known before the date of
registration.
4. Each student is permitted at most two attempts for each of
the two chosen topics. It is expected that every student aspiring to do Ph.D.
will successfully complete the qualifying examination requirement by the end of
the first academic year. In any case, the maximum time allotted for a student
to pass the qualifying examination shall be three semesters. If a student fails
to pass the examination in one or both of the chosen topics, then he/she may be
permitted a change of topic. However, the ceiling of maximum two attempts and
the overall ceiling of three semesters to complete the qualifying examination
requirement shall always remain in force.
5. A student may choose a Ph.D. thesis advisor after the
successful fulfillment of the qualifying examination requirement. Until that
time, the Head, Department of Mathematics shall be the guide for all official
purposes.
6. In case a student fails the qualifying examination
requirement, then he/she may be offered the option of transferring to M.Phil.
programme by continuing for about a semester so as to complete the requirements
for the degree of M.Phil. degree.
7. Notwithstanding the qualifying examination requirement,
the course requirement stipulated by the Academic Office for the Ph.D. (or the
M.Phil., as the case may be) programme will have to be satisfied. It may be
noted that currently the course requirement for Ph.D. students consist of
completing at least 34 course credits. However, students may register for up to
two 500level M.Sc. courses (for example, those that are relevant for the topic
of qualifiers) to partially satisfy the credit requirement.
SYLLABUS FOR QUALIFYING EXAMINATIONS
Topic
(i): ALGEBRA
Groups, Simple
groups and solvable groups, nilpotent groups, simplicity of alternating groups,
composition series, JordanHolder Theorem. Semidirect products. Free groups,
free abelian groups.
Rings, Examples
(including polynomial rings, formal power series rings, matrix rings and group
rings), ideals, prime and maximal ideals, rings of fractions, Chinese Remainder
Theorem for pairwise comaximal ideals. Euclidean Domains, Principal Ideal
Domains and Unique Factorizations Domains. Polynomial rings over UFD's.
Fields,
Characteristic and prime subfields, Field extensions, Finite, algebraic and
finitely generated field extensions, Classical ruler and compass constructions,
Splitting fields and normal extensions, algebraic closures. Finite fields,
Cyclotomic fields, Separable and inseparable extensions.
Galois groups,
Fundamental Theorem of Galois Theory, Composite extensions, Examples (including
cyclotomic extensions and extensions of finite fields).
Norm, trace and
discriminant.
Solvability by
radicals, Galois' Theorem on solvability.
Cyclic extensions, Abelian extensions, Transcendental
extensions.
Modules,
submodules, quotient modules and module homomorphisms.
Generation of
modules, direct sums and free modules. Tensor products of modules. Exact
sequences, projective modules.
Tensor
algebras, symmetric and exterior algebras.
Finitely
generated modules over principal ideal domains, invariant factors, elementary
divisors, rational canonical forms. Applications to finitely generated abelian
groups and linear transformations.
Noetherian
rings and modules, Hilbert basis theorem, Primary decomposition of ideals in
noetherian rings.
Integral
extensions, Goingup and Goingdown theorems, Extension and contraction of
prime ideals, Noether's Normalization Lemma, Hilbert's Nullstellensatz.
Localization of
rings and modules. Primary decompositions of modules.
Topic
(ii): ANALYSIS
Semialgebra,
Algebra, Monotone class, Sigmaalgebra, Monotone class theorem. Measure spaces.
Extension of
measures from algebras to the generated sigmaalgebras: Measurable sets;
Lebesgue Measure and its properties.
Measurable
functions and their properties; Integration and Convergence theorems.
Introduction to
L^{p}spaces, RieszFischer theorem; Riesz Representation theorem for L^{2}
spaces.
Absolute
continuity of measures, RadonNikodym theorem. Duals of L^{p}spaces.
Product measure
spaces, Fubini's theorem.
Fundamental
Theorem of Calculus for Lebesgue Integrals.
Normed spaces.
Continuity of linear maps. HahnBanach Extension and Separation Theorems.
Banach spaces. Dual spaces and transposes.
Uniform
Boundedness Principle and its applications. Closed Graph Theorem, Open Mapping
Theorem and their applications. Spectrum of a bounded operator. Examples of
compact operators on normed spaces.
Inner product
spaces, Hilbert spaces. Orthonormal basis. Projection theorem and Riesz
Representation Theorem for Hilbert spaces.
Topic (iii):
GEOMETRY and TOPOLOGY
Functions on
Euclidean spaces, continuity, differentiability; partial and directional
derivatives, Chain Rule, Inverse Function Theorem, Implicit Function
Theorem.
Riemann Integral
of realvalued functions on Euclidean spaces, measure zero sets, Fubini's
Theorem.
Partition of
unity, change of variables.
Integration on
chains, tensors, differential forms, Poincare Lemma, singular chains,
integration on chains, Stokes' Theorem for integrals of differential forms on
chains. (general version). Fundamental theorem of calculus.
Differentiable
manifolds (as subspaces of Euclidean spaces), differentiable functions on
manifolds, tangent spaces, vector fields, differential forms on manifolds,
orientations, integration on manifolds, Stokes' Theorem on manifolds.
Paths and
homotopy, homotopy equivalence, contractibility, deformation retracts.
Basic
constructions: cones, mapping cones, mapping cylinders, suspension.
Cell complexes,
subcomplexes, CW pairs.
Fundamental
groups. Examples (including the fundamental group of the circle) and
applications (including Fundamental Theorem of Algebra, Brouwer Fixed Point
Theorem and BorsukUlam Theorem, both in dimension two). Van Kampen's Theorem,
Covering spaces, lifting properties, deck transformations. Universal coverings
(existence theorem optional).
Simplicial
complexes, barycentric subdivision, stars and links, simplicial approximation.
Simplicial Homology. Singular Homology. MayerVietoris Sequences. Long exact
sequence of pairs and triples. Homotopy invariance and excision (without
proof).
Topic
(iv): DIFFERENTIAL EQUATIONS
Methods for
solving first order as well as second order equations, Power Series methods
with properties of Bessel functions and Legendre polynomials.
Existence and
Uniqueness of Initial Value Problems: Picard's and Peano's Theorems, Gronwall's
inequality, continuation of solutions and maximal interval of existence,
continuous dependence.
Higher order
linear differential equations and linear systems: fundamental solutions,
Wronskian, variation of constants, matrix exponential solution, behaviour of
solutions.
Two Dimensional
Autonomous Systems and Phase Space Analysis: critical points, proper and
improper nodes, spiral points and saddle points.
Asymptotic
Behavior: stability (linearized stability and Lyapunov methods).
Boundary Value
Problems for Second Order Equations: Green's function, Sturm comparision
theorems and oscillations, eigenvalue problems
Cauchy Problems
for First Order Hyperbolic Equations: method of characteristics, Monge cone.
Classification
of Second Order Partial Differential Equations: normal forms and
characteristics.
Initial and
Boundary Value Problems: LagrangeGreen's identity and uniqueness by energy
methods.
Stability
theory, energy conservation and dispersion equation: mean value property, weak
and strong maximum principle, Green's function, Poisson's formula, Dirichlet's
principle, existence of solution using Perron's method (without proof).
Heat equation:
initial value problem, fundamental solution, weak and strong maximum principle
and uniqueness results.
Wave equation:
uniqueness, D'Alembert's method, method of spherical means and Duhamel's
principle.
Methods of
separation of variables for heat, and wave equations.
Topic (v):
PROBABILITY
Probability
measure, probability space, construction of Lebesgue measure, extension
theorems, limit of events, BorelCantelli lemma.
Random
variables, Random vectors, distributions, multidimensional distributions,
independence.
Expectation,
change of variable theorem, convergence theorems.
Sequence of
random variables, modes of convergence. Moment generating function and
characteristics functions, inversion and uniqueness theorems, continuity
theorems, Weak and strong laws of large number, central limit theorem.
RadonNikodym
theorem, definition and properties of conditional expectation, conditional
distributions.
Stochastic
processes: Description and definition. Markov chains with finite and countably
infinite state spaces. Classification of states, irreducibility, ergodicity.
Basic limit theorems.
Markov
processes with discrete and continuous state spaces. Poisson process, pure
birth process, birth and death process. Brownian motion.
Applications to
queueing models and reliability theory.
Basic theory
and applications of renewal processes, stationary processes. Branching
processes. Markov Renewal and semiMarkov processes, regenerative processes.
Topic
(vi): STATISTICS
Distribution of
functions of random variables, Order Statistics. Estimation  loss function,
risk, minimum risk unbiased estimators, maximum likelihood estimation, method
of moments, Bayes estimation. Sufficient Statistics, completeness, Basu's
Theorem, exponential families, invariance and maximal invariant
statistics.
Testing of
Hypotheses  parametric and nonparametric problems, examples with data
analytic applications.
Confidence
Intervals.
Simple and
multiple linear regression models, estimation, tests and confidence regions.
Check for normality assumption. Likelihood ratio test, confidence intervals and
hypotheses tests; tests for distributional assumptions. Collinearity, outliers;
analysis of residuals, Selecting the best fit, regression equation,
transformation of response variables. Ridge's regression.
Ph.D. COURSES
The following courses are likely to be offered in each of the
corresponding semesters.
First
Semester
Course Code 
Name of the
Course 
L T P C 
MA 813 
Algebra
I 
3 0 0 6 
MA 819 
Measure
Theory 
3 0 0 6 
MA 815 
Differential
Topology 
3 0 0 6 
MA 817 
Partial
Differential Equations I 
3 0 0 6 
MA833 
Weak Convergence
and Martingale Theory 
3 0 0 6 
MA 821 
Theory
of Estimation 
3 0 0 6 
MA 861 
Combinatorics I 
3 0 0 6 
1. The credit requirements
for students having M.Sc. or equivalent qualification shall be 34 to 46
credits.
2. Credits acquired through
PG level courses shall be 24 or more.
3. Students may earn upto
a maximum of 8 credits through seminars which should be spread over two
semesters.
Second
Semester
Course Code 
Name of the
Course 
L T P C 
MA 812 
Algebra
II 
3 0 0 6 
MA 814 
Complex
Analysis 
3 0 0 6 
MA 816 
Algebraic
Topology 
3 0 0 6 
MA 818 
Partial
Differential Equations II 
3 0 0 6 
MA 820 
Stochastic
Processes 
3 0 0 6 
MA 822 
Testing
of Hypothesis 
3 0 0 6 
MA 824 
Functional
Analysis 
3 0 0 6 
MA 862 
Combinatorics II 
3 0 0 6 
Note: Each student must credit at least
3 of the 13 courses MA 813 – 822 & MA 824.
In addition to the above courses, some or all of the
following courses may be offered subject to sufficient demand from the students
and availability of faculty.
First
Semester
Course Code 
Name of the
Course 
L T P C 
MA 841 
Topics
in Algebra I 
3 0 0 6 
MA 843 
Topics
in Analysis I 
3 0 0 6 
MA 845 
Topics
in Combinatorics I 
3 0 0 6 
MA 847 
Topics
in Geometry I 
3 0 0 6 
MA849 
Topics
in Topology I 
3 0 0 6 
MA 851 
Topics
in Number Theory I 
3 0 0 6 
MA 853 
Topics
in Differential Equations I 
3 0 0 6 
MA 855 
Topics
in Numerical Analysis I 
3 0 0 6 
MA 857 
Topics in
Probability I 
3 0 0 6 
MA 859 
Topics
in Statistics I 
3 0 0 6 
MAS 801

Seminar

0 0 0 4 
Second
Semester
Course Code 
Name of the
Course 
L T P C 
MA 842 
Topics
in Algebra II 
3 0 0 6 
MA 844 
Topics
in Analysis II 
3 0 0 6 
MA 846 
Topics
in Combinatorics II 
3 0 0 6 
MA 848 
Topics
in Geometry II 
3 0 0 6 
MA850 
Topics in
Topology II 
3 0 0 6 
MA 852 
Topics
in Number Theory II 
3 0 0 6 
MA 854 
Topics
in Differential Equations II 
3 0 0 6 
MA 856 
Topics
in Numerical Analysis II 
3 0 0 6 
MA 858 
Topics
in Probability II 
3 0 0 6 
MA 860 
Topics
in Statistics II 
3 0 0 6 
MAS 802

Seminar

0 0 0 4 
Ph.D. COURSE CONTENTS
DEPARTMENTAL
COURSES
NOTE: Each course is
of 6 credits with the structure of 3006. A prerequisite for an even numbered
course is exposure to the preceding odd numbered course, except in the case of
MA 824 for which exposure to MA 819 shall be the prerequisite.
MA
813 Algebra I
Review of field and Galois theory: solvable and radical
extensions, Kummer theory, Galois cohomology and Hilbert's Theorem 90, Normal
Basis theorem.
Infinite Galois extensions: Krull topology,
projective limits, profinite groups, Fundamental Theorem of Galois theory for
infinite extensions.
Review of integral ring extensions: integral Galois
extensions, prime ideals in integral ring extensions, decomposition and inertia
groups, ramification index and residue class degree, Frobenius map, Dedekind
domains, unique factorisation of ideals.
Categories and functors: definitions and examples. Functors
and natural transformations, equivalence of categories,. Products and
coproducts, the hom functor, representable functors, universals and adjoints.
Direct and inverse limits. Free objects.
Homological algebra: Additive and abelian categories,
Complexes and homology, long exact sequences, homotopy, resolutions, derived
functors, Ext, Tor, cohomology of groups, extensions of groups.
Text/References:
S. Lang, Algebra, 3^{rd} Ed.,
Addison Wesley, 1993.
N. Jacobson, Basic Algebra, Vol. 1 and 2,
Hindustan Publishing Corporation, 1984.
MA
812 Algebra II
Valuations and completions: definitions, polynomials
in complete fields (Hensel's Lemma, Krasner's Lemma), finite dimensional
extensions of complete fields, local fields, discrete valuations rings.
Transcendental extensions: transcendence
bases, separating transcendence bases, Luroth's theorem. Derivations.
Artinian and Noetherian modules,
KrullSchmidt theorem, completely reducible modules, projective modules,
WedderburnArtin Theorem for simple rings.
Representations of finite groups: complete
reducibility, characters, orthogonal relations, induced modules, Frobenius
reciprocity, representations of the symmetric group.
Text/References:
S. Lang, Algebra, 3^{rd} Ed.,
Addison Wesley, 1993.
N. Jacobson, Basic Algebra, Vol. 1 and 2,
Hindustan Publishing Corporation, 1984.
J.P. Serre, Linear Representations of
Finite Groups, SpringerVerlag, 1977.
O. Zariski and P. Samuel, Commutative
Algebra, Vol.1 and 2, SpringerVerlag, 1975.
MA
819 Measure Theory
Review of measure theory: monotone
convergence theorem, dominated convergence theorem, complete measures. Borel
measures: Riesz representation theorem, Lebesgue measure on R^{k}, L^{p }spaces
Complex measures: total variation, absolute
continuity, RadonNikodym theorem, polar and Hahn decompositions, bounded
linear functionals on L^{p}, generalised Riesz representation theorem.
Differentiation: Maximal function, Lebesgue
points, absolute continuity of functions, fundamental theorem of calculus,
Jacobian of a differentiable transformation, change of variable formula.
Product measures: Fubini's theorem,
completion of product measures, convolutions, Fourier transform,
RiemannLebesgue lemma, inversion theorem, Plancherel theorem, L^{1 }as
a Banach algebra.
Content on a locally compact Hausdorff
space, existence and uniqueness of the Haar measure on a locally compact group.
Text/References:
K. Chandrasekharan, A Course on Topological
Groups, Hindustan Book Agency, 1996.
L. Nachbin, The Haar Integral, van
Nostrand, 1965.
I. K. Rana, An Introduction to Measure and
Integration, 2^{nd} Ed., American Mathematical Society, 2002.
H. L. Royden, Real Analysis, 3^{rd}
Ed., Prentice Hall of India, 1988.
W. Rudin, Real and Complex Analysis,
McGrawHill, 1987.
MA
814 Complex Analysis
Review of basic complex analysis: Cauchy's
theorem, Liouville's theorem, power series representation, open mapping
theorem, calculus of residues.
Harmonic functions, Poisson integral,
Harnack's theorem, Schwarz reflection principle.
Maximum modulus principle, Schwarz lemma,
PhragmenLindelof method,.
Runge's theorem, MittagLeffler theorem,
Weierstrass theorem, conformal equivalence, Riemann mapping theorem,
characterisation of simply connected regions, Jensen's formula.
Analytic continuation, monodromy theorem,
little Picard theorem.
Text/References:
L. V. Ahlfors, Complex Analysis,
McGrawHill, 1996.
S. Lang, Complex Analysis, 4th Ed.,
Springer, 1999.
D. H. Luecking and L. A. Rubel, Complex
Analysis: A Functional Analysis Approach, SpringerVerlag, 1984.
R. Narasimhan and Y. Nievergelt, Complex
Analysis in One Variable, Birkhäuser, 2001.
R. Remmert, Theory of Complex Functions,
Springer (India), 2005.
W. Rudin, Real and Complex Analysis, McGraw
Hill, 1987.
MA
815 Differential Topology
Review of differentiable manifolds, tangent
and cotangent bundles, tensors.
DeRham complex, Poincare's Lemma,
MayerVietoris sequences, cohomology with compact supports, degree of a map,
Poincare duality.
Vector bundles,
cohomology with vertical compact supports, Thom isomorphism, twisted DeRham
complex, Poincare duality for nonorientable manifolds.
Texts/References:
R. Bott and L. W. Tu , Differential Forms
in Algebraic Topology,
SpringerVerlag, New York, 1982.
L. Conlon, Differentiable manifolds, 2^{nd}
Ed., Birkhäuser, Boston, 2001.
G. E Bredon, Topology and Geometry,
SpringerVerlag, New York, 1997.
MA
816 Algebraic Topology
Paths and homotopy, homotopy equivalence,
contractibility, deformation retracts.
Basic constructions: cones, mapping cones,
mapping cylinders, suspension.
Cell complexes, subcomplexes, CW pairs.
Fundamental groups. Examples (including the
fundamental group of the circle) and applications (including Fundamental Theorem
of Algebra, Brouwer Fixed Point Theorem and BorsukUlam Theorem, both in
dimension two). Van Kampen's Theorem. Covering spaces, lifting properties, deck
transformations, universal coverings.
Simplicial complexes, barycentric
subdivision, stars and links, simplicial approximation. Simplicial Homology.
Singular Homology. MayerVietoris sequences. Long exact sequence of pairs and
triples. Homotopy invariance and excision.
Degree. Cellular Homology.
Applications of homology: JordanBrouwer
separation theorem, Invariance of dimension, Hopf's Theorem for commutative
division algebras with identity, BorsukUlam Theorem, Lefschetz Fixed Point
Theorem.
Optional Topics: Outline of the theory of:
cohomology groups, cup products,
Kunneth formulas, Poincare duality.
Texts/References:
M.J. Greenberg and J. R. Harper, Algebraic
Topology, Benjamin, 1981.
W. Fulton, Algebraic topology: A First
Course, SpringerVerlag, 1995.
A. Hatcher, Algebraic Topology, Cambridge
Univ. Press, Cambridge, 2002.
W. Massey, A Basic Course in Algebraic
Topology, SpringerVerlag, Berlin, 1991.
J.R. Munkres, Elements of Algebraic
Topology, Addison Wesley, 1984.
J.J. Rotman, An Introduction to Algebraic
Topology, Springer (India), 2004.
H. Seifert and W. Threlfall, A Textbook of
Topology, Academic Press, 1980.
MA
817 Partial Differential Equations I
Distribution Theory and Sobolev Spaces: Distributional
derivatives, Definitions and elementary properties of Sobolev Spaces,
Approximations by smooth functions, Traces, Imbedding Theorems (without proof),
RellichKondrachov Compactness Theorem.
Second Order Linear Elliptic Equations: Weak Solutions,
LaxMilgram Theorem, Existence and Regularity Results, Maximum Principles,
Eigenvalue Problems.
Second Order Linear Parabolic Equations: Existence of
weak solutions and Regularity Results, Maximum Principles.
Second Order Linear Hyperbolic Equations: Existence of
weak solutions and Regularity Results, Maximum Principles, Propagation of Disturbance.
Text/References:
S. Kesavan, Topics in Functional Analysis
and Applications, New Age International Pvt. Ltd., 1989.
L C. Evans, Partial Differential Equation,
American Mathematical Society, 1998.
M. Renardy and R. C. Rogers, An Introduction
to Partial Differential Equations,
SpringerVerlag, 2004.
G. B. Folland, Introduction to Partial
Differential Equations, 2^{nd} Ed., PrenticeHall of India, 1995.
R. C. McOwen, Partial Differential
Equations: Methods and Applications, 2^{nd} Ed., Pearson Education,
Inc., 2003.
MA
818 Partial Differential Equations II
Nonlinear FirstOrder Scalar Equations: Method of
Characteristics, Weak Solutions and Uniqueness for HamiltonJacobi Equations,
Scalar Conservation Laws: shocks and entropy condition, weak solutions and
uniqueness, and long time behavior.
Calculus of Variations: EulerLagrange Equation, Second Variations,
Existence of Minimizers: Coercivity, LowerSemicontinuity, Convexity, and
Constrained Minimization Problems.
HamiltonJacobi Equations: Viscosity Solutions, Uniqueness,
Applications to Control Theory and Dynamic Programming.
System of Conservation Laws: Theory of Shock
Waves, Traveling Waves, Entropy Criteria, Riemann Problem, Glimm Existence
Result for System of Two Conservation Laws.
Text/References:
L C. Evans, Partial Differential Equations,
American Mathematical Society, 1998.
M. Renardy and R. C. Rogers, An
Introduction to Partial Differential Equations, Springer, 2004.
M. Defermos, Hyperbolic Conservation Laws
in Continuum Physics, Springer, 2000.
B. Dacorogna, Direct Methods in Calculus of
Variation, Springer 1989.
P. Prasad and R. Ravindran, Partial
Differential Equations, Wiley Eastern, 1985.
J. Smoller, Shock Waves and ReactionDiffusion
Equations, Springer, 1993.
MA
833 Weak Convergence and Martingale Theory
Review of conditional expectation :
Conditional expectation and conditional probability, regular conditional
distributions, disintegration, conditional independence.
Martingales and Stopping times : Stopping
times, random time change, martingale property, optional sampling theorem,
maximum and upcrossing inequalities, martingale convergence theorem.
Gaussian processes and Brownian motion:
Symmetries of Gaussian distribution, existence and path properties of Brownian
motion, law of iterated logarithm.
Weak convergence in metric spaces with special reference to C[0, 1],
Martingale central limit theorem.
Texts/
References :
P. Billingsley, Convergence of probability
measures, Wiley, 1999.
K. R. Parthasarathy, Probability measures
on metric spaces, Academic press, 1967.
V S. Borkar, Probability theory : an
advanced course, Springer, NewYork, 1995.
A. W. VanderVaart and J. A. Wellner, Weak
convergence and empirical processes:
With applications to Statistics,
SpringerVerlag, 1996.
D. Williams, Probability with martingales,
Cambridge Mathematical textbooks, 1991.
MA
820 Stochastic Processes
Review of discrete time Markov chains:
Markov property and transition kernels, invariant distributions, recurrence and
transience, ergodic behaviour of irreducible chains.
Stationary processes and ergodic theory:
Stationarity, invariance and ergodicity, discrete and continuous time ergodic
theorems.
Poisson and pure jump Markov processes:
Random measures and Poisson point processes, mixed Poisson and binomial
processes, independence and symmetry criteria, Markov transition and rate
kernels, embedded Markov chains and explosion.
Levy processes: Levy processes and
subordinators, stable processes, infinitely divisible distributions.
Markov processes: Transition and
contraction operators, ratio ergodic theorem, spacetime invariance, Harris
recurrence and transience, existence and uniqueness of invariant measures.
Texts/
References :
O. Kallenberg, Foundations of Modern
Probability, 2^{nd} Ed., Springer, 2000.
R. B. Ash and M. F. Gardner, Topics in
Stochastic Processes, Academic Press, 1975.
D. W. Stroock, Markov processes from K. Ito's
Perspective, Annals of
Mathematics Studies, No. 155, Princeton
University Press, 2003.
MA
821 Theory of Estimation
Elements of decision theory such as complete
class theorem, admissibility of Bayes rule, Minmax Theorem
Review of sufficiency, consistency and
efficiency. UMVU estimators and their properties. Application to normal and
exponential one and two sample problems. Information inequality(multiple
parameter case) Equivariance, Invariance. Application to location and scale
families.
MRE estimation.
Bayes and minimax estimation for
exponential families. Admissibility of estimators, Blyth's ratio method, Karlin's
sufficient conditions.
Pitman's estimator and its properties,
Simultaneous estimation. Stein's phenomenon, Shrinkage estimation.
Texts/
References:
J. Berger, Statistical decision theory,
SpringerVerlag, 1980.
T. S. Ferguson, Mathematical Statistics: A
Decision Theoretic Approach,
Academic Press, 1967.
E. L. Lehmann, Theory of Statistical
Inference, Wiley, 1983.
S. Zacks, The Theory of Statistical
Inference, Wiley, 1971.
Jun Shao, Mathematical Statistics, 2^{nd}
Ed., Springer, 2003.
MA
822 Testing of Hypothesis
UMP tests, NeymannPearson fundamental
lemma, Distributions with ML ratio, Confidence bounds, Generalization of the
fundamental lemma.
Least favourable distributions,
applications to normal distribution.
Similarity and completeness, UMP unbiased
twosided tests, Applications to exponential families, FisherBeherns problem,
Unbiased confidence sets. Most powerful permutation and invariant tests,
Admissibility of tests, Chitests and invariance, The HuntStein theorem and
its applications.
Texts/References:
T. S. Ferguson, Mathematical Statistics: A
Decision Theoretic Approach,
Academic Press, 1967.
L. Le. Cam, Asymptotic in Statistics,
SpringerVerlag, 1990.
E. L. Lehmann, Testing Statistical
Hypotheses, Wiley, 1986.
Jun Shao, Mathematical Statistics, 2^{nd}
Ed., Springer, 2003.
MA
824 Functional Analysis
Review of normed linear spaces, HahnBanach
theorems, uniform boundedness principle, open mapping theorem, closed graph
theorem, Riesz representation theorem on Hilbert spaces.
Weak and weak* convergence, reflexivity in
the setting of normed linear spaces.
Compact operators, SturmLiouville
problems.
Spectral projections, spectral
decomposition theorem, spectral theorem for a bounded normal operator, unbounded
operators, spectral theorem for an unbounded normal operator.
Text/References:
M. Ahues, A. Largillier and B. V. Limaye,
Spectral Computations for Bounded Operators, Chapman & Hall/CRC, 2001.
J. B. Conway, Functional Analysis, 2nd Ed.,
SpringerVerlag, 1990.
S. Lang, Complex Analysis, 4th Ed.,
Springer, 1999.
B. V. Limaye, Functional Analysis, 2nd Ed.,
New Age International Publishers, 1996.
F. Riesz and B. SzNagy, Functional
Analysis, Dover Publications, 1990.
W. Rudin, Functional Analysis, Tata McGraw
Hill, 1974.
K. Yosida, Functional Analysis, 5th Ed.,
Narosa, 1979.
MA
841 Topics in Algebra I
A
selection of topics from the following:
Regular
sequences, grade and depth. Projective dimension, AuslanderBuchsbaum formula.
Koszul complex. Rank of modules.
BuchsbaumEisenbud acyclicity criterion. Graded rings and modules. Basic
properties of graded modules: associated primes, dimension etc.
Hensel's
Lemma, Newton' Theorem and Weierstrass Preparation Theorem.
Chevalley's
Theorem on invariants of a finite
pseudoreflection group acting on the polynomial ring.
The
Jacobian criterion for regularity.
Divisor class group of a noetherian normal domain and its properties
under ring extensions etc. Applications to unique factorization.
CohenMacaulay
rings. Homological
characterization of regular local rings.
Injective
hulls, Matlis Duality. Local cohomology. Basic properties. Invariance under
flat and finite base changes. Canonical module: Existence and basic
properties. Local duality
and applications. Canonical module of graded rings.
Text/References:
S. S.
Abhyankar, Lectures on Algebra, Vol. I, World
Scientific, Hackensack, NJ, 2006.
W.
Bruns and J. Herzog, CohenMacaulay Rings, Revised second edition, Cambridge
University Press, 1998
H.
Matsumura, Commutative Ring Theory, Cambridge University Press, 1989.
A
selection of topics from the following:
CohenMacaulay rings and modules, Canonical Module, Gorenstein
rings.
Hilbert functions and multiplicities, Macaulay's Theorem
StanleyReisner rings, shellability.
Semigroup rings and rings of invariants
Determinantal rings, Straightening law.
Big
CohenMacaulay modules, Hochster's finiteness theorem.
Texts/References:
S. S. Abhyankar,
Lectures on Algebra, Vol. I, World
Scientific, Hackensack, NJ,
2006.
W. Bruns and J. Herzog, CohenMacaulay Rings, Revised second edition, Cambridge
University
Press, 1998
H. Matsumura, Commutative Ring Theory, Cambridge
University Press, 1989.
MA
843 Topics in Analysis I
A
selection of topics from the following:
Singular Integrals (CalderonZygmund theory), the Kakeya
problem, the Uncertainty Principle, the almost everywhere convergence of
Fourier series, multilinear operators between Lp spaces.
Pseudodifferential operators, Index theorems.
Advanced complex analysis in one variable: Nevanlina theory,
the existence of quasiconformal maps, iterated polynomial maps, complex
dynamics, compact Riemann surfaces, the Corona theorem.
Holomorphic functions in several complex variables:
elementary properties of functions of several complex variables, analytic
continuation, subharmonic functions, Hartog's theorem, automorphisms of bounded
domains.
Texts/References:
R.C. Gunning, Introduction to holomorphic functions of
several variables. Vol. I. Function theory, Wadsworth & Brooks/Cole, 1990.
A.W. Knapp, Advanced real analysis, Birkhauser, 2005.
S. Lang and W. Cherry, Topics in Nevanlinna theory,
SpringerVerlag, 1990.
R. Narasimhan, Several complex variables, University of
Chicago Press, 1995.
E.M. Stein, Harmonic Analysis: Real Variable
Methods,Orthogonality, and Oscillatory Integrals, Princeton University Press,
1993.
S. Thangavelu, An Introduction to the Uncertainty Principle:
Hardy's Theorem on Lie Groups, Birkhauser, 2004.
MA
844 Topics in Analysis II
A
selection of topics from the following:
Fourier Series and
Fourier Transforms: Orthonormal
Sequences in Inner Product Spaces, Fourier Series, RiemannLebesgue Lemma,
Convergence/Divergence of Fourier Series, Fejer Theory, Fourier Transform,
Inversion Theorem, Approximate Identities, Plancherel Theorem
H^{p}
spaces: Harmonic
and Subharmonic Functions, H^{p} spaces, Nevanlinna Class of Functions,
Boundary Values, Nontangential Limits, F. and M. Riesz Theorem, Inner
Functions, Outer Functions, Factorization Theorems, Beurling's Theorem
Banach Algebras: Examples of Banach Algebras,
Spectrum, Gelfand Representation, C*Algebras, Positive Linear Functionals,
GelfandNaimark Representation
Perturbation
Theory for Linear Operators: Analyticity of the resolvent
operator, spectral projection and the weighted mean of the eigenvalues, The
method of majorizing series, Spectral Decomposition Theorem.
Spectral Approximation: Norm and nu convergence,
Iterative refinement methods such as the RayleighSchrodinger series and methods based on the
fixed point techniques, error estimates.
Approximate
solutions of Operator Equations: Galerkin, Iterated Galerkin and Nystrom methods, Condition
Numbers, Two Grid Methods.
Texts/References:
M. Ahues, A. Largillier, B.V. Limaye, Spectral Computation
for bounded operators, Chapman & Hall/CRC, 2001.
K.E. Atkinson, The Numerical Solution of Integral Equations
of the Second Kind, Cambridge University Press, 1997.
G. Bachman, L. Narici and E. Beckenstein, Fourier and
Wavelet Analysis, SpringerVerlag, 2000.
S. K. Berberian, Lectures in Functional Analysis and
Operator Theory, Narosa Publishing House, 1979.
F. Chatelin, Spectral Approximation of Linear Operators,
Academic Press, 1983.
J.B. Conway, A Course in Functional Analysis,
SpringerVerlag, 1985.
P.L. Duren, Theory of H^{p} spaces, Dover
Publications, 2000.
W. Hackbusch, Integral Equations: Theory and Numerical
Treatment, Birkhauser, 1995.
T.
Kato, Perturbation Theory for
Linear Operators, SpringerVerlag, 1995.
R. Kress, Linear Integral Equations, Second Edition,
SpringerVerlag, 1999.
P. Koosis, Introduction to H^{p}_{ }spaces,
2^{nd} Edition, Cambridge University Press, 1999.
C.S.
Kubrusly, An Introduction to Models and Decompositions in Operator Theory,
Birkhauser, 1997.
G.J. Murphy, C*Algebras and Operator Theory, Academic Press
Inc., 1990.
W. Rudin, Real and Complex Analysis, McGrawHill, 1987.
W.
Rudin, Functional Analysis, McGraw Hill, 1991.
A.
Vretblad, Fourier Analysis and its Applications, SpringerVerlag, 2005.
MA
845 Topics in Combinatorics I
A selection of topics from the following:
Basic Combinatorial Objects : Sets, multisets, partitions of sets, partitions
of numbers, finite vector spaces, permutations, graphs etc.
Basic Counting Coefficients: The twelve fold way, binomial,
qbinomial and the Stirling coefficients, permutation statistics, etc.
Sieve Methods : Principle of inclusionexclusion,
permutations with restricted positions, Signreversing involutions,
determinants etc.
Combinatorial reciprocity.
Theory of
Symmetric functions.
Texts/References:
C.
Berge, Principles of Combinatorics, Academic Press, 1972.
I.G. Macdonald,
Symmetric functions and Hall polynomials. Second edition, Oxford University Press, 1995.
R.P.
Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole, 1986.
MA
846 Topics in Combinatorics II
A
selection of topics from the following:
Partially ordered sets, Mobius
inversion.
Rational generating functions:
Ppartitions and linear Diophantine equations.
Polya theory and representation
theory of the symmetric group:
Combinatorial algorithms, and
symmetric functions.
Generating functions: Single and
multivariable Lagrange inversion.
Young
tableaux and plane partitions
Texts/References:
M. Aigner, Combinatorial Theory,
SpringerVerlag, New York, 1979.
I. G. Macdonald, Symmetric
functions and Hall polynomials. Second edition, Oxford University Press, New York, 1995.
B.E. Sagan, The Symmetric Group:
Representations, Combinatorial Algorithms and Symmetric Functions, Wadsworth
& Brooks/Cole, 1991.
R. P. Stanley, Enumerative
Combinatorics, Vol. I, Wadsworth and Brooks/Cole, Monterey, CA, 1986.
R. P.
Stanley, Enumerative Combinatorics, Vol. II, Cambridge
University Press, Cambridge, 1999.
MA 847 Topics in
Geometry I
A
selection of topics from the following:
Review
of the theory of curves and surfaces in the Euclidean 3space.
Differentiable manifolds, and Riemannian structures. Connections, and curvature
tensor.
The theorems of BonnetMeyers and Hadamard. Manifolds of constant curvature.
Texts/References:
J. M. Lee,
Riemannian Manifolds: An Introduction to Curvature, SpringerVerlag, New York, 1997.
W. M.
Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd
edition, Academic
Press, 2002.
M. Do
Carmo, Differential Geometry of Curves and Surfaces,
Prentice
Hall, 1976.
S. Kumaresan, A Course in Differential Geometry and Lie Groups,
Hindustan Book Agency, 2002.
J. Milnor, Morse Theory, Princeton University Press, 1963.
A
selection of topics from the following:
Affine and projective varieties, rational maps,
nonsingularity.
Algebraic Curves, Riemann Roch Theorem.
Sheaves and Schemes. Basic properties. Divisors and
Differentials.
Cohomology
of sheaves, Serre Duality Theorem.
Texts/References:
S. S. Abhyankar,
Algebraic Geometry for Scientists and Engineers, American Mathematical Society, Providence, RI, 1990.
D. Eisenbud and J. Harris, The Geometry of Schemes, SpringerVerlag, 2000.
R. Hartshorne, Algebraic Geometry, SpringerVerlag, 1977.
I. R. Shafarevich, Basic Algebraic Geometry, Vol. 1 and 2,
Second edition, SpringerVerlag, 1994.
A
selection of topics from the following:
CW complexes, Homotopy groups, Cellular Approximation.
Whitehead's theorem, Hurewicz theorem.
Excision, Fibre bundles, Long exact sequences.
Postnikov Towers, Obstruction Theory.
Stable
homotopy groups. Spectral Sequences, Serre Class of abelian groups.
Texts/References:
B.
Gray, Homotopy Theory, Academic Press, 1975.
A. Hatcher, Algebraic Topology, Cambridge University Press
2002.
G. W. Whitehead, Elements of Homotopy Theory, Springer
Verlag, 1978.
P. Hilton, Homotopy Theory and Duality, Gordon and Beach Sc.
Publishers, 1965.
N. Steenrod, The Topology of Fibre Bundles, 7^{th}
reprint, Princeton University Press, 1999.
R. M.
Switzer, Algebraic topology:
Homotopy and Homology, Springer Verlag, 2002.
Basics
of Topological groups, Lie group.
Group
actions, homogeneous spaces examples.
Gspaces,
existence of slice and tubes
Covering
homotopy theorem, Classification of GSpaces.
Finite
group actions, homology spheres
Gcoverings,
Cech theory
Locally
smooth actions, orbit types,
principal orbits
Actions
of tori.
Cohomology
structure of fixed point sets, Z_pactions projective spaces and product of
spheres.
Texts/References:
G. E. Bredon, Introduction to Compact Transformation Groups, Academic
Press 1972.
T.
Bröcker and T. tom Dieck, Representations of Compact Lie Groups, SpringerVerlag,
New York, 1985.
W. Y.
Hsiang, Cohomology Theory of Topological Transformation Groups,
SpringerVerlag, 1975.
A
selection of topics from the following:
Algebraic number theory, abelian and nonabelian reciprocity
laws, the Langlands programme, automorphic forms and representations.
The arithmetic of algebraic groups.
Arithmetic algebraic geometry: counting rational points of
varieties over finite fields
Galois representations and galois cohomology.
Additive
number theory: partitions, compositions, Goldbach problem.
Texts/References:
S.
Lang, Algebraic number theory., Second edition, SpringerVerlag, New York,
1994.
D. Bump, Automorphic forms and representations, Cambridge
University Press, Cambridge, 1997.
H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society, Providence, RI, 2004.
H.
Hida, Modular forms and Galois cohomology, Cambridge University Press, Cambridge, 2000.
MA 852 Topics in
Number Theory II
A
selection of topics from the following:
Harmonic analysis on Lie groups, Lfunctions, ladic
representations and motives.
Diophantine equations and the applications of Ktheory to
number theory.
Analytic number theory and transcendental methods.
Applications
of ergodic theory to number theory.
Texts/References:
S.
Lang, Algebraic number theory., Second edition, SpringerVerlag, New York,
1994.
D.
Bump, Automorphic forms and representations, Cambridge University Press,
Cambridge, 1997.
H.
Iwaniec and E. Kowalski, Analytic
number theory, American
Mathematical Society, Providence, RI,
2004.
H.
Hida, Modular forms and Galois cohomology, Cambridge University Press, Cambridge, 2000.
MA 853 Topics in
Differential Equations I
A
selection of topics from the following:
Schauder
theory, regularity for second order elliptic equations. Nonlinear analysis and
its applications to nonlinear PDEs: Fixed point methods, variational methods,
monotone iteration, degree theory.
Evolution
equations: Existence via semigroup theory
Nonlinear
Hyperbolic systems: Theory of well posedness, compensated compactness,
Young measures;
propagation of oscillations, weakly nonlinear geometric optics.
Texts/References:
D.
Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second
Order, SpringerVerlag, 1983.
P. Grisvard,
Elliptic Problems in Nonsmooth Domains, Pitman, 1984.
D.
Serre, Systems of Conservation Laws, Vols. 1, 2, Cambridge University Press, 2000.
L.
Evans, Weak Convergence Methods for Nonlinear PDEs, CBMS Regional Conference series in Math., American
Mathematical Society, Providence RI, 1990
A.
Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic
Structures, North Holland, 1978.
M.
Struwe, Variational Methods: Applications to nonlinear PDEs and Hamiltonian
systems, SpringerVerlag, 1990.
MA 854 Topics in
Differential Equations II
A
selection of topics from the following:
Diffeomorphisms
and flows: Elementary dynamics of diffeomorphisms, flows and differential
equations, conjugacy, equivalence of flows, Sternberg's theorem on smooth
conjugacy (statement only), Hamiltonian flows and Poincare maps.
Local
properties of flows and diffeomorphisms: Hyperbolic fixed points,
HartmanGrobman theorems for maps and flows, Normal forms for vector fields,
Centre manifolds.
Structural
stability and hyperbolicity: Structural stability for linear systems, Flows on
2dimensional manifolds, Peixoto's
characterisation of structural stability on unit disc, Anosov and Horseshoe diffeomorphisms, Homoclinic
points, Melnikov function.
Bifurcations
and Perturbations: Saddlenode and Hopf
bifurcations, AndronovHopf bifurcation, The logistic map, Arnold's
circle map; Perturbation theory: Melnikov's method for the study of
perturbation of completely
integrable systems.
Floquet
theory and Hill's equation and some of its applications.
Two
dimensional systems: PoincareBendixon theorem, Index of planar vector fields
and the Poincare Hopf index theorem for two dimensional manifolds.
Van
der Pol's equation, Duffing's equation, Lorenz's equation.
First
integrals and functional independence of first integrals, notion of complete integrability, Jacobi
multipliers, Liouville's theorem on preservation of phase volume, Jacobi's last multiplier theorem and its
applications.
Texts/References:
D.
K. Arrowsmith, C. M. Place: An Introduction to Dynamical Systems, Cambridge University Press,
1990.
C.
Chicone, Ordinary Differential Equations. SpringerVerlag, 1999.
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical
Systems, and Bifurcations of Vector Fields. SpringerVerlag, 2002.
P.Glendinning, Stability, instability
and chaos: An Introduction to the Theory
of Nonlinear Differential
Equations, Cambridge University Press, 1994.
J. Palis and W. C. de Melo, Geometric Theory of Dynamical
Systems, SpringerVerlag, 1982.
R. Grimshaw, Nonlinear Ordinary Differential Equations. CRC
press, 1991.
N.A. Magnitskii and S.V. Sidorov, New
Methods for Chaotic Dynamics,
World Scientific, 2006.
L. Perko, Differential Equations and Dynamical Systems,
SpringerVerlag, 2001.
A
selection of topics from the following:
Review
of finite difference methods for elliptic, parabolic and hyperbolic problems.
Stability, consistency and convergence theory.
Finite
difference schemes for scalar conservation laws (LaxFriedrichs, Upwind,
LaxWendroff, etc.), Conservative schemes and their numerical flux functions,
Consistency, LaxWendroff Theorem, CFL Condition, Nonlinear Stability and TVD
property, Monotone Difference schemes, Numerical entropy condition, Convergence
result.
Finite
difference Schemes for onedimensional system of conservation laws, approximate
Riemann solvers, Godunov's method, High resolution methods, Multidimensional approaches.
Large Scale
Scientific Computing: Classical
Iterative Methods for solving Linear systems, Large Sparse Linear Systems,
Storage Schemes, GMRES algorithm, Preconditioned Conjugate Gradient method and Multigrid
method, Newton's Method and some of its variations for solving nonlinear
systems.
Texts/References:
Axelsson, O. Iterative Solution Methods, Cambridge
University Press, 1994.
Briggs,
W. L., Henson, V. E. and McCormick, S. F. A Multigrid tutorial, SIAM, 2000.
Godlewski,
E. and Raviart, P. –A. Numerical Approximation of Hyperbolic Systems of
Conservation Laws, Springer, 1995.
Kroner,
D. Numerical Schemes for Conservation Laws. John Wiley, 1997.
LeVeque,
R. J. Finite Volume Methods for Hyperbolic Problems, Cambridge University
Press, 2002.
LeVeque,
R. J. Numerical Methods for
Conservation Laws. Birkhauser, 1992.
Quarteroni,
A. and Valli, A. Numerical Approximation of Partial Differential Equations,
Springer, 1997.
A
selection of topics from the following:
Mixed Finite Element Methods: Examples of mixed variational formulations
primal, dual formulations; abstract mixed formulations, discrete mixed
formulations, existenceuniqueness of solutions, convergence analysis,
implementation procedures.
Adaptive FEM: A study of Explicit A
posteriori error estimators, Implicit A posteriori estimators, Recovery based
error estimators, Goal Oriented adaptive mesh refinement for second order
elliptic boundary value problems.
Discontinuous Galerkin Methods
for second order elliptic boundary value problems:
Global element methods,
Symmetric Interior Penalty Method, Discontinuous hp Galerkin Method,
Nonsymmetric interior penalty method: Consistency, approximation properties,
existence and uniqueness of solutions, error estimates, implementation
procedures.
FEM for
parabolic problems: The standard Galerkin method,
semidiscretization in space. discretization in space and time, the
discontinuous Galerkin Method, a mixed method, implementation procedures.
Elements of Multigrid Methods: Multigrid Components 
Interpolation, restriction Coarsegrid correction, V, W, and FMG cycles,
Implementation, Convergence analysis, Performance diagnostics.
Texts/References:
Z.
Chen, Finite Element Methods And Their Applications, SpringerVerlag, New
York, 2005.
S.
C. Brenner and R. L. Scott, The Mathematical Theory of Finite Element Methods,
2^{nd} Edition, SpringerVerlag, New York, 2002.
M.
Ainsworth and J. T. Oden, A
Posteriori Error Estimation in Finite Element Analysis, John Wiley and Sons,
2000.
V. Thomee, Galerkin Finite
Element Methods for Parabolic Problems, 2^{nd} Edition,
SpringerVerlag, Berlin, 2006.
A
selection of topics from the following:
Stochastic Integrals : Ito integral, Ito's formula.
Stochastic differential equations
: strong and weak solutions,
Existence and uniqueness results,
Martingale formulation,
moment estimates, comparison theorems.
Texts/References:
I. Karatzas and S. E.
Shreve, Brownian motion and
stochastic calculus, 2^{nd} Edition, SpringerVerlag, 1991.
D. W. Stroock and S.R.S.
Varadhan, Multidimensional Diffusion Processes, SpringerVerlag, 1979.
A
selection of topics from the following:
Stochastic optimal control: compactness of laws, dynamic programming
principle.
Malliavin calculus and applications to
finance: WienerIto chaos
expansion, Shorohod integral, Integration by parts formula, Clark Ocone
formula and application to finance.
Texts/References:
V.S. Borkar, Optimal control of diffusion processes, Longman Scientific and Technical,
Harlow (copublished by John Wiley), 1989.
D. Nualart, The Malliavin calculus and related
topics, SpringerVerlag, 1995.
A
selection of topics from the following:
Univariate
Stochastic Ordershazard rate order, likelihood ratio order, mean residual rate
order. Univariate variability orders convex order, dispersive order,
peakedness order. Univariate
monotone convex and related orders. Multivariate stochastic orders. Multivariate
variability and related orders. Statistical Inference for stochastic ordering.
Applications in reliability theory, biology, economics and scheduling.
Texts/References:
J.
George Shanthikumar and Moshe Shaked (1994) Stochastic Orders and their
Applications, Academic press.
C.D.
Lai and M. Xie (2006) Stochastic Ageing and Dependence for Reliability,
Springer Verlag.
MA 860 Topics in
Statistics II
A
selection of topics from the following:
Inference
in Semiparametric models: Models
with infinite imensional parameters, Efficient estimation and the delta method,
Score and information operators, Estimating equations, Maximum Likelihood
estimation, Testing.
Generalized
linear models: Components of a GLM, estimation techniques, diagnostics, continuous
response models, Binomial response models, Poisson response models,
overdispersion, multivariate GLMs, quasi likelihoods, generalized estimating
equations, generalized linear mixed models, programming in R and SAS.
Texts/References:
A. W.
Van der Vaart, Asymptotic
Statistics, Cambridge University Press, 2000.
U.
Grenander, Abstract Inference, John Wiley, 1981.
P.
McCullagh and J. A. Nelder, Generalized Linear
Models, 2^{nd} Edition, Chapman and Hall/CRC, 1994.
L.
Fahrmeir and G. Tutz, Multivariate Statistical Modeling based on Generalized
Linear Models, 2^{nd} Edition, SpringerVerlag, 1994.
R. H.
Myers, D. C. Montgomery and G. Geoffrey Vining, Generalized Linear Models with
applications in Engineering and Sciences, WileyInterscience, 2001.