ANNUAL FOUNDATION SCHOOL (AFS)  II (2014)
Supported by National Centre for Mathematics
Venue: KSOM, Kozhikode
0128 May, 2014
Conveners  

M Manickam Professor & Director(i/c) Kerala School of Mathematics, Kozhikode 
B Ramakrishnan Professor HRI, Ahmedabad 
Description: The main objectives of AFS are to bring up students with diverse background to a common level and help them acquire basic knowledge in algebra, analysis and topology. This programme is organised by National Centre for Mathematics (NCM).
Syllabus :
Ring Theory.
(1) Modules over Principal Ideal Domains Modules, direct sums, free modules, nitely generated modules over a PID, structure of nitely generated abelian groups, rational and Jordan canonical form.
(2) Basics Commutative rings, nil radical, Jacobson radical, localization of rings and modules, Noetherian rings, primary decomposition of ideals and modules.
(3) Integral extensions of rings, Going up and going down theorems, niteness of integral closure, discrete valuation rings, Krull's normality criterion, Noether normalization lemma, Hilbert's Nullstellensatz
(4) Semisimple rings, Wedderburn's Theorem, Rings with chain conditions and Artin's theorem, Wedderburn's main theorem,
Functional Analysis.
(1) Normed linear spaces, Continuous linear transformations, application
to dierential equations, HahnBanach theoremsanalytic and geometric versions, vector valued integration.
(2) Bounded Linear maps on Banach Spaces Baire's theorem and applications: Uniform boundedness principle and application to Fourier series, Open mapping and closed graph theorems, annihilators, complemented subspaces, unbounded operators and adjoints
(3) Bounded linear functionals Weak and weak* topologies, Applications to re exive separable spaces, Uniformly convex spaces, Application to calculus of variations
(4) Hilbert spaces, Riesz representation theorem, LaxMilgram lemma and application to variational inequalities, Orthonormal bases, Applications to Fourier series and examples of special functions like Legendre and Hermite polynomials.
Differential Topology.
(1) Review of dierential calculus of several variables: Inverse and implicit function theorems. Richness of smooth functions; smooth partition of unity, Submanifolds of Euclidean spaces (without and with boundary) Tangent space, embeddings, immersions and submersions, Regular values, preimage theorem, Transversality and Stability. [The above material should be supported amply by exercises and examples from matrix groups.]
(2) Abstract topological and smooth manifolds, partition of unity, Fundamental gluing lemma with criterion for Hausdorness of the quotient, classication of 1manifolds. Denition of a vector bundle and tangent bundle as an example. Sard's theorem. Easy Whitney embedding theorems.
(3) Vector elds and isotopies Normal bundle and Tubular neighbourhood theorem. Orientation on manifolds and on normal bundles. Vector elds. Isotopy extension theorem. Disc Theorem. Collar neighbourhood theorem.
(4) Intersection Theory: Transverse homotopy theorem and oriented intersection number. Degree of maps both oriented and non oriented cases, winding number, Jordan Brouwer separation theorem, Borsuk
Ulam theorem.
Resource Persons
Prof. B Ramakrishnan, HRI, Allahabad
Prof. M. Manickam, KSOM, Kozhikode
Dr. Kavita Sutar, CMI, Chennai
Dr. Priyavrat Deshpande, CMI, Chennai
Prof. Purusottam Rath, CMI, Chennai
Prof. Sanoli Gun, IMSc, Chennai
Dr. Manoj Kummini, CMI, Chennai
Dr. Krishna Hanumanthu,CMI, Chennai
Prof. R. Thangadurai,HRI, Allahabad
Prof. D.Surya Ramana ,HRI, Allahabad
Prof. Satya Deo ,HRI, Allahabad
ATM Workshop on Graph Theory(2014)
Supported by National Centre for Mathematics
Venue: KSOM, Kozhikode
1722 March, 2014
Conveners  

M Manickam Professor & Director(i/c) Kerala School of Mathematics, Kozhikode 
S S Sane Professor Department of Mathematics, IIT Bombay 
Description and salient features of the syllabus: This workshop covered some topics in Graph Theory, including both algorithmic and nonalgorithmic that are not normally covered in a first course on Graph Theory. The topics include spectral graph theory, some topics in algebraic graph theory leading to Tutte polynomial of a graph, extremal graph theory including the Szemeredi regularity lemma and some topics in algorithmic graph theory including reducibility of 3SAT.
Resource Persons
Professor R. Balakrishnan, Department of Mathematics, Bharathidasan University, Trichy
Professor Ajit Diwan, Department of Computer Science, I.I.T. Bombay, Mumbai
Professor S.A. Choudum, (Retired) Professor, I.I.T. Madras, (at present in) Bangalore
Annual Foundation School  Part I (2013)
Supported by National Centre for Mathematics
Venue: KSOM, Kozhikode
0228 December, 2013
Group Theory:
Speaker: B. Ramakrishnan
Group actions, Sylow Theory, direct and semidirect products, simplicity of the alternating groups, solvable groups, pgroups, nilpotent groups, JordanHolder theorem.
Speaker: M. Manickam
Free groups, generators and relations, finite subgroups of SO(3), SU(2), simplicity of PSL(V).
Representations and characters of finite groups: Maschke’s Theorem, Schur’s lemma, characters, orthogonality relations, character tables of some groups, Burnside’s theorem.
Speaker: R. Thangadurai
Abstract measures, outer measure, completion of a measure, construction of the Lebesgue measure, nonmeasurable sets.
Measurable functions, approximation by simple functions, Cantor function, almost uniform convergence, Egoroff and Lusin’s theorems, convergence in measure.
Speaker: P. K. Ratnakumar
Integration, monotone and dominated convergence theorems, comparison with the Riemann integral, signed measures and RadonNikodym theorem.
Speaker: K. Sandeep
Fubini’s theorem, Lp  spaces.
Differential Topology:
Speakers: A. J. Jayanthan
Review of differential calculus of several variables: Inverse and Implicit function theorems. , Richness of smooth functions; Smooth partition of unity, Submanifolds of Euclidean spaces (without and with boundary), Tangent space, embeddings, immersions and submersions, Regular values, preimage theorem , Transversality and Stability
Speakers: P. K. Ratnakumar
Topological and smooth manifolds, partition of unity , Fundamental gluing lemma and classication of 1manifolds, Vector bundle; Tangent bundle, Morse  Sard theorem, Easy Whitney embedding theorems.
Speakers: P. Sankaran
Orientation on manifolds. , Transverse Homotopy theorem and oriented intersection number , Degree of maps both oriented and non oriented case, Winding number, Jordan Brouwer Separation theorem, Borsuk  Ulam Theorem, Vector eld and isotopies (statement of theorems only) with application to Hopf Degree theorem.
Speakers: G. Santhanam
Morse functions, Morse Lemma, Connected sum, attaching handles , Handle decompostion theorem, Application to smooth classication of compact smooth surfaces.
Workshop and International Conference on Automorphic Forms and Number Theory
Venue: KSOM, Kozhikode
30^{th} August  3^{rd} September, 2013
Objective
Recent developments in various branches of mathematics and physics show an increasing
interest in the explicit construction of Jacobi forms, in particular, Jacobi forms of matrix (or,
equivalently, lattice) index. This has to do with their appearance in quantum field theory, module spaces of surfaces, infinite dimensional Lie algebras. Though Jacobi forms of matrix index are
known since quite long it is only recently that there was an essential breakthrough in the theory. For example, recent work of Gritsenko, Skoruppa, Zagier shows that Jacobi forms can be
constructed in a surprisingly explicit way. These constructions in turn are closely related to
classical problems in the arithmetic theory of lattices and in the theory of trigonometric
polynomials.
For getting interested researchers or graduate students up to date in recent developments we
have conducted a 3 days instructional workshop on Jacobi forms of lattice index, followed by a 2
days conference, where participants of the workshop got an opportunity to report on
their own research projects for initiating discussions, obtaining feedback or help.
Part 1: Workshop on Jacobi forms of lattice index
Organizers: Prof. M. Manickam, Prof. Ramakrishnan B., Prof. Nils Peter Skoruppa
Lecturers: Hatice Boylan (Istanbul Üniversitesi and MaxPlanckInstitute Bonn), Fabien Cléry (University of Siegen), NilsPeter Skoruppa (University of Siegen and MaxPlanckInstitute
Bonn)
Overview: The workshop aimed to give a careful and thorough introduction into the theory of
Jacobi forms of lattice index with emphasis on explicit constructions, in particular, for low
weights and maximal lattices as index. In addition, it showed how Jacobi forms can be
used to construct other types of automorphic forms via the socalled additive and multiplicative
liftings.
Subjects:
● Basic notions: Lattices, shadows, discriminant modules, definition of Jacobi forms,
discussion of the definition, first examples, functorial properties
● Jacobi forms as theta functions: vector valued modular forms, relation between Jacobi
forms and vector valued modular forms, dimension formulas
● Explicit constructions of Jacobi forms: Taylor expansion around 0, Theta blocks,
invariants of Weil representations
● Maximal lattices as index, explicit description of forms of singular and critical weight and
maximal index
● Additive and multiplicative liftings, product expansions
Schedule:
Friday to Sunday: each day 3 lectures of 90 min and 60 minutes for discussions, questions and
problem session
Part 2: Conference on Automorphic Forms and Number Theory
List of Speakers
1  Mr. Ali Ajouz  Siegen University, Germany 
2  Dr. Anandavardhanan .U.K  IIT, Mumbai 
3  Dr. Brundaban Sahu  NISER, Bhubaneshwar 
4  Dr. Fabien Cléry  France 
5  Dr. Hatice Boylan  Istanbul University and MaxPlanck Institute for Mathematics, Bonn 
6  Dr. Jaban Meher  IMSc., Chennai 
7  Dr. Jagathesan .T  RKM vivekananda College, Chennai 
8  Dr. Jayakumar R  RKM vivekananda College, Chennai 
9  Ms. Jisna P  KSOM, Kozhikode 
10  Dr. Karam Deo Shankhadhar  IMSc., Chennai 
11  Dr. Kumarasamy .K  RKM vivekananda College, Chennai 
12  Dr. Manickam .M  KSOM, Kozhikode 
13  Dr. Nils Peter Skoruppa  Siegen University, Germany 
14  Mr. Rahothaman .R  Sona College of Technology, Salem 
15  Dr. Ramakrishnan .B  HRI, Allahabad 
16  Mr. Shankar P  KSOM, Kozhikode 
17  Mr. Srivatsa .V  KSOM, Kozhikode 
18  Dr. Sujay Ashok  IMSc. Chennai 
19  Ms. Tamil Selvi  Alpha College of Engineering, Chennai 
The programme was partially suported by HRI, Allahabad
Instructional Workshop on Logic and Set Theory
Venue: KSOM, Kozhikode
18^{th} February  1^{st} March, 2013
The workshop is meant to benefit research scholars and young teachers of Mathematics, Statistics and Computer Science.
The course will cover fundamentals of propositional logic, first order logic, completeness theorem, model theory with applications in number theory, algebra and geometry.
Cardinal arithmetic, partiallyordered sets, wellordered sets, transfinite induction, ordinal numbers, axiom of choice and its equivalent forms such as Zorn's lemma and wellordering principle and their applications in mathematics, ZermeloFraenkel axioms.
Resource Persons
S. M. Srivastava (ISI, Kolkata)  
H. Sarbadhikari (ISI, Kolkata)  
B.V.Rao (CMI, Chennai)  Lecture Notes  
N. Raja (TIFR, Mumbai)  
R. Ramanujam (IMSc, Chennai). 
Workshop on Number Theory and Dynamical Systems
Venue: KSOM, Kozhikode
0408 February, 2013
Topics Covered:
Linear recurrent sequences and iterations of linear maps
Diophantine approximation and dynamical systems
Continued fractions and the geodesic flow
Scientific Committee
Prof. Yann Bugeaud, Strasbourg University, Mathematics, 7, rue Rene Descartes, 67084 STRASBOURG Cede, France
Prof. Pietro Corvaja, Dipartimento di Matematica e Informatica, University of Udine, Italy.
Prof. S.G. Dani, Indian Institute of Technology, Mumbai, India
Prof. Michel Waldschmidt, Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, Paris 6, France
Local Organising Committee
Prof. M Manickam, Kerala School of Mathematics, Kozhikode, India
Prof. Michel Waldschmidt, Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, Paris 6, France
ATMW Hilbert Modular forms and varieties (2013)
Venue: KSOM, Kozhikode
2131 January, 2013
Organiser
National Centre for Mathematics, a joint centre of IIT Mumbai and TIFR, Mumbai
Brief description of the workshop
The goal of the workshop will be to introduce researchers from scratch to some of the basic concepts in the theory of automorphic forms and varieties attached to GLn over totally real fields. We shall quickly treat some of the basic concepts over the first two or three days, and reserve the latter part of the first week for more advanced topics, some of these may include: padic Hilbert modular forms, Herzig's classification of the mod p representations of GLn over a local field and Taylor's recent construction of Galois representations for GLn over totally real fields.During the second week, there will be a short conference to collect together local experts in the area of modular forms and related areas of number theory. We will in particular concentrate on representation theoretic and padic aspects of the theory of automorphic forms. The conference will have several one hour research level talks every day.
School on Modular Forms
Venue: KSOM, Kozhikode
03  19 October, 2012
In connection with the National Mathematical Year celebrations we had conducted a Workshop on Theory of Prime Numbers and Related Areas from 0610 May 2012 at KSOM. The workshop,School on Modular Forms conducted from 03rd October, 2012 to 19th october 2012 was also related to Ramanujan's work.
Topics Covered:
Basic of modular forms for the full modular group and for its congruence subgroups and related areas.
Lectures delivered by:
Prof. M. Manickam, Kerala School of Mathematics, Kozhikode
Prof. B. Ramakrishnan, Harish Chandra Research Institute, Allahabad.
Prof. NilsPeter Skoruppa, Siegen University, Germany
Special Lectures delivered by:  

Prof. Ananda Vardhanan, IIT, Mumbai  
Prof. Eknath Ghate, TIFR, Mumbai  
Prof. A. Raghuram, IISER, Pune  
Prof. J. Sengupta, TIFR, Mumbai  
Dr. R. Thangadurai, HRI, Allahabad.  
Dr. Brundaban Sahu, NISER, Bhubaneshwar 
Theory of Prime Numbers and Related Areas
Venue: KSOM, Kozhikode
06  10 May, 2012
The Workshop was meant for research fellows, young teachers, PG and bright UG students of Mathematics. The course program covered: Ramanujan's Proof of Betrand's Postulate, Prime Number Theorem, Highly Composite Numbers and its applications and Ramanujan's way of summation.
Resource Persons:
Prof. Ram Murty 

Dr. Anirban Mukhopadyaya Institute of Mathematical Sciences 

Dr. D. S. Ramana 

Dr. R. Thangadurai HarishChandra Research Institute 
Instructional Workshop on Ergodic Theory
Venue: KSOM, Kozhikode
17 October  04 November 2011
The Workshop was meant to benefit research fellows and young teachers of Mathematics and Statistics. The course program covered: Basic Ergodic Theory and Topological Dynamics, Probabilistic Systems, Spectral Theory of Dynamical Systems, Ergodic Theory with groups and Diophantine Approximations, Ergodic Theory in Geometry.
Resource Persons:
M. G. Nadkarni (IIT, Indore)  Lecture Notes  
S. G. Dani (TIFR, Mumbai)  
S. C. Bagchi (ISI, Kolkata)  
Siddharta Bhattacharya (TIFR, Mumbai)  Lecture Notes  
B. V. Rao (CMI, Chennai)  Lecture Notes  
J. Dani (University of Mumbai)  
C. S. Aravinda (TIFR, Bangalore)  
Shiva Shankar (CMI, Chennai)  
J. Mathew (KSOM, Kozhikode) 
School on Algebra, Analysis & Topology
Venue: KSOM, Kozhikode
0107 September, 2011
The School was aimed at M.Sc. Mathematics students of Kerala. The course covered some topics of Algebra, Analysis and Topology which are not usually covered in the M.Sc. curriculum
Resource Persons: Basudeb Dutta (I.I.Sc., Bangalore) 
Workshop on Functional Analysis and Harmonic Analysis
Venue: KSOM, Kozhikode
01 June 2011  10 June 2011
The workshop was aimed at senior postgraduate students with aptitude for research and junior research fellows with research interests broadly in Analysis. The workshop also benefit young teachers of Mathematics teaching at postgraduate level.
Resource Persons Prof. Gadadhar Misra (IISc, Bangalore) 
School on Analysis, Algebra & Topology
Venue: KSOM, Kozhikode
13 Dec 2010  07 Jan 2011
The programme was aimed at junior research fellows and final year postgraduate students in Mathematics from Universities/Colleges in Kerala. Those who have completed postgraduate programme recently and intending to pursue mathematics as a career also appllied. The School was conducted by eminent researchers from Universities/Institutions in India and covered some advanced topics in Algebra, Topology and Analysis which are generally considered to be prerequisites for research scholars in all disciplines of Mathematics. The course consist of rigorous class room lectures and problem solving tutorials.
Resource Persons
U. K. Anandavardhanan , IIT Bombay
Manoj Kumar Keshari , IIT Bombay
Balwant Singh , CBS Bombay
Sudhir Ghorpade , IIT Bombay
Raja Sreedharan , TIFR Bombay
Rudra Pada Sarkar, ISI, Kolkatta
A K Nandakumaran, IISc Bangalore
K Sandeep, TIFR Bangalore
P K Rathnakumar, HRI Allahabad
P K Rathnakumar, HRI Allahabad
Mahuya Datta, ISI, Kolkatta
V Uma , IIT Madras
Amit Hogadi , TIFR Bombay
A J Parameswaran, KSOM
Advanced Instructional School on Schemes & Cohomology
Venue: KSOM, Kozhikode
28 June  16 July, 2010
Conveners: A. J. Parameswaran & V. Balaji
Advanced Instructional School on Schemes and Cohomology
The AIS was directed towards making young researchers in India familiar with the developments in Algebraic Geometry, with special emphasis on Grothendieck’s programme. The three week programme was primarily aimed at early researchers in this field to become familiar and users of the tools and techniques in this subject. To acquire a good knowledge of modern algebraic geometry it is essential to see the “local” aspects coming from Commutative algebra in its interplay in geometry as well as the immense machinery of cohomology. These two themes were the main ones for the workshop. To give a flavour of the manner in which these tools have lead to fundamental theorems in algebraic geometry, the final week stress on some research themes. A team of active researchers in the field had participated for this purpose.
Resource persons
TEV Balaji V. Balaji Vivek Mallik
D. S. Nagaraj Suresh Naik ,
A. J. Parameswaran, K. N. Raghavan
Unity of Mathematics Lectures
Nitin Nitsure, Kapil Paranjape, S. Ramanan,
Pramathanath Sastry , C. S. Seshadri, C. S. Rajan
Jugal Verma, Shiva Shankar
Summer School on Probability: Foundations & Applications
Venue: KSOM, Kozhikode
17 May  04 June, 2010
The Summer School was meant to benefit senior postgraduate students, junior research
fellows and young teachers of Mathematics, Statistics and Computer Science. The
course covered fundamentals of discrete and continuous probability models, law
of large numbers, weak convergence and limit theorems, conditional probability,
Markov Chains and Martingales, large deviations and entropy. It also covered
some applications to number theory, graph theory, computer science and other related
fields.
Resource Persons
M. G. Nadkarni (Univesity of Mumbai)  Lecture Notes  
B. V. Rao (C. M. I., Chennai)  Lecture Notes  
B. Rajeev (I. S. I., Bangalore)  Lecture Notes  
Inder Rana (I. I. T., Mumbai)  Lecture Notes  
R. L. Karandikar (C. M. I., Chennai)  Lecture Notes  
S Ramasubramanian (I. S. I. Bangalore)  Lecture Notes  
Workshop on Statistics
Venue: KSOM, Kozhikode
0506 March, 2010
The workshop was meant exclusively for the Scientists of KSCSTE and the R&D institutions under KSCSTE.
Resource Persons
T.Krishnan obtained his Master's and Ph.D. degrees from the Indian Statistical Institute, Kolkata. He has been on the faculty of this Institute since 1965 until his retirement in 1998 as a Professor of Applied Statistics. After retirement he worked as a fulltime consultant for Cranes Software International Limited, Bangalore, on the development of a statistical software SYSTAT for eight years. Currently he is working for Strand Life Sciences, a Bioinformatics company in Bangalore as a Statistician. His book (with G.J. McLachlan) on the EM Algorithm is a definitive and standard reference on the subject.
R.V.Ramamoorthi obtained his Master's degree from Utkal University and his Ph.D. degree from the Indian Statistical Institute, Kolkata. Since 1982 he has been on the faculty of the Michigan State University, East Lansing, Michigan, U.S.A. where he is currently a Professor of Statistics. He is currently on leave from this position and is a visiting professor at the Indian Institute of Science, Bangalore. . His earlier research was on sufficiency and decision theory. For the last few years he has been working on Bayesian Nonparametric Inference and his book (with J.K.Ghosh) on Bayesian Nonparametrics is an authoritative book on the subject.
Time Schedule
Friday, 05 March 2010
09:3010:00  Inauguration  Dr. K. V. Jayakumar  
10:0011:15  Introduction to Data Analysis  TK (T. Krishnan)  Lecture Notes 
11:3012:45  Basic Probability  RVR (R. V. Ramamoorthi)  Lecture Notes 
12:4514:00  Lunch  
14:1515;30  Descriptive and Graphical Statistics  TK  
15;4517:00  Sampling Distributions and Confidence Intervals  RVR  Lecture Notes 
Saturday, 06 March 2010
10:0011:15  Hypothesis Testing  RVR  Lecture Notes 
11:3012:45  Analysis of Variance and Regression  TK  Lecture Notes 
12:4514:00  Lunch  
14:1515;30  Experimental Designs and Survey Sampling  TK  
15;4517:00  ComputerIntensive Statistical Methods  RVR, TK 
Workshop on Functional Analysis and Harmonic Analysis
Venue: KSOM, Kozhikode
110, February 2010
The workshop was meant to benefit senior postgraduate students, junior research fellows and young teachers of Mathematics and aims to cover the necessary background material for understanding, pursuing and teaching advanced level topics in the two main areas of the workshop. The workshop is meant for senior MSc. students, Junior Research Scholars and young teachers from university departments and col leges from the country.
Topics covered:
Functional analysis: Banach algebras, maximal ideals, GelfandNaimark theorem, spectral theorem
Harmonic analysis: Fourier series, Fourier transforms and PaleyWiener theorems
Resource persons
Prof. Gadadhar Mishra, Indian Institute of Science, Bangalore
Prof. E. K. Narayanan, Indian Institute of Science, Bangalore
Prof. Alladi Sitaram, Indian Institute of Science, Bangalore
Prof. V. S. Sunder, Institute for Mathematical Sciences, Chennai
Prof. B. V. Rajarama Bhat, Indian Statistical Institute, Bangalore
Instructional Workshop on Spectral Theorem
Venue: KSOM, Kozhikode
15th and 16th June 2009
CoSponsored by KSCSTE
Day 1, June 15, 2009, Monday:
Registration : 8:30 a.m.
Inauguration : 10:00 a.m. – 11:00 a.m.
Tea : 11:00 a.m.  11:30 a.m.
Lecture 1, 11:30 a. m12.30 p.m.: Spectral theorem for normal matrices – Dr. Rajarama Bhat
Lecture 2, 12:30 p.m1:30 p.m: Spectral theorem for compact selfadjoint operators
 Dr. G. Ramesh
Lunch break: 1:30 p.m2:30 p.m
Lecture 3, 2:30 p.m3:30 p.m: Topologies of B (H) –Dr. A K Vijayarajan
Tea: 3:30 p.m 4:00 p.m.
Lecture 4, 4:00 p.m.5:00 p.m.: General spectral theorem for bounded operatorsI Dr. Rajarama Bhat
Day 2, June 16, 2009, Tuesday:
Lecture 1, 9:30 a.m10:30 a.m.: General spectral theorem for bounded operators –II Dr. Rajarama Bhat
Tea break: 10:30 a.m11:00 a.m.
Lecture 2, 11:00 a.m12:00 p.m: Double Commutant Theorem  Dr. A K Vijayarajan
Discussions: 12:00 p.m12:30 p.m
Lunch break: 12:30 p.m2:00 p.m
Lecture 3, 2:00 p.m3:00 p.m: Trace Class Operators and Duality Dr. G. Ramesh
Tea break: 3:00 p.m3:30 p.m
Lecture 4, 3:30 p.m4:30 p.m: Some Applications of spectral Theorem
Discussion: 4:30 p.m.5:00 p.m. Dr. Rajarama Bhat
Resource persons & Speakers
Name 
Affiliation 
Designation 
Address/email 
Dr. B. V. Rajarama Bhat 
Indian Statistical Institute 
Professor 
Indian Statistical
Institute, R. V. College(PO) Bangalore560059 
Dr. G. Ramesh 
Indian Statistical Institute 
Postdoctoral Fellow 
Indian Statistical
Institute, R. V. College(PO) 
Dr. Vijayarajan A. K. 
Kerala School of Mathematics 
Associate Professor 
Kerala School of Mathematics 