Number Theory and Discrete Mathematics

313 Pages·2002·8.82 MB·English

by  M. Waldschmidt (auth.)

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Number Theory
and Discrete
Mathematics
A.K. Agarwal
Bruce C. Berndt
Christian F. Krattenthaler
Gary L. Mullen
K. Ramachandra
Michel Waldschmidt
Editors
Springer Basel AG
Editors' address:
AK. Agarwal
Centre for Advanced Study in Mathematics
Panjab University
Chandigarh 160014
India
e-mail: [email protected]
2000 Mathematical Subject Classification 05-06; 11-06; 05A18,11Fxx, 11Gxx, 11 Jxx,
11Mxx,11Rxx14Gxx, 14H52
f
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C,. USA
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Number theory and discrete mathematics / A. K. Agarwal... ed..
(Trends in mathematics)
ISBN 978-3-0348-9481-4 ISBN 978-3-0348-8223-1 (eBook)
DOI 10.1007/978-3-0348-8223-1
This work is subject to copyright. All rights are reserved, whether the whole or part of the
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©2002 Springer Basel AG
Originally published by Birkhäuser Verlag, Basel - Boston - Berlin in 2002
Softcover reprint of the hardcover 1st edition 2002
for exclusive distribution worldwide except India
ISBN 978-3-0348-9481-4
98765432 1 www.birkhauser-science.cmo
Contents
List of Participants.... .................... ....... ...... ................. ......... ... ............. ..... ....... ix
Preface .............................................................................................................. xv
Multiple Polylogarithms: An Introduction
M. Waldschmidt ............................................................................................... .
A (Conjectural) 113-phenomenon for the Number of Rhombus Tilings
of a Hexagon which contain a Fixed Rhombus
C. Krattenthaler................................................................................................ 13
The Influence of Carr's Synopsis on Ramanujan
Bruce C. Berndt....... ...... ......... .......... ........ ...... ..... .......... ...................... ........ ..... 31
A Bailey Lemma from the Quintuple Product
George E. Andrews........................................................................................... 37
Little Flowers to O.H. Hardy (07-02-1877-01-12-1947)
K. Ramachandra ............... ............................................. .................................. 47
Rogers-Ramanujan Type Identities for Burge's Restricted Partition
Pairs Via Restricted Frobenius Partitions
A.K. Agarwal and Padmavathamma.. ................. ..... ........................................ 53
On q-additive and q-multiplicative Functions
1. Katai.............................................................................................................. 61
Antimagic Labeling of Complete m-ary Trees
PD. Chawathe and Vijaya Krishna.... .............. ....................... ......... ................ 77
Some Recent Advances on Symmetric, Quasi-Symmetric
and Quasi-Multiple Designs
Sharad Sane ............................................ ,. ...... ............... .......... ......... ................ 81
On T-core Partitions and Quadratic Forms
Neville Robbins.................... ........................... .................................................. 89
Observations on Some Algebraic Equations Associated with
Ramanujan's Work
Michele Elia...................................................................................................... 101
vi Contents
On Rapidly Convergent Series for Dirichlet L-function Values
Via the Modular Relation
S. Kanemitsu, Y. Tanigawa and M Yoshimoto ...... ... ................... ..... ... ... ......... 113
On a Conjecture of Andrews-II
Padmavathamma and M Ruby Salestina........................................................ 135
Integrity of P2 X Pn
PD. Chawathe and S.A. Shende .... ...... ........ .................. .......... ........ ...... .......... 149
A Note on Cordial Labelings of Multiple Shells
Mahesh Andar, Samina Boxwala and N.B. Limaye ........................................ 157
A Report on Additive Complements of the Squares
D.S. Ramana .................................................................................................... 161
Transcendental Infinite Sums and Some Related Questions
Sukumar Das Adhikari ............................. ..................... .................................. 169
The Lehmer Problem on the Euler Totient: A Pendora's Box
of Unsolvable Problems
M. V. Subbarao ................................. ................................. ............................... 179
The Problems Solved by Ramanujan in the Journal of the Indian
Mathematical Society
Bruce C. Berndt................................................................................................ 189
On the gcd and Icm of Matrices Over Dedekind Domains
V.C. Nanda ....................................................................................................... 201
The Billiard Ball Motion Problem I: A Markoff Type Chain
for the Octahedron in !,)l3
R.J. Hans-Gill ................................................................................................... 213
Hilbert's Seventeenth Problem and Pfister's Work on Quadratic Forms
A.R. Rajwade. ................................................................................................... 225
Certain Representations of Mock-Theta Functions
R. Y. Denis, S.N. Singh and D. Sulata.............. ...... ........ ................ ...... ............. 231
Bi-Graceful Graphs
M. Murugan ...................................................................................................... 243
Wheels, Cages and Cubes
G. Sudhakara .................................................................................................... 251
Contents vii
Relevance ofSrinivasa Ramanujan at the Dawn of the New Millennium
K. Srinivasa Rao ............................................................................................... 261
Number of Solutions of Equations over Finite Fields
and a Conjecture of Lang and Weil
Sudhir R. Ghorpade and Gilles Lachaud .. ... ...... ... ..... ....... ...... .............. ... .... 269
On An Additive Question
S. Srinivasan .................................................................................................... 293
n-Colour Partitions
A.K. Agarwal ................................................................................................... 301
1st Row: Sushma, Vandana, Rachna, Mrs. R.I. Hans-Gill, Mrs. Madhu Raka,
Mr. Manohar Madan
2nd Row: Mr. A.K. Agarwal, Mr. Imre Katai, Mr. R.P. Bambah,
Mr. K. Ramachandara, Mr. Michele Elia,
Mr. Michele Waldschmidt, Mr. K. Srinivasa Rao, N. Saradha,
G. Kaur, A. Prasad
3rd Row: Mrs. Padmavathamma, Ms. Ruby Salestina. M,
Mr. Pranjal Rajkhowa, Mr. N. Sankaran,
Mr. S. Ismail Mohideen, Mr. D.V. Shah, Mr. Marcus
D. Bhoominathan, PJ.A. Alphonse, Mr. T.N. lanaki Raman,
Mr. Sudhir R. Ghorpade, Mr. S.D. Adhikari
4th Row: Mrs. Nirmala Limaye, Mrs. P.D. Chawathe,Ms. Aditi Bapat,
Tarkeshwar Singh, Mr. Sampat Kumar, Mr. Christian
F. Krattenthaler, Mr. Remy Y. Denis, Mr. R.N. Maggo,
Mr. V. V. Rane, Mr. A.K. Bhandari
5th Row: Mr. Anirban Mukhopadhyay, Mr. D.S. Khassa, Mr. P.K. Deb,
Mr. Neville Robbins, Mr. S.S. Sane, Mr. S.A. Katre,
Mr. Y.K. Grover
6th Row: Mr. Santosh Shende, Mr. T.N. Shorey, Mr. Surya Ramana,
Mr. Sinnou David
7th Row: Amrit Pal Singh, Amarinder Singh
List of Participants xi
Prof. Kanemitsu, S. Prof. Luthar, I.S.
Deptt. of Electrical Engineering H.No. 256 Sector 6
Univ. ofKinki lizuka Panchkula
Fukooka 820 0011 Haryana
Japan
Prof. Madan, M.L.
Prof. Katai, Imre
Mathematics Department
Department of Computer Algebra
Ohio State University
1117 Budapest
Columbus
PAZMA NYPETR SETANY
Ohio 43210
Hungary
USA
Prof. Katre, S.A.
Mr. Maggo, R.N.
Department of Mathematics
874, Sector 43A
University of Pune
Chandigarh 160022
Pune 411007
Prof. Khanduja, Sudesh Kaur Mr. Mohideen, Ismail S.
Department of Mathematics Department of Mathematics
Panjab University Jamal Mohamed College
Chandigarh Trichy-20
Tamilnadu
Prof. Khassa, D.S.
Department of Mathematics Mr. Mukhopadhyay,A.
Panjab University The Harish-Chandra
Chandigarh Research Institute
Chhatnag Road, Jhusi
Prof. Krattenthaler, Christian F. Allahabad 2111019
Institute of Mathematik
Strudlhof gasse 4 (Strudlhof)
Dr. Mukku, C.S.
A-1090 WIEN
Department of Mathematics
Austria Panjab University
Chandigarh
Prof. Lachaud, Gilles
Directeur de Recherches au CNRS
Dr. Murugan, M.
Institute de Math. Ematiques
Department of Mathematics
de luminy, UPR 9016
The American College
163, Avenue De Luminy
Madurai 625002 (Tamilnadu)
Case 930, 13288
Marseille Cedex 9 Prof. Nanda, V.C.
France 600, Sector 16-D
Chandigarh
Prof. Limaye, N.B.
Department of Mathematics Ms. Noha, Eftekhri
University of Mumbai Department of Mathematics
Vidya Nagri Panjab University
Mumbai 400 098 Chandigarh
xii List of Participants
Prof. Padmavathamma Prof, Rane, V.V.
Department of Studies in Mathematics Director
University of Mysore Institute of Sciences
Mysore 6 Mumbai 32
Dr. Pedgaonkar, Anil Anant
Dr. Rani, Urmila
12-A, Gagan Mahal
D.A.V. College
Sir Ponch Khanwala Road, Worli
Sector 10
Mumbai-25
Chandigarh
Dr. Prasad, Anjana Wirmani
Department of Mathematics Prof. Rao, K. Srinivasa
University of Mumbai Institute of Mathematical Sciences
Vidya Nagari CIT Campus
Mumbai 400 098 Chennai 600 013
Dr. Pusri, Anju
Prof. Robbins, Neville
Department of Mathematics
Mathematics Department
Punjab University
San Francisco State
Chandigarh 160 014
University
San Francisco
Mr. Rajkhowa, P.
CA 94132
Senior Lecturer in Mathematics
USA
Cotton College
Guwahati 781001
Ms Ruby Salestina. M
Prof. Rajwade, A.R. Department of Studies
Department of Mathematics in Mathematics
Punjab University University of Mysore
Chandigarh 160 014 Mysore 6
Prot Raka, Madhu
Prof. S.S. Sane
Department of Mathematics
Department of Mathematics
Punjab University
University of Mumbai
Chandigarh 160 014
Mumbai 400 098
Prof. Ramachandara, K.
Prof. Sampathkumar, E.
Tata Institute
Department of Studies
of Fundamental Research
in Mathematics
Post Box No. 1234
University of Mysore
Bangalore 560 012
Mysore 6
Dr. Raman, T.N. Janaki
Department of Mathematics & Prof. Sanakran, N.
Computer Applications 516, I Cross 8th Main
Regional Engineering College Koramangala Block IV
Trichy (Tamilnadu) Bangalore 560034