AMC2005

A MC
20 0 5
Fourth Asian Mathematical
Conference
20-23 July
National University of Singapore
The Asian Mathematical Conference series (AMC) was initiated by the South East Asian
Mathematical Society (SEAMS) as a platform to showcase talents from Asian countries
and to encourage academic exchange within the region.
AMC2005 is held in conjunction with
the Centennial Celebrations at the National University of Singapore
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AMC2005, Singapore
DONOR & SPONSORS FROM EXTERNAL ORGANIZATIONS
We thank these organizations for their generous support:
SEAMS
South East Asian Mathematical Society
UNESCO Office, Jakarta
Lee Foundation
Centre International de Mathématiques Pures et Appliquées
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AMC2005, Singapore
SPONSORS FROM NATIONAL UNIVERSITY OF SINGAPORE
We thank these units for their generous support:
Faculty of Science
Institute for Mathematical Sciences
NUS Centennial Celebrations
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AMC2005, Singapore
ORGANIZERS
Department of Mathematics
Department of Statistics & Applied
Probability
Institute for Mathematical Sciences
SEAMS
South East Asian Mathematical Society
Singapore Mathematical Society
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AMC2005, Singapore
COMMITTEES
INTERNATIONAL SCIENTIFIC COMMITTEE
1.
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Kenji Ueno, Kyoto University, Japan (Chairman)
Jaigyoung Choe, Seoul National University, South Korea
Romanov Vladimir Gavrilovich, Sobolev Institute of Mathematics, Russia
Masaki Kashiwara, Research Institute for Mathematical Sciences, Japan
Jian-Shu Li, Hong Kong University of Science and Technology, Hong Kong SAR
Bong Lian, National University of Singapore, Singapore
Zhi Ming Ma, Institute of Applied Mathematics, Chinese Academy of Sciences,
China
Harald Niederreiter, National University of Singapore, Singapore
S. Ramanan, Chennai Mathematical Institute, India
David Siegmund, Stanford University, USA
Gilbert Strang, Massachusetts Institute of Technology, USA
Polly Wee Sy, University of the Philippines at Diliman, Philippines
Eng Chye Tan, National University of Singapore, Singapore (Secretary)
Lo Yang, Institute of Mathematics and System Sciences, Chinese Academy of
Sciences, China (Past Chairman – Ex-Officio)
STEERING COMMITTEE
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11.
Eng Chye Tan, National University of Singapore, Singapore (Chairman)
Wanida Hemakul, Chulalongkorn University, Thailand
Myung-Hwan Kim, Seoul National University, South Korea
Rosihan M Ali, Universiti Sains Malaysia, Malaysia
Blessida Raposa, De La Salle University, Philippines
Zuowei Shen, National University of Singapore, Singapore (Secretary)
Kar Ping Shum, The Chinese University of Hong Kong SAR
Sri Wahyuni, Gadjah Mada University, Indonesia
Toshikazu Sunada, Meiji University, Japan
Do Long Van, Hanoi Institute of Mathematics, Vietnam
Jiping Zhang, Peking University, China
LOCAL ORGANIZING COMMITTEE
1.
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12.
Eng Chye Tan, National University of Singapore, Singapore (Chairman)
Anthony Kuk, National University of Singapore, Singapore
Peng Yee Lee, National Institute of Education, Singapore
Ping Lin, National University of Singapore, Singapore (Treasurer)
San Ling, Nanyang Technological University, Singapore (Secretary)
Wei Liem Loh, National University of Singapore, Singapore
Peter Yu Hin Pang, National University of Singapore, Singapore
Stella Pang, National University of Singapore, Singapore (Administrative Officer)
Yeneng Sun, National University of Singapore, Singapore
Kim Chuan Toh, National University of Singapore, Singapore (Editorial Coordinator)
Khoon Yoong Wong, National Institute of Education, Singapore
Chengbo Zhu, National University of Singapore, Singapore (Programme
Coordinator)
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AMC2005, Singapore
PLENARY LECTURES
Format on this page:
Speaker’s name (in alphabetical order of family name)
Affiliation, Region
Title of lecture
Click on [abstract] view the corresponding PDF page. Full papers are not available.
Noga ALON
Tel Aviv University, Israel
(n, d, λ)-graphs in combinatorics and complexity
[abstract]
Tony F CHAN
University of California at Los Angeles, USA
Combining PDE and wavelet techniques for image processing
[abstract]
Ching-Shui CHENG
[abstract]
University of California at Berkeley, USA/ Institute of Statistics, Academia Sinica, Taiwan
Orthogonal arrays as factorial designs
Chi Tat CHONG
National University of Singapore, Singapore
Reverse mathematics and logical analysis of Ramsey’s theorem
[abstract]
Martin GOLUBITSKY
University of Houston, USA
Coupled cell systems: theory and examples
[abstract]
Lei GUO
Chinese Academy of Science, China
Feedback and uncertainty
[abstract]
Masaki KASHIWARA
Research Institute for Mathematical Sciences, Japan
Crystal bases for quantized affine algebras
[abstract]
Ngaiming MOK
[abstract]
The University of Hong Kong, Hong Kong SAR
Geometry of the Bergman metric and total geodesy of local holomorphic isometries
between bounded symmetric domains
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AMC2005, Singapore
INVITED TALKS
Format on this page:
Topic
Speaker’s name (in alphabetical order of family name)
Affiliation, Region
Click on [abstract] or [paper] view the corresponding PDF page. Not all papers are available.
Algebra & Group Theory
S.M. BHATWADEKAR
Tata Institute of Fundamental Research, India
[abstract] [paper]
Yupapom KEMPRASIT
Chulalongkorn University, Thailand
[abstract] [paper]
Masahiko MIYAMOTO
University of Tsukuba, Japan
[abstract] [paper]
Andrew RAJAH
Universiti Sains Malaysia, Malaysia
[abstract]
Ngo Viet TRUNG
Mathematical Sciences Research Institute, Vietnam
[abstract]
Jie-Tai YU
The University of Hong Kong, Hong Kong SAR
[abstract]
Algebraic Geometry
Jun-Muk HWANG
Korea Institute for Advanced Study, South Korea
[abstract] [paper]
V.B. MEHTA
Tata Institute of Fundamental Research, India
[abstract] [paper]
Shigeru MUKAI
Nagoya University, Japan
[abstract]
Kang ZUO
Mainz University, Germany
[abstract]
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AMC2005, Singapore
INVITED TALKS
Format on this page:
Topic
Speaker’s name (in alphabetical order of family name)
Affiliation, Region
Click on [abstract] or [paper] to view the corresponding PDF page. Not all papers are available.
Analysis
Victor DIDENKO
Universiti Brunei Darussalam, Brunei
[abstract] [paper]
Hendra GUNAWAN
Bandung Institute of Technology, Indonesia
[abstract] [paper]
Suthep SUANTAI
Chiangmai University, Thailand
[abstract] [paper]
Xiangyu ZHOU
Chinese Academy of Sciences, China
[abstract]
Applications of Mathematics in Sciences
Chao HSIUNG
National Health Research Institute, Taiwan
[abstract]
Xing JIN
National University of Singapore, Singapore
[abstract] [paper]
Yongvimol LENBURI
Mahidol University, Thailand
[abstract] [paper]
Yasuyuki KAWAHIGASHI
University of Tokyo, Japan
[abstract] [paper]
Xing-Bin PAN
East China Normal University, China
[abstract] [paper]
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AMC2005, Singapore
INVITED TALKS
Format on this page:
Topic
Speaker’s name (in alphabetical order of family name)
Affiliation, Region
Click on [abstract] or [paper] to view the corresponding PDF page. Not all papers are available.
Combinatorics & Graph Theory
Nawarat ANANCHUEN
Silpakorn University, Thailand
[abstract] [paper]
Edy Tri BASKORO
Bandung Institute of Technology, Indonesia
[abstract] [paper]
Masatoshi NOUMI
Kobe University, Japan
[abstract]
Arlene A. PASCASIO
De La Salle University, Philippines
[abstract]
Differential Equations
Jaigyoung CHOE
Seoul National University, South Korea
[abstract] [paper]
Vladimir ROMANOV
Sobolev Institute of Mathematics, Russia
[abstract] [paper]
Tong YANG
City University of Hong Kong, Hong Kong SAR
[abstract] [paper]
Geometry & Topology
Indranil BISWAS
Tata Institute of Fundamental Research, India
[abstract] [paper]
Wing Keung TO
National University of Singapore, Singapore
[abstract] [paper]
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AMC2005, Singapore
INVITED TALKS
Format on this page:
Topic
Speaker’s name (in alphabetical order of family name)
Affiliation, Region
Click on [abstract] or [paper] to view the corresponding PDF page. Not all papers are available.
Weiping ZHANG
Nankai Institute of Mathematics, Nankai University, China
[abstract]
Lie Theory
Jingsong HUANG
Hong Kong University of Science & Technology, Hong Kong SAR
[abstract] [paper]
Seok-Jin KANG
Seoul National University, South Korea
[abstract] [paper]
Toshiyuki KOBAYASHI
Research Institute of Mathematical Sciences, Japan
[abstract]
Hiraku NAKAJIMA
Kyoto University, Japan
[abstract]
Cheng-Bo ZHU
National University of Singapore, Singapore
[abstract] [paper]
Logic and Computing
Toshiyasu ARAI
Kobe University, Japan
[abstract] [paper]
Qi FENG
Academy of Mathematics and Systems Science / Tsinghua University, China
[abstract]
Nathan LINIAL
The Hebrew University of Jerusalem, Israel
[abstract]
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AMC2005, Singapore
INVITED TALKS
Format on this page:
Topic
Speaker’s name (in alphabetical order of family name)
Affiliation, Region
Click on [abstract] or [paper] to view the corresponding PDF page. Not all papers are available.
Mathematics Education
Noriko H. ARAI
National Institute of Informatics, Japan
[abstract] [paper]
Berinderjeet KAUR
National Institute of Education, Nanyang Technological University, Singapore
[abstract]
Fou-Lai LIN
National Taiwan Normal University, Taiwan
[abstract] [paper]
Man Keung SIU
University of Hong Kong, Hong Kong SAR
[abstract] [paper]
Khoon Yoong WONG
[abstract] [paper]
National Institute of Education, Nanyang Technological University, Singapore
Number Theory & Applications
Heng Huat CHAN
National University of Singapore, Singapore
[abstract] [paper]
Makoto MATSUMOTO
Hiroshima University, Japan
[abstract] [paper]
Rama PARIMALA
Tata Institute of Fundamental Research, India
[abstract]
Dipendra PRASAD
Tata Institute of Fundamental Research, India
[abstract]
Chaoping XING
National University of Singapore, Singapore
[abstract] [paper]
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AMC2005, Singapore
INVITED TALKS
Format on this page:
Topic
Speaker’s name (in alphabetical order of family name)
Affiliation, Region
Click on [abstract] or [paper] to view the corresponding PDF page. Not all papers are available.
Numerical Analysis
Ping LIN
National University of Singapore, Singapore
Zongmin WU
Fudan University, China
Jungho YOON
Ewha Womans University, South Korea
[abstract]
[abstract] [paper]
[abstract]
Operations Research
Masakazu KOJIMA
Tokyo Institute of Technology, Japan
[abstract] [paper]
Kim Chuan TOH
National University of Singapore, Singapore
[abstract] [paper]
Xiang Sun ZHANG
Chinese Academy of Sciences, China
[abstract] [paper]
Xun Yu ZHOU
Chinese University of Hong Kong, Hong Kong SAR
[abstract]
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AMC2005, Singapore
INVITED TALKS
Format on this page:
Topic
Speaker’s name (in alphabetical order of family name)
Affiliation, Region
Click on [abstract] or [paper] to view the corresponding PDF page. Not all papers are available.
Probability Stochastic Processes
Igor S. BORISOV
Sobolev Institute of Mathematics, Russia
Cheng-Der FUH
Institute of Statistics, Academia Sinica, Taiwan
[abstract] [paper]
[abstract]
Qiman SHAO
Hong Kong University of Science and Technology, Hong Kong SAR
[abstract] [paper]
Fengyu WANG
Beijing Normal University, China
[abstract] [paper]
Scientific Computation
Xiao-Ping WANG
Hong Kong University of Science and Technology, Hong Kong SAR
Shao Liang ZHANG
University of Tokyo, Japan
[abstract]
[abstract] [paper]
Statistics
Arup BOSE
Indian Statistical, Institute Calcutta, India
[abstract]
Probal CHAUDHURI
Indian Statistical Institute, Calcutta, India
[abstract]
Anthony KUK
National University of Singapore, Singapore
[abstract]
Benjamin YAKIR
Hebrew University of Jerusalem, Israel
[abstract] [paper]
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AMC2005, Singapore
SHORT COMMUNICATIONS
Format on this page:
Topic
Contributor’s name (in alphabetical order of family name)
Affiliation, Region
Title of presentation
Click on title of presentation to view the corresponding PDF page. Only abstracts are available.
Algebra and Group Theory
Sriwulan ADJI
Univesiti Sains Malaysia, Malaysia
The twisted Toeplitz algebras
Christine ANTONIO
Sophia University, Japan
On the group structure of the Jacobian of some hyperelliptic curve over several Fp
Viatcheslav Alexandrovitch ARTAMONOV
Moscow State University, Russia
Actions of pointed Hopf algebras on quantum torus
Muhamad ASHIQ
National University of Sciences and Technology, Pakistan
Actions of G(3,3)=<u,v:u3=v3=1> on Q*(Ö-n)
Guiyun CHEN
Southwest China Normal University, China
Characterization of sporadic simple groups
Miyeon KWON
University of Wisconsin-Platteville, USA
Notes on W-type non-associative algebras I
William E. LONGSTAFF
University of Western Australia, Australia
Lengths of pairs of complex 5 × 5 matrices
Qaiser MUSHTAQ
Quaid-i-Azam University, Pakistan
Subgroups of PGL(2,q) as homomorphic images of group D (2,3,7)
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SHORT COMMUNICATIONS
Format on this page:
Topic
Contributor’s name (in alphabetical order of family name)
Affiliation, Region
Title of presentation
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Ki-Bong NAM
University of Wisconsin – Whitewater, USA
Non-associative algebras with right identities
Sansanee NENTHEIN
Chulalongkorn University, Thailand
Some BQ-rings of linear transformations
Sajee PIANSKOOL
Chulalongkorn University, Thailand
Divisibility of some hypergroups defined from groups
Nor Haniza SARMIN
Universiti Teknologi Malaysia, Malaysia
Irreducible representations of some two-groups
K P SHUM
The Chinese University of Hong Kong, Hong Kong SAR
On p.p. rings which are reduced
Chitlada SOMSUP
Mahidol University, Thailand
On the structure of some matrix ring related to injectivity and its generalisations
Budi SURODJO
Gadjah Mada University, Indonesia
Coalgebra of generalized power series rings
George SZETO
Bradley University, USA
On Galois extensions automorphism group as Galois group
Phi Hung TONG VIET
University of Natural Sciences, Vietnam
On the subnormaliser condition for subgroups
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AMC2005, Singapore
SHORT COMMUNICATIONS
Format on this page:
Topic
Contributor’s name (in alphabetical order of family name)
Affiliation, Region
Title of presentation
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Maliwan TUNAPAN
Mahidol University, Thailand
On the relations between the classes of QF3-rings and hereditary rings
Madana Swamy UPPASETTY
Andhra University, India
Sheaves of algebras over locally Boolean spaces
George WILLIS
University of Newcastle, Australia
Totally disconnected locally compact groups and their automorphisms
Larry Lianyong XUE
Bradley University, USA
On characterizations of a commutator Galois extension
Analysis
Rosihan M. ALI
Universiti Sains Malaysia, Malaysia
Subordination and superordination of analytic functions associated with linear operators
Juancho Arranz COLLERA
University of the Philippines, Philippines
Perturbations of differential expressions with logarithmically decaying coefficients
preserving the nullities
Dilip Kumar GANGULY
University of Calcutta, India
On subseries of divergent series
Christiana Rini INDRATI
Gadjah Mada University, Indonesia
Dominated convergence theorems involving small Riemann sums
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AMC2005, Singapore
SHORT COMMUNICATIONS
Format on this page:
Topic
Contributor’s name (in alphabetical order of family name)
Affiliation, Region
Title of presentation
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Dohan KIM
Seoul National University, South Korea
Conditionally positive definite (Fourier) hyperfunctions
Denny LEUNG
National University of Singapore, Singapore
Extension of functions with small oscillation
Sornsak THIANWAN
Chiangmai University, Thailand
Some convexity properties of Orlicz-direct-sum of Banach spaces
Tin Lam TOH
National Institute of Education, Nanyang Technological University, Singapore
On Henstock’s version of multiple stochastic integral
Widodo
Gadjah Mada University, Indonesia
Measure entropy of discrete dynamical systems
Applications of Mathematics
Min DAI
National University of Singapore, Singapore
A free boundary problem arising from the pricing of strike reset options
Jose Maria L. ESCANER IV
University of the Philippines, Diliman, Philippines
Chaos Behaviour in a Brusselator Reaction Mechanism
Vinod MISHRA
Sant Longowal Institute of Engineering & Technology, India
Wavelet solutions of certain ill-posed problems
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SHORT COMMUNICATIONS
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Topic
Contributor’s name (in alphabetical order of family name)
Affiliation, Region
Title of presentation
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Hyungju PARK
Korea Institute for Advanced Study, South Korea
Optimization issues in the construction of multidimensional wavelets
Rattikan SAELIMY
Prince of Songkla University (Pattani Campus), Thailand
On the absence of arbitrage opportunity for the fractional Black-Scholes model
Combinatorics and Graph Theory
Evangeline P. BAUTISTA
Ateneo de Manila University, Philippines
S-extremal additive codes over GF(4)
Maria Cristeta N. CUARESMA
University of the Philippines Los Baños, Philippines
Homogenous factorisations of the Johnson graphs
Frederic EZERMAN
Ateneo de Manila University, Philippines
A new quantum code of length 20
Loeky HARYANTO
Delft University of Technology, Netherlands
Applying the structures of standard gray codes in constructing certain types of snakes
Yvette F. LIM
De La Salle University, Philippines
Unfolding complete graphs, paths, and cycle
Manuel Joseph C. LOQUIAS
University of the Philippines, Diliman, Philippines
On the construction of semiperfect colorings of symmetrical patterns
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SHORT COMMUNICATIONS
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Affiliation, Region
Title of presentation
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Bernardo R. MARQUEZ
Ateneo de Naga University, Philippines
Self-dual codes over Z2 x Z2 and groups of 2 x 2 binary matrices
Felix P. MUGA II
Ateneo de Manila University, Philippines
On factorial numbers
Arunkumar R. PATIL
Indian Institute of Technology, Bombay, India
Generalised spectrum of Grassmann codes
Blessilda P. RAPOSA
De La Salle University, Manila, Philippines
Edge-reduction number of graphs
Leonor A. RUIVIVAR
De La Salle University, Manila, Philippines
Subdivision number: powers of paths and cycles
Bernhard SCHMIDT
Nanyang Technological University, Singapore
Cyclic projective planes
I Nengah SUPARTA
Delft University of Technology, Netherlands
A construction of balanced maximum counting sequences
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AMC2005, Singapore
SHORT COMMUNICATIONS
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Topic
Contributor’s name (in alphabetical order of family name)
Affiliation, Region
Title of presentation
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Differential Equations
Sofia BOROK
Ben-Gurion University of the Negev, Israel
On the solutions of "ghost" manifolds problem
Roath CHAN
Royal Academy of Cambodia, Cambodia
Resolution on n-order functionally - differential equations with operator coefficients and
delayed variables in Hilbert space
Daoyuan FANG
Zhejiang University, China
WP or IP issues for nonlinear wave equation
Joanna GOARD
University of Wollongong, Australia
How to incorporate non-invariant boundary conditions in invariant solutions
Munira ISMAIL
University Teknologi Malaysia, Malaysia
An integral equation for a particular solution of a non-uniquely solvable interior Riemann
problem on a region with corners
Junjie LI
Zhejiang University, China
Qualitative properties for solutions of a fourth order degenerate parabolic equation in
higher space dimensions
Yunguang LU
University of Science and Technology, China
Applications of the compensated compactness method on hyperbolic conservation
systems
Anh Tuan NGUYEN
Ho Chi Minh City University of Pedagogy, Vietnam
A class of boundary value problem for high order differential equations
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SHORT COMMUNICATIONS
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Affiliation, Region
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Carlene P. ARCEO
University of the Philippines, Diliman, Philippines
Existence and uniqueness of the solution to a volevic system of linear equations with
general singularity
Hui-Jiang ZHAO
The Chinese Academy of Sciences, China
Global stability of elementary waves for certain dissipative hyperbolic conservation laws
Geometry and Topology
Raveendranath Amasebail AITHAL
University of Mumbai, India
On two functionals connected to the Laplacian in a class of doubly connected domains in
space-forms
Falleh R. M. AL-SOLAMY
King AbdulAziz University, Saudi Arabia
CR-submanifolds of generalized Sasakian space form
Wen-Haw CHEN
Tung-Hai University, Taiwan
Curvature, fundamental groups and Hausdorff distance
Satoshi ISHIWATA
University of Tsukuba, Japan
Long time asymptotics of the heat semigroup on nilpotent covering manifolds
Lie Theory
Anthony DOOLEY
University of New South Wales, Australia
Intertwining operators, H-type groups and transference
Anh Vu LE
Ho Chi Minh University of Pedagogy, Vietnam
On a class of solvable Lie algebras and groups of dimension 5
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Jiachen YE
Tongji University, China
Some results on the irreducible modules for the algebraic groups and corresponding Lie
algebras of type A
Logic and Computing
Paul M E SHUTLER
National Institute of Education, Nanyang Technological University, Singapore
A linear time knowledge based sorting algorithm
Guohua WU
Nanyang Technological University, Singapore
Intervals containing exactly one c.e. degree
Mathematical Biology
Sittipong RUKTAMATAKUL
Mahidol University, Thailand
Wave front solutions of continuous neural networks
Mathematics Education
Deane ARGANBRIGHT
Korea Advanced Institute of Science and Technology, South Korea
Creative mathematical modeling and visualization via spreadsheets
Marleonie M. BAUYOT
San Pedro College, Davao City, Philippines
Predictive-model of Numerical Ability Test (NAT) and Achievement Test (AT) in college
algebra as basis for construction of module for bridging program
Hailiza KAMARULHAILI
Universiti Sains Malaysia, Malaysia
Graphing calculator as a teaching and learning aid for secondary school students and
teachers
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Elvin J. MOORE
King Mongkut's Institute of Technology North Bangkok, Thailand
Teaching applications of planar graphs with Maple
Number Theory and Applications
Meng Fai LIM
National University of Singapore, Singapore
Some characteristics classes in number theory
Fidel R. NEMENZO
University of the Philippines, Philippines
On the class number and Sylow 2-group of the ideal class group of quadratic fields
Siti Hasana SAPAR
Universiti Putra Malaysia, Malaysia
On the cardinality of the set of solutions to congruence equation associated with seventh
degree form
Frank STEPHAN
National University of Singapore, Singapore
Presentations of trivial reals and Kolmogorov complexity
Pee Choon TOH
National University of Singapore, Singapore
Classes of identities involving the Dedekind eta function and Eisenstein series
Operations Research
Cheng-Feng HU
I-Shou University, Taiwan
Resolution of the system of fuzzy integer inequalities
Karthik Balkrishnan NATARAJAN
National University of Singapore, Singapore
Some new asymptotic bounds for combinatorial optimization problems (revised talk)
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Sirirat WONGPRAKORNKUL
Khon Kaen University, Thailand
Benders’ partitioning algorithm for an integrated one-dimensional cutting stock and
transportation problem
Fengming XU
Xi’an Jiaotong University, China
A feasible direction algorithm without line search for solving Max-Bisection problems
Probability and Stochastic Process
Chih-Chung CHANG
National Taiwan University, Taiwan
On the occupation time large deviations of symmetric simple exclusion process
Eveyth DELIGERO
Keio University, Japan
The non-Archimedean metric Diophantine approximations
Patrick MULDOWNEY
University of Ulster, North Ireland
Probability and stochastic processes using a Riemann-type integration
Scientific Computation
Md. Shafiqul ISLAM
University of Dhaka, Bangladesh
A novel approach on explicit integration formulae for linear convex quadrilateral finite
elements
Delin CHU
National University of Singapore, Singapore
Singular value assignment with low rank matrices
Changqiu JIN
Hong Kong University of Science and Technology, Hong Kong SAR
Gas-kinetic BGK schemes on moving and adaptive meshes
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Tieh Yong KOH
Nanyang Technological University, Singapore
A geometric viewpoint of geophysical fluid dyamics in the tropics?
Bipin KUMAR
National University of Singapore, Singapore
Krylov solvers for grain growth simulation using phase-field modeling
Christian LICHT
Universite Montpellier II, France
Various modelings of fluid/structure interactions
Leng Leng LIM
Massey University, New Zealand
Modelling volcanic ashfall using partial differential equations
Anton PURNAMA
Sultan Qaboos University, Sultanate of Oman
The effect of desalination plants on the salinity of the Arabian Gulf waters
Jianxian QIU
Nanjing University, China
Simulations of multi-phase flow using Runge-Kutta discontinuous Galerkin methods with
conservative approach of interfaces
Vasily P. SHAPEEV
Russian Academy of Science, Russia
Implicit finite-difference scheme with approximation error O(r4, h8) for the heat
conduction equation
Alexander SHAPEEV
National University of Singapore, Singapore
Application of wavelet transform to problems involving integral kernels
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Bibhya Nand SHARMA
University of the South Pacific, Fiji Islands
Controlled steering, obstacle avoidance, and posture-stabilisation of car-like mobile
robots via a Lyapunov-based approach
Manuel TORRILHON
Hong Kong University of Science and Technology, Hong Kong SAR
Stability and consistency of kinetic upwinding for advection-diffusion-equations
Kim Tuan VU
University of West Georgia, USA
Irregular sampling
Jagath K WIJERATHNA
University of Colombo, Sri Lanka
The memory effect in fast impregnation processes
Statistics
Bruce BROWN
National University of Singapore, Singapore
Semi-parametric inference for non-parametric parameter measures
Subhash Ajay CHANDRA
University of the South Pacific, Fiji Islands
Asymptotic distribution of Cramer-von Mises statistics for ARCH processes
Sri Haryatmi KARTIKO
Gadjah Mada University, Indonesia
Simulation study for partially linear model with random covariates
Efang KONG
National University of Singapore, Singapore
Variable selection for the single-index model
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Yue LI
National University of Singapore, Singapore
Resampling the generalized estimating functions for analysis of longitudinal data
Dedi ROSADI
Gadjah Mada University, Indonesia
Identification of moving average process with infinite variance
Choon Peng TAN
Universiti Tunku Abdul Rahman, Malaysia
Some properties of the Fisher index of discrimination
Subanar
Gadjah Mada University, Indonesia
Monte Carlo simulation study of the neural network linearity test for time series
Copyright Reserved
(n, d, λ)-graphs in Combinatorics and Complexity
N. Alon
Schools of Mathematics and Computer Science,
Raymond and Beverly Sackler Faculty of Exact Sciences,
Tel Aviv University, Tel Aviv 69978, Israel ([email protected])
An (n, d, λ)-graph is a d-regular graph on n vertices so that the absolute value of each eigenvalue
of its adjacency matrix, besides the largest one, is at most λ. I will survey some of the remarkable
pseudo-random properties of such graphs in which λ is much smaller than d, describe various
constructions, and present several applications of these graphs in the solution of problems in
Extremal Combinatorics, Combinatorial Geometry and Theoretical Computer Science.
Orthogonal Arrays as Factorial Designs
Ching-Shui Cheng
Institute of Statistical Science Academia Sinica, Taiwan ([email protected])
Several important combinatorial structures arise in statistical design of experiments, including
orthogonal arrays which are widely used for conducting factorial experiments in various scientific
and industrial investigations. I will review some recent developments in the construction of
efficient factorial designs. Some results from coding theory and finite projective geometry play
important roles.
Reverse Mathematics and Logical Analysis of Ramsey’s Theorem
Chi Tat Chong
Department of Mathematics, National University of Singapore,
2 Science Drive 2, Singapore 117543 ([email protected])
Reverse mathematics asks the fundamental question: Which set existence axiom is necessary and
sufficient to prove a mathematical statement? The study was initiated by Harvey Friedman and
Stephen Simpson to investigate the relationship between set existence axioms and mathematical
theorems. Its origin dates back to the work of Hilbert and Bernays on the foundations of
mathematics. Research over the last several decades has identified a class of key set existence
axioms which characterize the strengths of many classical mathematical theorems in terms of
logical equivalence. Our talk will give a review of the various axiom systems, collectively called
subsystems of second order arithmetic, that have been extensively studied in recent years. A
mathematical theorem which is currently under active investigation by logicians, and which
has wide applications in mathematics such as combinatorics, dynamical systems, geometry of
2
Banach spaces and logic, is the theorem first formulated by F P Ramsey in 1930: if the k-tuples
of IN are given m colors, then there is an infinite set A ⊂ IN all of whose k-tuples have the same
color. We will present Ramsey’s Theorem as a case study in reverse mathematics, and examine
its computational strength in the sense of computability theory, and discuss some recent results.
Combining PDE and Wavelet Techniques for Image Processing
Tony F. Chan
Department of Mathematics, UCLA, Los Angeles, CA 90095 ([email protected])
Wavelet and PDE techniques are among the most popular and powerful mathematical techniques
for image processing. Superficially, they seem to be based on completely different philosophies
and techniques and indeed the respective research communities have been quite independent of
each other. In this talk, I’ll show, by several examples of our own works and others, how a
synergistic combination of these two approaches can lead to powerful techniques that can reap
the special advantageous properties of each.
Coupled Cell Systems: Theory and Examples
Martin Golubitsky
Department of Mathematics, University of Houston,
Houston, TX 77204-3008, USA ([email protected])
Research supported in part by NSF through grant DMS-0244529.
Joint work with Ian Stewart.
A coupled cell system is a collection of interacting dynamical systems. Coupled cell models
assume that the output from each cell is important and that signals from two or more cells can
be compared so that patterns of synchrony can emerge. We ask: How much of the qualitative
dynamics observed in coupled cells is the product of network architecture and how much depends
on the specific equations?
The ideas will be illustrated through a series of examples and three theorems. The first theo
rem classifies spatio-temporal symmetries of periodic solutions; the second gives necessary and
sufficient conditions for synchrony in terms of network architecture; and the third shows that
synchronous dynamics may itself be viewed as a coupled cell system through a quotient construc
tion. We also show how nongeneric bifurcations with nongeneric results arise in bifurcations in
coupled systems.
3
Feedback and Uncertainty
Lei Guo
Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing ([email protected])
Feedback is ubiquitous, and exists in almost all goal-directed behaviors. It is indispensable to the
human intelligence, and is important in learning, adaptation, organization and evolution, etc.
Feedback is the most important concept in control systems, where the main purpose of feedback
is to deal with uncertainties in complex dynamical systems. The uncertainties of a system are
usually classified into two types: internal (structure) and external (disturbance) uncertainties,
depending on the specific dynamical systems to be controlled. Feedback needs information, and
there are also two types of information in a control system: a priori information and posteriori
information. It is the posteriori information that makes it possible for the feedback to reduce
the influence of the uncertainties on control systems. Two of the fundamental questions in
control theory are: How much uncertainty can be dealt with by feedback? What are the limits
of feedback? These are conundrums, despite of the considerable progress in control theory
over the past several decades. To explore the maximum capability and potential limits of the
feedback mechanism, we have to place ourselves in a framework that is somewhat beyond those
of the classical ones (including robust and adaptive control), since we need to study the full
capability of the feedback mechanism which includes all (nonlinear and time-varying) causal
mappings, and are not only restricted to a fixed feedback law or a set of specific feedback
laws. We need not only to answer what the feedback can do, but also to answer, the more
important and difficult question, what the feedback cannot do. In this talk, we will first present
a mathematical framework for investigating the quantitative relationship between the capability
of feedback and the size of uncertainty, and will then give a survey of some basic ideas and
results obtained recently towards understanding the maximum capability (and potential limits)
of the feedback mechanism in dealing with structural uncertainties. In particular, for several
basic classes of discrete-time (or sampled-data) uncertain nonlinear dynamical control systems
with either parametric or nonparametric uncertainties, we will present several ”Critical Values”
or “Impossibility Theorems” concerning the maximum capability of feedback in dealing with
uncertainties.
4
Crystal Bases for Quantized Affine Algebras
Masaki Kashiwara
Research Institute for Mathematical Sciences, Kyoto University,
Kyoto, 606–8502, Japan ([email protected])
In this talk, I start from the definition of crystal bases and then finite-dimensional representations
of quantum affine algebras.
The finite-dimensional representations of quantized affine algebras U (g) are extensively studied
in connection with exactly solvable models. The crystal bases are one of the tool to study those
problems. It is expected that there exists a “good” finite-dimensional U (g)-module W (m�k )
with a multiple m�k of a fundamental weight �k as an extremal weight. This module is good
in the sense that it is irreducible and it has a crystal base and moreover a global basis.
In the untwisted case, its conjectural character formula is given by Kirillov–Reshetikhin, and its
conjectural fusion construction is given by Kuniba–Nakanishi–Suzuki. It is proved by Nakajima
that the fusion construction gives irreducible modules with the expected character in the simply
laced case, and by Chari in some cases. It is also expected that any “good” finite-dimensional
U (g)-module is a tensor product of modules of the above type.
Geometry of the Bergman metric and local holomorphic isometries between
bounded symmetric domains
Ngaiming Mok
Department of Mathematics, The University of Hong Kong, ([email protected])
By the celebrated work of Calabi every germ of holomorphic isometry of a simply-connected
complete Kähler manifold into the projective space P n extends to a global isometry. The
same remains true when the target manifold is replaced by an irreducible compact Hermitian
symmetric manifold X, since X can be isometrically embedded into some P n .
For the dual situation of germs of holomorphic immersions between bounded symmetric domains
it is commonly believed that the situation is more rigid. The question is to characterize such
maps and to find conditions which force the map to be totally geodesic.
In a recent work of Clozel-Ullmo in relation to arithmetic problems on bounded symmetric
domains they found interesting applications of solutions of special cases of the problem to arith
metic questions on quotients of bounded symmetric domains by torsion-free discrete groups of
automorphisms. They solved the problem for certain germs of holomorphic maps of the unit
disk into a product of unit disks which are isometric with respect to the Bergman metric up
to normalizing constants. For the proof following Calabi they made use of identities of Kähler
potential functions arising from isometries up to scalar constants.
5
We are now able to understand the general situation by developing first of all a method of
analytic continuation. Starting with the same functional identities and polarizing we obtain an
infinite number of holomorphic identities, and the first question is to determine whether the
functional identities are sufficiently nondegenerate to force analytic continuation. Once we have
algebraic extension, the functional identities force the extended map to be a proper algebraic
map between bounded symmetric domains, and the question is study the asymptotic behavior
of such maps.
For the first question we develop a method by studying deformations of solutions of the holomor
phic functional identities, and force algebraic extension by showing that, in the event that there
are nontrivial deformations the germ of holomorphic isometry must take values on hyperplane
sections of the embedding of the domain into the infinite-dimensional projective space P This
statement applies to all germs of holomorphic isometries (up to scalar constants) with respect to
the Bergman metric. The hyperplane sections that we obtain correspond to extremal functions
with respect to the Bergman metric. In the event that the Bergman kernel function K(z, w) is
rational in (z, w), they will yield algebraic equations satisfied by the germ of map. This is the
case for bounded symmetric domains.
∞
As to the rest, it amounts to a study of the geometry of bounded symmetric domains with
respect to Bergman metrics. In particular, we will make use of our earlier works using the
Poincaré-Lelong equations in relation to the question of characterization of certain compact
holomorphic geodesic curves on quotients of bounded symmetric domains.
6
Connected domination critical graphs with cutvertices
Nawarat Ananchuen
Department of Mathematics, Silpakorn University,
Nakorn Pathom, Thailand 73000 ([email protected])
A subset of vertices D of a graph G is a dominating set for G if every vertex not in D is
adjacent to one in D. A dominating set for G is a connected dominating set if it induces
a connected subgraph of G. The connected domination number of G, denoted by γc , is the
minimum cardinality of a connected dominating set. Graph G is said to be k − γc − critical if
γc (G) = k but γc (G + e) < k for each edge e ∈
/ E(G). In this paper, we present the structure of
connected domination critical graphs with cutvertices. We also establish a characterization of
3 − γc − critical graphs with cutvertices.
Let’s Blend Online and Offline; ”Write-in-Math!”
Noriko H. Arai
National Institute of Informatics,
2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan ([email protected])
E-Classroom is a project to create an advanced learning community beyond usual curricula on
the Internet in cooperation not only with teachers but also with mathematicians and computer
scientists. E-Classroom provides an advanced math course called ”Write-in-Math!” for talented
youth. ”Write-in-Math” is managed by a team of mathematicians, logicians, computer scientists
and high school teachers. Three classes of the first year high school students participated in
the activity of ”Write-in-Math!” in the school year of 2004. The activity of ”Write-in-Math!” is
favorably accepted by both students and teachers. In this paper, we describe how we blended
the online activity provided by ”Write-in-Math!” and the offline regular classroom activity.
The activity of ”Write-in-Math” influenced students’ writing behavior. Our data shows that
the more students attend ”Write-in Math!” course, they improved logical way of thinking and
presentation, and they attained better online communication methodology.
An expository survey on epsilon substitution method
Toshiyasu Arai
Graduate School of Science and Technology, Kobe University,
Rokko-dai, Nada-ku, Kobe 657-8501, Japan ([email protected])
Epsilon substitution method is a method proposed by D. Hilbert to prove the consistency of (for
mal) theories. The idea behind the method is that one could replace consistently transfinite/non
computable objects as a figure of speech by finitary/computable ones as far as transfinite ones
8
are finitely presented as axioms of a theory. In other words, the replacement (epsilon substitu
tion) depends on contexts, i.e., formal proofs in which axioms for the transfinite objects occur. If
this attempt would be successfully accomplished, then the (1-)consistency of the theory follows.
For example, for the first order arithmetic PA, replace each existential formula ∃xF [x] by
F [�x.F [x]], where the epsilon term �x.F [x] intends to denote the least number x satisfying
F [x] if such a number exists. Otherwise it denotes an arbitrary object, e.g., zero. Then PA is
interpretable in a ’propositional calculus’ having the epsilon axioms :
F [t] → t �< �x.F [x] ∧ F [�x.F [x]].
The problem is to find a solving substitution which assigns numerical values to epsilon terms
and under which all the epsilon axioms occurring in a given proof are true.
Hilbert’s Ansatz is, starting with the null substitution S 0 which assigns zero whatever, to ap
proximate a solution by correcting false values step by step, and thereby generate the process
S 0 , S 1 , . . .. The problem is to show that the process terminates.
In this talk we expound basic ideas of the epsilon substitution method à la Ackermann [W.
Ackermann, Zur Widerspruchsfreiheit der Zahlentheorie. Math. Ann. vol. 117(1940), pp.
162-194], examine its proof-theoretic consequences, and report recent progress on the subject.
On the structure of almost Moore digraphs containing selfrepeats
E. T. Baskoro
Department of Mathematics, Institut Teknologi Bandung,
Jl. Ganesa 10 Bandung 40132, Indonesia ([email protected])
Joint work with Y.M. Cholily (Institut Teknologi Bandung) and M. Miller (University of Bal
larat).
An almost Moore digraph of degree d and diameter k, denoted by (d, k)-digraph, is a diregular
digraph of degree d > 1, diameter k > 1 and the number of vertices is d + d2 + · · · + dk (one less
than the Moore bound). In this talk, we study the existence of almost Moore digraphs of degree
d ≥ 4, diameter k ≥ 3 and containing a cycle of k. In particular, we will derive the structure of
permutation cycles of the vertices if such a digraph exists.
Projective modules over smooth real affine algebras
S. M. Bhatwadekar
School of Mathematics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005 India ([email protected])
9
Let A be a smooth affine algebra of dimension n over a field k and P be a finitely generated
projective A-module of rank n. In the case k = C, the field of complex numbers, a result of
Murthy says that P splits off a free summand of rank one if and only if the top Chern class
Cn (P ) of P in CH0 (A) is zero. However, in the case k = R, the field of real numbers, this
result in not true in general. For example, if A is the coordinate ring of an even dimensional
real sphere and P is the tangent bundle, then it is well known that P does not split off a free
summand of rank one even though Cn (P ) = 0. Incidentally, all the known examples of projective
modules exhibiting such a behaviour are over even dimensional real affine algebras. Therefore,
it is natural to ask :
Let A be a smooth affine algebra of odd dimension over the field of real numbers and let P be a
projective A-module of rank n. Does P split off a free summand of rank one if Cn (P ) = 0?
In my talk I will address this question and show that the question has an affirmative answer in
many cases.
Notes on holomorphic principal bundles over a compact Kähler manifold
Indranil Biswas
School of Mathematics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Bombay 400005, India ([email protected])
Our aim is to review some recent results on holomorphic principal bundles over a compact
Kähler manifold. We describe a necessary and sufficient condition for a holomorphic principal
G–bundle over a compact Riemann surface to admit a holomorphic connection. Another result
of the existence of connections is the following: A principal G–bundle EG over a compact Kähler
manifold admits a Hermitian–Einstein connection if and only if EG is polystable. Let G be a
connected reductive linear algebraic group over the field of complex numbers. Fix a parabolic
subgroup P ⊂ G without any simple factor and also fix a character χ of P such that χ is trivial
on the center Z(G) ⊂ G, and the restriction of χ to the parabolic subgroup of each simple factor
of G/Z(G) defined by P is nontrivial and antidominant. Let EG be a principal G–bundle over
a connected projective manifold M . Then the following four statements are equivalent:
1. The G–bundle EG is semistable and the second Chern class
c2 (ad(EG )) ∈ H 4 (M, Q)
vanishes.
2. The associated line bundle Lχ := (EG × Cχ )/P over EG /P for the character χ is numer
ically effective.
3. For every pair of the form (Y , ψ), where Y is a compact connected Riemann surface and
ψ : Y −→ M
10
a holomorphic map, and every holomorphic reduction EP ⊂ ψ ∗ EG of structure group to P
of the principal G–bundle ψ ∗ EG over Y , the associated line bundle EP (χ) = (EP ×Cχ )/P
over Y is of nonnegative degree.
4. For any pair (Y , ψ) as in (3), the G–bundle ψ ∗ EG over Y is semistable.
A generalization to principal bundles of the Atiyah–Krull–Remak theorem for vector bundles is
described.
Stochastic integrals of nonrandom kernels for nonorthogonal noises and
asymptotic analysis of canonical von Mises statistics of dependent observations
I. S. Borisov
Sobolev Institute of Mathematics,
630090 Novosibirsk, Russia ([email protected])
Joint work with A. A. Bystrov (Sobolev Institute of Mathematics).
Let X be a nonempty set and M be a semi-ring of its subsets with identity. Let {µ(A); A ∈ M}
be a random process defined on a probability space {Ω, Θ, P} and satisfying the condition
Eµ2 (A) < ∞ for all A ∈ M. The process µ(A) is called an elementary stochastic measure or a
noise if µ(A1 ∪ A2 ) = µ(A1 ) + µ(A2 ) P-a.s. for all subsets satisfying the conditions A1 ∩ A2 = ∅
and A1 ∪ A2 ∈ M. We study L2 -construction of stochastic integrals of the form
�
f (t)µ(dt),
X
where f (t) is a nonrandom σ(M)-measurable kernel function and σ(M) is the minimal σ-field
generated by M, in the case when the noise µ does not satisfy the classical orthogonality
condition Eµ(A1 )µ(A2 ) = m(A1 ∩ A2 ) , where m(A) is a measure on σ(M). As examples, we
consider several classes of random processes (not necessarily Gaussian!) with nonorthogonal
increments on the real line generating the corresponding elementary stochastic measures, for
which the stochastic integral exists under minimal restrictions on the kernel functions.
As an application of the above-mentioned general scheme, we also study multiple stochastic
integrals of the form
�
f (t1 , . . . , tn )µ(dt1 ) . . . µ(dtn )
Xn
and compare this construction with the corresponding results in [1] and [2].
In the second part of the talk we discuss some applications of this construction to asymptotic
analysis of normalized canonical von Mises statistics and U-statistics based on samples from a
stationary sequence of observations under some dependency conditions.
11
[1] Cambanis, S., Huang, S. T. Stochastic and multiple Wiener integrals for Gaussian processes
— Ann. Probab., 1978, v.6, p. 585-614.
[2] Dasgupta, A., Kallianpur, G. Multiple fractional integrals — Probab. Th. Rel. Fields, 1999,
v. 115, No. 4, p. 505-526.
Record Values
Arup Bose
Indian Statistical Institute, Calcutta, India ([email protected])
We address the question of convergence of appropriately normalised partial sum of upper/lower
records. It is shown that all nonnegative infinitely divisible distributions with continuous Levy
measure arise as the limit of partial sums of records. We also find conditions under which the
partial sum process of records converges to a Gaussian process.
New series for π,
1
1
and 2
π
π
Heng-Huat Chan
Department of Mathematics, National University of Singapore,
2 Science Drive 2, Singapore 117543 ([email protected])
1
1
In this survey talk, we present new series associated with π,
and 2 which are recently
π
π
discovered by various mathematicians.
Statistical learning from distributions of DNA words
Probal Chaudhuri
Indian Statistical Institute, Calcutta ([email protected])
I will begin with some examples from molecular biology to demonstrate how statistical analysis of
the distributions of oligonucleotides can lead to significant biological discoveries. The problems
considered can be formulated as statistical learning problems that are often partially supervised
in nature and requires ”selection of variables” i.e., selection of oligonucleotides. This motivated
the development of certain probabilistic models for DNA data. These models are derived from
a stochastic replication model for DNA sequences and have several interesting features, which
will be discussed together with some related results.
12
Total curvature and isoperimetric inequality
Jaigyoung Choe
Department of Mathematics, Seoul National University,
Seoul, 151-742, South Korea ([email protected])
Joint work with Mohammad Ghomi (Georgia Institute of Technology) and Manuel Ritoré (Uni
versidad de Granada).
We prove that if Σ is a compact hypersurface in Euclidean space Rn , its boundary lies on
the boundary of a convex body C, and meets C orthogonally from the outside, then the total
positive curvature of Σ is bigger than or equal to half the area of the sphere S n−1 . Also we
obtain necessary and sufficient conditions for the equality to hold. Using this total curvature
estimate we then prove that the area of a hypersurface Σ which traps a given volume outside
a convex body C in Rn is bigger than or equal to the area of a hemisphere which traps the
same volume on one side of a hyperplane. Further, when C has smooth boundary ∂C, we
show that equality holds if and only if Σ is a hemisphere which meets ∂C orthogonally. This
relative isoperimetric inequality can also be obtained in three and four-dimensional Riemannian
manifolds of nonpositive curvature.
Spectral radius of refinement and subdivision operators for some classes of
dilation matrices
V. D. Didenko
Department of Mathematics, Universiti Brunei Darussalam,
Bandar Seri Begawan, BE1410 Brunei ([email protected])
A matrix M ∈ Z s×s is called power diagonal if there is a positive integer k such that
M k = diag (µ1 , µ2 , . . . , µs ).
Note that the set of power diagonal matrices contains, in particular, some classes of isotropic
matrices.
In this work we estimate spectral radius of refinement operator RaM and subdivision operator
s
s
m
SaM , defined respectively on the Hilbert spaces Lm
2 (R ) and l2 (Z ), m ≥ 1 by
�
RaM ϕ :=
ak ϕ(M · −k),
s
k∈Z
�
� M �
Sa ξ j :=
aj−M k ξk , j ∈ Z s ,
s
k∈Z
where ak are the Fourier coefficients of a matrix function a ∈ Lm×m
(T ), and M is similar to a
∞
power diagonal matrix.
13
The Inner Model Program
Qi Feng
Chinese Academy of Sciences, China ([email protected])
More than sixty years ago, K Gödel constructed the first inner model of set theory, L, the
universe of constructible sets, and proved that (ZF + V = L)L , (AC)L , (GCH)L , namely,
the Axiom of Choice and the General Continuum Hypothesis are relatively consistent to ZF,
Zemerlo–Frankel Axioms of Sets. Since 1970’s, the inner model program started to evolve to
adapt to various large cardinals in order to gain deeper understanding of higher infinity. In this
talk, I would like to present a big picture upto date of this program to the general mathematical
community of Asia by explaining the fundamentals and key results in this study.
Renewal theory and ruin probabilities in Markov random walks
Cheng-Der Fuh
Institute of Statistical Science, Academia Sinica,
Taipei, Taiwan 11529 ([email protected])
Let {Xn , n ≥ 0} be a Markov chain on a general state space X with transition probability P
and stationary probability π. Suppose an additive component Sn takes values in the real line R,
is adjoined to the chain such that {(Xn , Sn ), n ≥ 0} is a Markov random walk. In this talk, we
present an uniform Markov renewal theorem with an estimate on the rate of convergence. This
result is applied to boundary crossing problems for {(Xn , Sn ), n ≥ 0}. To be more precise, for
given b ≥ 0, define the stopping time τ = τ (b) = inf{n : Sn > b}. When a drift µ of the random
walk Sn is 0, we derive a one-term Edgeworth type asymptotic expansions for the first passage
probabilities Pπ {τ < m} and Pπ {τ < m, Sm < c}, where m ≤ ∞, c ≤ b and Pπ denotes the
probability under the initial distribution π. When µ �= 0, Brownian approximations for the first
passage probabilities with correction terms are derived. Applications to sequential estimation
and truncated tests in random coefficient models and first passage times in products of random
matrices are also given.
Generalized fractional integral operators and their modified versions
Hendra Gunawan
Department of Mathematics, Bandung Institute of Technology,
Bandung 40132, Indonesia ([email protected])
14
Associated to a function ρ : (0, ∞) → (0, ∞), let Iρ be the operator defined on a suitable function
space by
�
ρ(|x − y |)
Iρ f (x) :=
f (y) dy,
n
Rn |x − y|
and I˜ρ be the modified version of Iρ given by
�
� �
ρ(|x − y |) ρ(|y|)(1 − χB0 (y))
˜
Iρ f (x) :=
−
f (y) dy.
|x − y |n
|y |n
Rn
For ρ(t) = tα , 0 < α < n, the operator Iρ is nothing but the fractional integral operator
or the Riesz potential, which is known to be bounded from Lp (Rn ) to Lq (Rn ) provided that
1/p − 1/q = α/n.
Next, for 1 ≤ p < ∞ and a function φ : (0, ∞) → (0, ∞), we define the generalized Morrey space
Mpφ = Mpφ (Rn ) by
Mpφ
�
:= f ∈ Lploc : sup
B
�
�1/p
�
�
1
1
p
|f (y)| dy
<∞
φ(B) |B| B
and the generalized Campanato space Lpφ = Lpφ (Rn ) by
Lpφ
�
:= f ∈ Lploc : sup
B
1 � 1
φ(B) |B|
�
|f (y) − fB |p dy
�1/p
�
<∞ ,
B
where the supremum is taken over all open balls B = B(a, r) in Rn , |B| denotes the Lebesgue
measure of B, φ(B) = φ(r), and fB is the average of f over B.
In this talk, we discuss the boundedness of Iρ and I˜ρ on generalized Morrey spaces and on
generalized Campanato spaces, respectively. Under some conditions on ρ, φ, and ψ, we prove
that Iρ is bounded from Mpφ to Mqψ , while I˜ρ is bounded from Lpφ to Lqψ for 1 < p < q < ∞.
Related results were proved earlier by E. Nakai [Taiwanese J. Math. 5 (2001), 587–602] and
Eridani [Tamkang J. Math. 33 (2002), 335–340]. Most of the results presented in this talk is
joint with Eridani and E. Nakai, and has been published recently in Sci. Math. Jpn. 60 (2004),
539–550.
Applications of Statistics in Genetic Studies-Gene Mapping for Complex Diseases
C. A. Hsiung
Division of Biostatistics and Bioinformatics, National Health Research Institutes (NHRI),
35 Keyan Road, Zhunan Town, Miaoli County 350, Taiwan ([email protected])
Research supported in part by NHRI through grants BS-090 094-PP-01 and NIH.
Joint work with Y. F. Chiu, C. F. Hsiao, W. C. Wang, C. C. Wen, I. S. Chang and SAPPHIRe
Study Group.
15
We have conducted an international collaborative genetic study ”SAPPHIRe” (Stanford Asian
Pacific Program in Hypertension and Insulin Resistance). The objectives are to map and iden
tify genetic loci underlying hypertension in Chinese and Japanese. Hypertension is known to
be a complex disease, with genes and environmental factors interacting to control risk of this
disease. We have built up an infrastructure to facilitate this gene-mapping study. We identify
chromosomal regions that may harbor genes influencing not only blood pressure but also lipids
and insulin resistance related variables. We also study if the candidate genes are associated with
the hypertension or metabolic syndrome or insulin resistance. For the linkage analysis, the study
design was based on the method of ”ascertaining sibpairs with extremely discordant or highly
concordant phenotypes to obtain greater power to detect linkage”. We suggest some systematic
way to decide how to select the discordant sibpairs or concordant sibpairs in linkage analysis
based on their phenotype trait values. We also propose a multi-point linkage analysis method,
which outperforms existing methods under the situation of existence of linkage disequilibrium.
In this talk we will also present the genome-wide scan results for metabolic phenotypes as well as
association studies between single nucleotide polymorphisms / haplotypes and complex diseases.
Characters and Dirac cohomology
Jing-Song Huang
Department of Mathematics, Hong Kong University of Science and technology,
Clear Water Bay, Kowloon, Hong Kong SAR, China ([email protected])
Dirac cohomology is a new tool to study irreducible unitary and more general admissible rep
resentations of reductive Lie groups. Usage of this tool has been successful to obtain simpler
proofs of or sharpen some classical theorems in representation theory. The more classical theory
of characters of admissible representations developed by Harish-Chandra and others provides
deep insights into structures of the representations. The aim of this talk is to discuss the con
nection of Dirac cohomology and characters of admissible representations.
Space of surjective morphisms between projective varieties
Jun-Muk Hwang
Korea Institute for Advanced Study,
207-43 Cheongnyangni-dong, Seoul 130-722, Korea ([email protected])
Let X and Y be two complex projective varieties. Denote by Homs (X, Y ) the set of surjective
morphisms from X to Y . Each irreducible component of Homs (X, Y ) has the structure of a
quasi-projective algebraic variety. A natural question to ask is what kind of an algebraic variety
this is. Given a surjective morphism f : X → Y , the question can be reformulated as describing
all possible deformations of f as morphisms between X and Y . In this talk, we will survey some
recent results on this question. In particular, our joint-work with S. Kebekus and T. Peternell
16
in the case when Y is of nonnegative Kodaira dimension and our joint-work with N. Mok in the
case when Y is a Fano manifold of Picard number 1 will be discussed.
A State-Space Partitioning Method for Pricing High-Dimensional American-Style
Options
Xing Jin
Department of Mathematics, National University of Singapore,
2 Science Drive 2, Singapore 117543 ([email protected])
Joint work with H.H. Tan (National University of Singapore) and J.H. Sun (Development Bank
of Singapore).
The pricing of American-style options by simulation-based methods is an important but difficult
task primarily due to the feature of early exercise, particularly for high-dimensional derivatives.
In this paper, a bundling method based on quasi-Monte Carlo sequences is proposed to price highdimensional American-style options. The proposed method extends Tilley’s bundling algorithm
to higher-dimensional situations. By using low-discrepancy points, this approach partitions
the state space and forms bundles. A dynamic programming algorithm is then applied to the
bundles to estimate the continuation value of an American-style option. A convergence proof
of the algorithm is provided. A variety of examples with up to 15 dimensions are investigated
numerically and the algorithm is able to produce computationally efficient results with good
accuracy.
Crystal bases for quantum affine algebras and combinatorics of Young walls
Seok-Jin Kang
Department of Mathematics, Seoul National University,
Seoul 151-742, Korea ([email protected])
The crystal basis theory developed by Kashiwara provides a powerful combinatorial tool to study
the representations of quantum groups. A crystal basis can be understood as a basis at q=0
and is given a structure of colored oriented graph, called the crystal graph. The crystal graphs
have many nice combinatorial features reflecting the internak structure of integrable modules
over quantum groups. In particular, crystal bases have extremely nice behavior with respect to
taking the tensor product.
In this talk, we will focus on crystal basis theory for quantum affine algebras and combinatorics
of Young walls. The Young walls consist of colored blocks with various shapes and can be viewed
as generalizations of Young diagrams. The crystal graphs for integrable highest weight modules
are realized as the affine crystals consisting of reduced proper Young walls.
17
We will also discuss the motivation behind combinatorics of Young walls and further develop
ments.
Performance of Singapore’s Pupils in TIMSS Some pedagogical practices of
Mathematics Teachers in Singapore Schools
Berinderjeet Kaur
National Institute of Education, Singapore ([email protected])
Eighth graders from Singapore were top in mathematics three times in a row for the international
study TIMSS (Trends in Mathematics and Science Study), in 1995, 1999 and 2003 (Beaton et
al. 1996, Mullis et al. 2000, Mullis et al. 2004). Similarly fourth graders from Singapore were
top in mathematics two times in a row for the same international study, TIMSS in 1995 and
2003 (Mullis et al. 1997, Mullis et al. 2004).
The commendable performance of pupils from Singapore has drawn a lot of attention to the
small island and its educational system, particularly to the teaching of mathematics in schools.
In the TIMSS national reports for Singapore (Ministry of Education 1996, 1997, 2000, 2004)
several possible contributing factors for high performance of pupils were stated. These were:
1. The Education System
2. Curriculum and Curriculum Implementation
3. Qualification and Working Ethos of Teachers
4. School Organization and Environment
5. Availability of Resources for Teaching and Learning
6. Pupils’ Attitudes Towards Mathematics
7. Pupils’ Educational Aspirations and Home Resources
A study by the American Institutes for Research (Ginsburg et al. 2005) comparing the teach
ing of elementary school mathematics in the United States and Singapore has found that the
strengths of Singapore are in the following areas:
1. Frameworks (one would call this syllabuses in Singapore)
2. Textbooks
3. Teaching & professional development of teachers
4. Assessment: Singapore uses more challenging tests and utilizes a value-added approach
that rewards schools and in turn teachers for individual student progress over time. In this
paper some pedagogical practices of mathematics teachers in Singapore schools will be explored.
18
Classification of conformal field theories with operator algebras
Yasuyuki Kawahigashi
Department of Mathematical Sciences, University of Tokyo,
Komaba, Tokyo, 153-8914, JAPAN ([email protected])
Research supported in part by JSPS.
Joint work with Roberto Longo (Università di Roma, “Tor Vergata”).
In an operator algebraic approach to quantum field theory, one chiral conformal field theory is
described with one family of operator algebras on a vacuum Hilbert space and such a family
is called a local conformal net of factors. A local conformal net of factors naturally brings
a unitary representation of the Virasoro algebra and it produces an invariant called a central
charge c, which is a positive real number. We have obtained a complete classification of local
conformal nets for the case c < 1. The classification list consists of the Virasoro nets, which arise
directly from unitary representations of the Virasoro algebra, their simple current extensions of
index 2, and four exceptionals at the central charges 21/22, 25/26, 144/145, 154/155. Three of
these four are coset models, but the other is a new example. This is the first classification
result in the operator algebraic approach to quantum field theory. Using this, we also obtain
classification results for full conformal field theory and boundary conformal field theory within
operator algebraic framework, where a certain new problem on 2-cohomology vanishing for a
tensor category is solved.
The technique used for this classification is based on representation theory of local conformal nets
of factors. The notion of complete rationality, introduced by Kawahigashi-Longo-Müger, and
the method of α-induction, introduced by Longo-Rehren and studied by Xu and BöckenhauerEvans-Kawahigashi, play fundamental roles.
This approach is closely related to theory of vertex operator algebras, another approach to
quantum field theory using infinite dimensional algebraic systems. Our classification result
above for the case c < 1 is immediately gives a classification result of extensions of the Virasoro
vertex operator algebras with c < 1, since the both classification problems can be stated in
terms of tensor categories.
We also persue this analogy with theory of vertex operator algebras, and construct and study
a local conformal net of factors corresponding to the moonshine vertex operator algebra, whose
automorphism group is the monster group, the largest among the 26 sporadic finite simple
groups.
Order-Preserving Transformation Semigroups Whose Bi-ideals and Quasi-ideals
Coincide
Yupaporn Kemprasit
Department of Mathematics, Faculty of Science, Chulalongkorn University,
19
Bangkok 10330, Thailand ([email protected])
Let BQ denote the class of all semigroups whose bi-ideals and quasi-ideals coincide. It is known
that BQ contains all regular semigroups. However, a semigroup in BQ need not be regular. For
an interval X in R, the order-preserving full transformation semigroup on X, OT (X) is known
to be regular if and only if X is closed and bounded. Hence if X is a closed and bounded interval
in R, then OT (X) ∈ BQ. The purpose of this paper is to show that for a nontrivial interval
X of a subfield F of R, OT (X) ∈ BQ if and only if F = R and X is closed and bounded.
Consequently, this condition is also necessary and sufficient for te regularity of OT (X).
Multiplicity-free theorem and visible actions on complex manifolds
Toshiyuki Kobayashi
RIMS, Kyoto University,
Sakyo-ku, Kyoto, 606-8502, Japan ([email protected])
Based on the Lie group action on a complex manifold such that its generic orbits intersect with a
totally real submanifold, we provide a new geometric principle that leads us to multiplicity-free
theorems of representations realized on holomorphic vector bundles.
Sums of squares relaxation of polynomial optimization problems
Masakazu Kojima
Department of Mathematical and Computing Science, Tokyo Institute of Technology,
2-12-1-W8-29, Oh-Okayama, Meguro, Tokyo 152-8552, Japan ([email protected])
Given a real-valued continuous function (objective function) defined on the n-dimensional Euclid
ean space and a subset of the space (a constraint set), we consider a problem of finding a point
in the constraint set that minimizes the objective function.This is a fundamental problem in the
field of mathematical programming, and numerical computation of such a point is sometimes
called global optimization. Without any additional assumptions, this problem is too difficult to
analyze and to design efficient numerical methods. A polynomial optimization problem (POP)
is a problem of minimizing a (real valued multivariate) polynomial objective function over a
constraint set described by a finite set of polynomial inequalities. The POP covers many impor
tant nonconvex optimization problems such as 0-1 integer programs and quadratic programs. It
serves as a unified and general mathematical model for global optimization. If only polynomials
are considered in global optimization, many profound results from algebra can be used to analyze
the problem and introduce efficient numerical methods. In recent years, sums of squares (SOS)
relaxation for POPs has been proposed and studied extensively. The fundamental idea behind
the SOS relaxation lies on a simple fact that a sum of squares of finitely many polynomials
is a nonnegative polynomial (but the converse is not true in general). A hierarchy of convex
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relaxation problems with increasing sizes is constructed in the SOS relaxation of a POP. Each
relaxation problem can be numerically solved as a semidefinite programming problem. Under
a moderate assumption, a convex relaxation problem with a finite size in the hierarchy attains
the exact optimal value of the original POP. The main purpose of this talk it to present the
effectiveness of the SOS relaxation when solving POPs in theory and practice.
Hybrid Pairwise Likelihood Methods
Anthony Y. C. Kuk
Department of Statistics & Applied Probability, National University of Singapore,
6 Science Drive 2, Singapore 117546
In many situations, it is difficult to specify correctly the multivariate joint density of the ob
served data, or the joint density could be analytically intractable, whereas the bivariate density
is usually more manageable. To facilitate statistical inference in situations like these, it has been
proposed that one could replace the unavailable or intractable joint likelihood by the pairwise
likelihood constructed by summing the log-likelihood contribution of every pair of observations.
There are considerable interests in pairwise likelihood methods recently, with applications in
spatial statistics, clustered and longitudinal data analysis, multivariate failure time distribution,
multivariate extremes, estimation of recombination rates from gene sequences, and the popula
tion genetic analysis of single nucleotide polymorphism (SNP) data. Obviously, if the marginal
distributions of the data are known up to dimensions 3 and 4, then one ought to be able to do
better, for example, by working out the optimal way to combine the pairwise likelihood score
functions, but this begs the question of why we resort to pairwise likelihood in the first place. A
more relevant question is whether we can improve the pairwise likelihood method in the situation
that it is designed for, namely, when only the bivariate distributions are known. To this end, we
propose a hybrid method whereby the marginal regression parameters, for a given association
parameter value, are estimated by solving an optimal estimating equation constructed from the
univariate score functions or moment equations; whereas the association parameter is estimated
from pairwise likelihood. It is shown that the alternating logistic regression procedure proposed
in the literature for multivariate binary data is a special case of this hybrid method. However,
it is also shown that an extension of alternating logistic regression that has appeared in the
literature to handle multivariate ordinal data is not likelihood based and can be improved upon
by following the proposed hybrid pairwise likelihood approach. The superiority of the suggested
hybrid approach over ordinary pairwise likelihood approach can be established for the case of ex
ponential family. Connection is also made to the IFM (inference functions for margins) method
proposed in the literature of multivariate distribution with given marginals. The superiority of
the hybrid pairwise likelihood approach is further demonstrated in a non-exponential setting
using simulations from a serially correlated gamma frailty model for longitudinal count data
with over-dispersed negative binomial marginals. A potential application of the hybrid pairwise
likelihood approach to the analysis of clustered data with informative cluster sizes will also be
mentioned.
21
Study of Pattern Selection in Nonlinear Phenomena by Weakly Nonlinear
Stability Analysis
Y. Lenbury
Department of Mathematics, Faculty of Science, Mahidol University,
Rama 6 Road, Bangkok, 10400 Thailand ([email protected])
Joint work with A. Pansuwan (Mahidol University) and D. J. Wollkind (Washington State
University).
The problem of predicting pattern selection in nonlinear phenomena can be studied through
the weakly nonlinear stability analysis which has been found to be suitable since it allows one
to deduce quantitative relationships between system parameters and stable patterns which are
valuable for experimental design and difficult to accomplish using simulation alone. The method
incorporates the nonlinearities of the relevant model, while basically pivoting the perturbation
procedure about the critical point of linear stability theory. We present its application to the
investigation of spontaneous stationary equilibrium pattern development on metallic or semi
conductor solid surfaces during ion-sputtered erosion at normal incidence. The process can
be represented by a damped Kuramoto-Sivashinsky nonlinear partial differential time-evolution
equation for the interfacial deviation from a planar surface which includes a deterministic ionbombardment arrival time and is defined on an unbounded spatial domain.
Secondary School Students Learning and Understanding the Validity Of
Conditional Statements
Fou-Lai Lin
Department of Mathematics, National Taiwan Normal University (linfl@math.ntnu.edu.tw)
Based on a national survey of mathematical proof and proving of Taiwanese junior high students,
we have observed that (1) students reasoning performances were influenced by their awareness
of the correctness of the conditional statements, (2) over half of the students considered a
statement and its converse to be the same, (3) Over one third of students who make their
decision empirically on the validity of conditional statements, i.e producing supporting or counter
examples. Taiwan junior high students have difficulties in distinguishing between the validity
of arguments and the truth of assertions. It was noticed that many senior high school students
also could not make the distinguishment while learning mathematical induction and proof by
contradiction. So learning to understand the validity of conditional statements is a crucial issue
for secondary mathematics education. This paper aims (1) to analyze, to what extent, the
Venn diagram can be used to represent what guide students think while they are conducting
justification on mathematical statements, and (2) to argue that understanding the validity of
conditional statements is either a sudden enlightment process or gradual enlightment process,
different rational in the two distinguished schools of Zen . The argument will be based on certain
teaching experiments.
22
Operator Splitting and Fast C 0 Finite Element Methods for Liquid Crystal Flows
Ping Lin
Department of Mathematics, National University of Singapore,
2 Science Drive 2, Singapore 117543 ([email protected])
In this talk we present a C 0 finite element method for a 2D hydrodynamic liquid crystal model
which is simpler than existing C 1 element methods and mixed element formulation and the
energy law is formally derived as well. The formulation is verified by comparing its results with
those obtained by C 1 elements and by mixed formulation. A splitting method combined with
only a few fixed point iteration for the penalty term of the director field is applied to reduce
the size of the stiffness matrix and to keep the stiffness matrix time-independent. The latter
avoids solving a linear system at every time step and largely reduces the computational time,
especially when direct linear system solvers are used. Through numerical experiments of a few
other splittings and explicit-implicit strategies we recommend a fast and reliable algorithm for
this model. A number of examples are computed to demonstrate the algorithm.
Lifts of Graphs
N. Linial
School of Computer Science and Engineering, The Hebrew University,
Jerusalem 91904, Israel ([email protected])
Covering maps have been extensively studied by topologists. A map ϕ : V (G) → V (H) between
the vertex sets of two graphs is a covering map if it maps the neighbor set ΓH (x) of every
x ∈ V (H) one-to-one onto the neighbor set of its image ΓG (ϕ(x)). It is an easy fact that every
covering map onto a connected graph has a fold number, i.e., there is an integer n such that ϕ
is n : 1. If there exists an n-fold covering map from H to G, we say that H is an n-lift of G.
In concrete combinatorial terms an n-lift H of a graph G = (V, E) is specified by associating
a permutation πe ∈ Sn with every edge e = xy ∈ E(G). The vertex set of H is V × [n] and
its edge set is {(x, i)(y, πe (i))|xy ∈ E(G), i = 1, . . . , n}. The set {(x, i)|i = 1, . . . , n} ⊆ V (H) is
denoted by Fx , and the covering map simply sends all of Fx to x for every x ∈ V (G).
The questions that we investigate in this area fall into three main categories: (i) Probabilistic,
(ii) Extremal, and (iii) Algorithmic. The probabilistic/asymptotic aspect of the theory is cur
rently the most advanced, though much remains unknown here as well. In a random lift, the
permutations πe are selected at random. It turns out that several interesting graph properties
exhibit an asymptotic zero/one law. For example, if δ is the smallest vertex degree in G, then
clearly the vertex connectivity of any lift of G cannot exceed δ. However, if δ ≥ 3, then asymp
totically almost every lift of G is δ-connected. Also, for every G and large even n either almost
every n-lift of G has a perfect matching or almost none has it.
23
Using lifts, we were recently able to explicitly construct Nearly Ramanujan Graphs. These are
d-regular graphs the second eigenvalue of which is almost as small as possible. Such graphs are
optimal expanders and their study is a deep and beautiful research area that connects between
graph theory, representation theory, number theory and more. We construct these graphs by
successive 2-lift steps. We still do not know how to generate Ramanujan graphs this way, and
this quest leads to a fascinating problem that remains open.
Recently, Khot has introduced the ”unique games conjecture”. He and others have shown how
to derive from this conjecture a variety of hardness-of-approximation results in computational
complexity. In the present context this question is a search problem in which one seeks a dense
section in a lifted graph i.e., a set of vertices in H that contains exactly one vertex in each of the
sets Fx . It is reasonable to expect that other computational problems on graph lifts will play a
significant role in the theory of algorithms.
The action of the Galois group on the Lie algebra of the fundamental groups
Makoto Matsumoto
Department of Mathematics, Hiroshima University,
1-3-1 Kagamiyama, Higashi Hiroshima 729-1104 Japan ([email protected])
Let X be a geometrically connected algebraic variety defined over a field K ⊂ C with a Krational base point x. Then, there is a natural action of the absolute Galois group GK of K on
the profinite completion Π̂ of the fundamental group of X.
There is a Lie-algebraization L of Π̂, known as the continuous Malcev completion over the �-adic
field Q� , on which GK acts by functoriality. That is, we have a group homomorphism
ρX,x : GK → AutL
which is a Galois representation on an infinite dimensional Lie algebra over Q� .
Even when X is the projective line minus three points with K = Q, this action is known to have
rich mathematical structure, by the studies of Anderson, Coleman, Deligne, Ihara, and others.
For example, they proved that the extensions appearing in L as GQ -modules contain all Soulé’s
cocycles (generators of one dimensional vector spaces H 1 (GQ , Q� (m))).
Deligne-Ihara’s conjecture claims that the Zariski closure of the Galois image in AutL (naturally
considered as the set of Q� points of an affine group scheme) is generated by some elements
corresponding to the Soulé’s cocycles, and moreover they are free generators.
We review how Soulé’s cocycles appears in this representation (but by a direct computation,
simpler than the original methods), and then explain about the generation-part of the conjecture
(a joint work with R. Hain in 2000).
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Singularities of modulii spaces of vector bundles on curves
Vikram B. Mehta
Tata Institute of Fundamental Research,
Homi Bhabha Road, Bombay 400005, India
Joint work with Venkata Balaji (August Universitat).
We study the modulii spaces of vector bundles over a family of curves. We prove that these
modulii spaces specialize correctly and they have canonical singularites in char 0 and strongly
F -regular in char p.
McKay’s E8 -observation, a lattice VOA VE8 and Coxeter elements
Masahiko Miyamoto
Institute of Mathematics, University of Tsukuba,
305, Tsukuba Japan ([email protected])
In my talk, I will present three sets of dihedral groups associated with the E8 -diagram and
explain their connection via vertex operator algebras (shortly VOA). One is from the Monster
simple group M, another is from a lattice VOA and the last is from Weyl groups. They connect
each other via Miyamoto involutions given by 2-dim. Ising models.
It is well known that the largest sporadic finite simple groups, the Monster simple group M,
has mysterious properties. One of them is a McKay’s E8 -observation. The products of two
2A-involutions of Monster simple group M fall into one of nine conjugacy classes:
1A − 2A − 3A − 4A − 5A − 6A − 4B − 2B
|
3C
where nX denotes a conjugacy class of order n. Their numerical labels coincide with the mul
tiplicities of simple roots in a primitive isotropic element in an extended E8 -diagram Ê8 .
Recently, Lam, Yamada and Yamauchi
found similar phenomenon in an automorphism group
√
of a lattice VOA V√2E8 , where 2E8 denotes a root lattice of type E8 with an inner product
multiplied by 2. For each node pX in Ê8 , the sub Dynkin diagram Ê8 − {pX} generates a root
τ √which acts on the weight
sublattice HpX which has an index p in E8 . Consider√an involution
√
one space (V√2E8 )1 as −1. Using a natural map ϕ : 2E8 → 2E8 / 2HpX ∼
= Z/pZ, we have
√
α
2π
−1ϕ(α)/p
α
an automorphism θpX of order p by θpX (e ) = e
e . Then �τ, θpX � ∼
= D2p satisfy the
similar properties as the dihederal groups generated by 2A-involutions in M do.
There is another set of Dihedral groups induced from the Ê8 -diagram. For a sub Dynkin diagram
X of Ê8 , divide the set of nodes X into X�= A ∪ B so that A and
� B have no edges. Then the
products of orthogonal reflections σA = x∈A Ref x and σB = x∈B Ref x2n+1 are involutions
and σA σB is a Coxeter element of X.
25
The main result is to show that these sets of Dihedral groups are linked together and the LYYphenomenon coincides perfectly to the McKay’s E8 -observation via orbifold theories of Niemeier
lattice VOAs.
On the original fourteenth problem of Hilbert
Shigeru Mukai
RIMS, Kyoto University,
Japan ([email protected])
Let V be a finite dimensional linear representation of an algebraic group G and consider the
induced action of G on the ring k[V ] of polynomial functions on V . The original fourteenth
problem asks whether k[V ]G , the ring of invariants, is finitely generated. It is so for arbitrary V
if G is reductive or the one dimensional additive group. In this talk I will discuss this problem
including the recent development on the following:
Problem. Is the ring of invariants finitely generated for the two dimensional additive group G?
Instanton counting
Hiraku Nakajima
Department of Mathematics, Kyoto University,
Kyoto, 606-8502, JAPAN ([email protected])
Nekrasov’s deformed partition functions of 5-dimensional SUSY Yang-Mills theories are given by
the characters of coordinate rings of instanton moduli spaces over R4 with respect to the natural
torus action. They satisfy functional equations (called blowup equations), which charcterize
them. We prove that their certain limits are given by period integrals of Seiberg-Witten curves.
Almost periodic and almost automorphic solutions of differential equations
Van Minh Nguyen
Department of Mathematics, University of West Georgia,
Carrollton, GA 30118, USA ([email protected])
In this talk we present recent developments on the study of almost periodic and almost automor
phic solutions of the evolution equations of the form u̇(t) = A(t)u(t) + f (t) (∗), where A(t) is a
26
(in general unbounded) linear operator on a Banach space that generates a periodic evolutionary
process and f is almost periodic or almost automorphic. The results are necessary and sufficient
conditions for the existence and uniqueness of mild solutions in terms of the spectrum of f and
the one determined by the equation. The method of study include (but not limited to) the
spectral theory of functions, evolution semigroups, sums of commuting operators. Applications
to ODE, FDE and PDE are also discussed.
Tropical combinatorics of Young tableaux
Masatoshi Noumi
Department of Mathematics, Kobe University,
Rokko, Kobe 657-8501, Japan ([email protected])
Joint work with Yasuhiko Yamada.
I will discuss how the Robinson-Schensted algorithm for Young tableaux can be formulated in
terms of piecewise linear and totally positive birational transformations. Also it will be shown
that there is a remarkable relationship between the combinatorics of Young tableaux and the
discrete Toda equation.
Nucleation of Smectics and Critical Magnetic Fields
Xingbin Pan
Department of Mathematics, East China Normal University,
Shanghai 200062, China ([email protected]).
Research supported in part by CNSF.
Landau-de Gennes model has been used to describe phase transitions of liquid crystals. We
shall examine the concentration behaviors of minimizers of some variational problems related to
the nucleation of smectics, and discuss possible analogies with the Meissner-normal transition
of type I superconductors and with surface nucleation of type II superconductors. We shall also
use the Landau-de Gennes model to examine the magnetic field-induced instabilities in liquid
crystals, and discuss some critical phenomena.
27
Arithmetic properties of certain 2-dimensional fields
Rama Parimala
Tata Institute of Fundamental Research,
Homi Bhabha Road, Bombay 400005, India ([email protected])
Let K be a number field. Then it is a classical theorem of Hasse- Brauer-Noether that every
central division algebra over K is cyclic with its exponent coinciding with the index. We exhibit
a class of 2-dimensional fields where a similar property holds for division algebras. We explain
consequences of such a result in the solution of Conjecture II of Serre concerning triviality
of principal homogeneous spaces under semi-simple simply connected linear algebraic groups
defined over a perfect field of cohomological dimension 2.
The Pseudo Primitive Idempotents of a Distance-Regular Graph
Arlene A. Pascasio
Department of Mathematics De La Salle University - Manila,
2401 Taft Avenue Malate, Manila 1004 Philippines ([email protected])
Joint work with Paul Terwilliger (University of Wisconsin - Madison)
Let Γ denote a distance-regular graph with diameter D ≥ 3 and Bose-Mesner algebra M. The
concept of a pseudo primitive idempotent was introduced by Weng and Terwilliger in [European
J. Combin. 25 (2004) pp. 287-298]. This concept is defined as follows. Let θ denote a real
number. A nonzero element E in M is called a pseudo primitive idempotent of Γ associated
with θ whenever (A − θI)E ∈ RAD , where A and AD are the adjacency matrix and the Dth
distance matrix of Γ, respectively. Let E and F denote pseudo primitive idempotents of Γ. We
say this pair is tight whenever the entrywise product E ◦ F is a pseudo primitive idempotent.
We determine all the tight pairs of pseudo primitive idempotents of Γ.
Period Integrals and L-functions
Dipendra Prasad
School of Mathematics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Bombay 400005, India ([email protected])
Let G be a reductive algebraic group over Q, and H a subgroup of G defined over Q. For an
automorphic form f on G, one can consider its integral on H(Q)H(A), called the period integral.
It is a question of much number theoretic interest to classify those automorphic representations
28
on G for which the period integral is nonzero for some element of the representation space.
There are several instances when the nonvanishing is controlled by an L-function: in some cases,
by the existense of a pole, and in some other situation by the non-vanishing of a central critical
value of an L-function. We will present a survey of known results.
From Moufang Loops to Groups: A Journey
Andrew Rajah
School of Mathematical Sciences, Universiti Sains Malaysia,
11800 USM, Penang, Malaysia ([email protected])
Let S be a non-empty set and ∗ a function from S ×S to S. Then �S, ∗� is called a binary system.
If ∗ maps (a, b) ∈ S × S to c ∈ S, we shall write a ∗ b = c. If specification of any two of the
elements x, y, z ∈ S in the equation x ∗ y = z uniquely determines the third element, then �S, ∗�
is called a quasigroup. If further there exists an identity element in S, then �S, ∗� is called a loop.
A loop is called a group if it satisfies the associative law (x∗y)∗z = x∗(y ∗z). On the other hand,
a loop is called a Moufang loop if it satisfies the Moufang identity (x∗y)∗(z ∗x) = (x∗(y ∗z))∗x.
It can be easily shown that all groups are Moufang loops. However, it is possible to construct
Moufang loops that are not groups - the smallest example is one containing exactly 12 elements,
that is, a nonassociative Moufang loop of order 12. Our study concerns the question: ”For what
values of n does there exist a nonassociative Moufang loop of order n?”, and the equivalent
question: ”For what values of n must all Moufang loops of order n be groups?”. We shall
present the latest results as well as certain techniques used in determining them.
Uniqueness and stability in inverse problems for hyperbolic equations
V. G. Romanov
Sobolev Institute of Mathematics,
Acad. Koptyug prospect, 4; 630090 Novosibirsk, Russia ([email protected])
Some methods for an investigation of inverse problems for hyperbolic equations and systems
of electrodynamics and elasticity are discussed. We consider the ray expansion of solutions to
the equations in a neighborhood of a characteristic cone and connections between coecients of
this expansion and unknown coecients of the equation that to be recovered from a dynamical
data on a boundary of a compact domain. The arising problems are closely related to some
problems of the integral geometry on geodesics of the Riemannian metric determined by the
leading part of the equation. Uniqueness and stability results to inverse problems are given.
We discuss the method based on stability estimates for solutions to the Cauchy problem for the
hyperbolic equation with data on a lateral boundary of a cylindrical domain. These estimates
are given up to the characteristic surface. Combining the estimates with the ray expansion of
solutions in a neighborhood of the characteristic surface we get a possibility to obtain stability
estimates for inverse problems that use the Cauchy data on the lateral boundary for a nite
time. The method works well for the media close to homogeneous. We give some applications
29
of this method for a series of inverse problems related to second order hyperbolic equations and
equations of electrodynamics. Finally, we consider the problem of a continuation of the Cauchy
data given on a boundary of a time-spatial half-space inside the half-space. We suppose that the
physical medium contains a nite number of a compact disconnected non-homogeneities. After
the continuation of the given Cauchy data on boundaries of the compact domains, these data are
used for recovering desired parameters of the medium. Some uniqueness theorems are presented.
Self-normalized Limit Theory and Its Recent Development
Qi-Man Shao
Department of Mathematics, Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hang Kong; and
Department of Mathematics, University of Oregon,
Eugene, OR 97403 ([email protected])
The normalizing constants in classical limit theorems are usually sequences of real numbers.
Moment conditions or other related assumptions are necessary and sufficient for many classical
limit theorems. However, the situation becomes very different when the normalizing constants
are sequences of random variables. A self-normalized large deviation holds without any moment
conditions. A self-normalized law of the iterated logarithm remains valid for all distributions
in the domain of attraction of a normal or stable law. This reveals that the self-normalization
preserves much better properties than deterministic normalization does. In this talk we shall
review recent development on the self-normalized limit theorems. The focus will be on the
Cramér type large deviations for independent random variables and non-uniform exponential
inequalities for self-normalized processes and for Studentized U-statistics.
Mathematics in Teaching and Teaching of Mathematics
Man Keung Siu
Department of Mathematics, The University of Hong Kong,
Pokfulam Road, Hong Kong SAR, China ([email protected])
Joint work with C.I. Fung (Hong Kong Institute of Education).
Will a mathematics major that graduates with academic distinction be a successful school teacher
in mathematics? Maybe, but not always, why? We attempt to find out what kind of frame of
mind and focal point of attention, and related to that what kind of mathematical knowledge,
a teacher in school mathematics would do well to possess and to be nurtured in. We believe
school teachers in mathematics should have research experience in mathematics, which, though
similar in spirit as that of a researcher in mathematics, can be quite different in form and
content, because a school teacher has to explain mathematics in a language and at a level of
30
sophistication suitable to the mental development of school pupils. Mathematics learnt in the
university provides the background and the general upbringing in the discipline, but it needs
research experience of the kind mentioned above to enable a teacher to work on designing the
teaching sequence in class. We will illustrate with many examples. The theoretical basis of this
attempt is an assimilation of the art of teaching and problem solving of George Pólya, the process
of mathematising of Hans Freudenthal, and the theory of substantial learning environment of
Erich Wittmann.
On New Fixed-Point Iterations for Asymptotically Nonexpansive Mapping in
Banach Spaces
Suthep Suantai
Department of Mathematics, Chiang Mai University,
Chiang Mai, 50200, Thailand([email protected])
The main purpose of this paper is to give weak and strong convergence theorems of a new
three-step iterative scheme for asymptotically nonexpansive mappings in Banach spaces and we
also give several weak and strong convergence theorems of the three-step iterative scheme with
errors for asymptotically nonexpansive mappings in Banach spaces. Mann-type and Ishikawa
-type iterations are included by the new iterative scheme. The results obtained in this paper
extend and improve the recent ones announced by Xu and Noor, Ishikawa, and several recent
results in this area.
Singular potential functions on Hermitian symmetric manifolds and some
applications
Wing-Keung To
Department of Mathematics, National University of Singapore,
2 Science Drive 2, Singapore 117543 ([email protected])
In this talk, I will discuss some joint works with J.-M. Hwang on the construction of certain po
tential functions with isolated singular points on Hermitian symmetric manifolds. In particular,
I will discuss some applications of such constructions such as bounding Seshardri constants and
volume of analytic subvarieties of Hermitian symmetric manifolds. Recent applications of the
latter results in bounding level structures of families of abelian varieties will also be discussed.
31
A semidefinite programming based approach for anchor-free 3D graph realization
Kim-Chuan Toh
Department of Mathematics, National University of Singapore,
2 Science Drive 2, Singapore 117543 ([email protected])
Joint work with Pratik Biswas, Tzu-Chen Liang, and Yinyu Ye (Stanford University).
We propose a distributed algorithm for solving anchor-free Euclidean metric realization problems
arising from 3D graphs. In our distributed algorithm, the graph is first subdivided into smaller
subgraphs using intelligent clustering methods. Then a semidefinite programming relaxation
and gradient search method is used to localize each subgraph. Finally, a stitching algorithm is
used to find affine maps between adjacent clusters and the positions of all points in a global
coordinate system are then derived. In particular, we apply our method to the problem of
finding the 3D molecular configurations of proteins based on a limited number of given pairwise
distances between atoms. The protein molecules, all with known molecular configurations, are
taken from the Protein Data Bank. Our algorithm is able to reconstruct reliably and efficiently
the configurations of large protein molecules from limited number of given pairwise distances.
Mixed multiplicities of ideals versus mixed volumes of polytopes
Ngo Viet Trung
Institute of Mathematics,
Viên Tóan Hoc, 18 Hoang Quôc Viêt, 10307 Hanoi, Vietnam (nvtrung@@math.ac.vn)
Joint work with J. Verma.
The classical Bezout’s theorem can be generalized as follows:
1
±1
Bernstein’s Theorem. Let f1 , ..., fn be Laurent polynomials in C[x±
1 , ..., xn ] with finitely
many common zeros in the torus C n . Then the number of common zeros of f1 , ..., fn in C n is
bounded above by the mixed volume the Newton polytopes of f1 , ..., fn . Moreover, this bound
is attained for a generic choice of coeffcients inf1 , ..., fn .
Bernstein’s theorem is a beautiful example of the interaction between algebra and combinatorics.
However, the original proof has more or less a combinatorial flavor. The relationship between
toric varieties and multigraded rings suggests that mixed multiplicities of ideals may be used to
give an algebraic proof. We will encounter two problems which are of independent interest:
• Can one interpret the number of common zeros of Laurent polynomials in the torus as
mixed multiplicity of ideals?
• Does there exist any relationship between mixed multiplicities of ideals and mixed volume
of polytopes?
32
These problems can be solved satisfactorily, and we will obtain thereby an algebraic proof
for Bernstein’s theorem which uses mixed multiplicities of ideals in a similar way as Samuel’s
multiplicity for Bezout’s theorem.
Dimension-Free Harnack Inequality for Diffusion Semigroups and Applications
Feng-Yu Wang
School of Mathematical Sciences, Beijing Normal University,
Beijing, 100875, The People’s Republic of China ([email protected])
By using the coupling method and Girsanov transformations, the dimension-free Harnack in
equality is established for diffusion semigroups with curvature not necessarily bounded below.
Applications of this inequality to heat kernel estimates, contractivity properties of semigroups
and log-Sobolev inequalities, are presented. Furthermore, some infinite-dimensional version of
the Harnack inequality and applications are also introduced.
The moving contact line on patterned surfaces
Xiao-Ping Wang
Department of Mathematics, Hong Kong University of Science and Technology,
([email protected])
Numerical simulations of the moving contact line over the chemically patterned surfaces were
carried out using a newly proposed model. These surface inhomogeneities lead to stick-slip
motion of the contact line, the capillary wave of the interface. We also investigate the associated
power-law behavior of the moving contact line dissipation as a function of the velocity.
Add values to mathematics instruction: A Singapore initiative
Khoon Yoong Wong
National Institute of Education, Nanyang Technological University,
Singapore ([email protected]).
The prevailing public image of mathematics is an objective, abstract, and inhuman subject.
Mathematicians are perceived to be human beings born with special talents in logical reasoning
and symbolic manipulations. Mathematical results are thought to be valid across the whole uni
verse. Although mathematics has been successfully applied to the hard sciences and engineering
that deal with ”objective” reality, it is less effective in modeling social, political, economic, and
33
psychological phenomena that involve human interactions and human values. Applications of
mathematics to everyday life are often limited to basic number operations, percentages, volumes,
and time measurements.
The above widespread characterization of mathematics is now being scrutinized from two trends.
The first trend is the emergence of the ethnomathematics or cultural mathematics movement,
first proposed by D’Ambrosio in 1984. Ethnomathematics is the study of the mathematics de
veloped and used by different cultural groups across the world from ancient to modern times.
Bishop (1991) considered six types of mathematical activities, namely counting, measuring, lo
cating, designing, explaining, and playing. This trend has widened the scope about mathematics
and its applications.
The second trend extends ethnomathematics by embedding cultural values into mathematics in
struction. In Singapore, this is referred to as infusing ”National Education” into school subjects
including mathematics. ”National education” refers to educating Singapore students about the
Singapore nationhood and its values. The Singapore mathematics syllabus (2000) states that,
”National Education can be integrated into instruction by drawing examples from the prevailing
national and current issues during mathematics lessons. These examples can be expressed in
the problem context during problem solving or incorporated into practical work”. The author
has developed the following NE ME matrix to help teachers conceptualize this infusion. For
example, calculation of the Body Mass Index (BMI) is a simple ”content” task under ”personal
environment”. Collating BMI values from a class, producing pictorial representations of the
data, and relating BMI to other variables is a ”process” task. Discussing the impacts of and
ways to deal with obesity in Singapore and globesity adds a ”values” component. These math
ematical tasks make use of real-life data and events. Students must learn to be critical about
the mathematics found in the media and Internet.
Mathematics Education (ME)
Content Process
Culture
National
Education (NE)
Environment:
Personal
National
Global
Values:
Personal
National
Universal
34
Best quasi- interpolation and scattered data quasi- interpolation
ZongMin Wu
Shanghai Key Lab. for Contemporary Applied Mathematics,
Department of Mathematics, Fudan University ([email protected])
Quasi- interpolation is very useful in the study of the approximation theory and its applications,
since the method can yield solutions directly and does not require solving any linear system
of equations. However, quasi- interpolation is usually discussed only for gridded data in the
literature. In this paper we shall introduce a generalized Strang- Fix condition, which is related
to non- stationary quasi- interpolation. Based on the discussion of the generalized Strang- Fix
condition we shall generalize our quasi- interpolation scheme for multivariate scattered data,
too.
A Characterization Of Quantum Codes And Constructions
Chaoping Xing
Department of Mathematics, National University of Singapore,
2 Science Drive 2, Singapore 117543 ([email protected])
We give a characterization of quantum codes in the talk. This characterization converts the
algebraic structure of quantum codes into a combinatorial structure and makes it possible to
employ classical codes and boolean functions to construct good quantum codes. In addition,
we make use of classical algebraic geometry codes to get the best known asymptotic bound on
quantum codes through this characterization.
On the distribution of linkage statistics for partially informative markers
Benjamin Yakir
Department of Statistics, The Hebrew University of Jerusalem,
Mount Scopus, Jerusalem 91905, Israel ([email protected])
Joint work with David Siegmund (Stanford University).
Nonparametric linkage analysis in genetic studies is based on the examination of relations of
chromosomal identity by decent (IBD) between related individuals. Unfortunately, at the mole
cular level these relations of IBD can only be inferred from genotypic marker data, which may
be consistent with more than a single pattern of inheritance. Hidden Markov Models (HMM)
are used for the computation of scanning statistics, which are the tools for the investigation of
linkage.
35
At this talk we will examine the distribution of the inferred scanning process, and compare it
to the distribution of the unobserved process. Analytical theory for the approximation of the
distribution of the inferred process has similarities to the analytical theory for the unobserved
process, but some major differences exists. The similarities and differences will be discussed.
The Contact Discontinuity for Gas Motions
Tong Yang
Liu Bie Ju Centre of Mathematics, City University of Hong Kong ([email protected])
Joint work with Feimin Huang (Acadmia Sinica) and Zhouping Xin (Chinese University of Hong
Kong).
The contact discontinuity is one of the basic wave patterns in gas motions. The stability of
contact discontinuities with general perturbations is a long standing open problem. One of the
reasons is that contact discontinuities are linearly degenerate waves in the nonlinear settings,
like the Navier-Stokes equations and the Boltzmann equation. The nonlinear diffusion waves
generated by the perturbations in sound-wave families couple and interact with the contact
discontinuity and then cause analytic difficulties. Another reason is that in contrast to the basic
nonlinear waves, shock waves and rarefaction waves, for which the corresponding characteristic
speeds are strictly monotone, the characteristic speed is constant across a contact discontinuity,
and the derivative of contact wave decays slower than the one for rarefaction wave. In this talk,
we will present our recent result on obtaining the time asymptotic stability of a damped contact
wave pattern with an convergence rate for the Navier-Stokes equations and the Boltzmann
equation in a uniform way. One of the key observations is that even though the energy estimate
1
involving the lower order may grow at the rate (1 + t) 2 , it can be compensated by the decay
1
in the energy estimate for derivatives which is of the order of (1 + t)− 2 . Thus, these reciprocal
order of decay rates for the time evolution of the perturbation are essential to close the priori
estimate containing the uniform bounds of the L∞ norm on the lower order estimate and then
it gives the decay of the solution to the contact wave pattern.
Stationary Subdivision Schemes Reproducing Polynomials
Jungho Yoon
Department of Mathematics, Ewha Womans University,
Seoul, 120–750, S. Korea ([email protected]; [email protected])
Joint work with S. Choi (Duksung W. University), B. Lee (Dongseo University) and Y. Lee
(Ewha Womans University).
36
A new class of subdivision schemes is presented. Each scheme in this class is a quasi-interpolatory
scheme with a tension parameter, which reproduces polynomials up to a certain degree. We find
that these schemes extend and unify not only the well-known Deslauriers-Dubuc interpolatory
scheme but the quadratic and cubic B-spline schemes. This paper analyze their convergence,
smoothness and accuracy. It is proved that the proposed schemes provide at least the same
or better smoothness and accuracy than the aforementioned schemes, when all the schemes are
based on the same polynomial space. We also observe with some numerical examples that, by
choosing an appropriate tension parameter, our new scheme can remove undesirable artifacts
which usually appear to interpolatory schemes with irregularly distributed control points.
On Cancellation Problems
Jie-Tai Yu
Department of Mathematics, The University of Hong Kong, ([email protected])
Joint work with A. Belov (International University of Bremen) and L. Makar-Limanov (Bar-Ilan
University).
We are going to discuss the ‘embedding part’ of the cancellation problem, and will also report
some new positive results for Cancellation Conjecture of Zariski for rings and fields in low
dimensional cases.
Bi-CR: An iterative method based on an A-Biorthogonalization process for
nonsymmetric linear systems
Shao-Liang Zhang
Department of Applied Physics, The University of Tokyo,
Hongo, 7-3-1, Bunkyo-ku, Tokyo, 113-8656, Japan ([email protected])
Joint work with Tomohiro Sogabe.
The Conjugate Gradient (CG) method and the Conjugate Residual (CR) method are well- known
Krylov subspace methods for solving symmetric (positive denite) linear systems. For solving
nonsymmetric linear systems, Fletcher extended CG to nonsymmetric linear systems. However,
the extended algorithm, known as Bi-CG, ofter shows irregular convergence behavior in the
residual norm. The purpose of this paper is to extend CR to nonsymmetric linear systems based
on an A-biorthogonalozation process. Since CR takes a minimum norm residual approach, the
extended CR algorithm, named Bi-CR, can be expected to give smoother convergence behavior
than Bi-CG. Numerical experiments show that Bi-CR is often more ecient than Bi-CG. Key
words. CG, CR, Bi-CG, nonsymmetric linear systems, Krylov subspace methods, bi- Lanczos
algorithm, two-term recurrence relations.
37
Bergman kernel and symplectic reduction
Weiping Zhang
Nankai Institute of Mathematics, Nankai University,
Tianjin 300071, P.R. China ([email protected])
We describe joint results with Xiaonan Ma on the asymptotic behaviors of Bergman kernels on
symplectic manifolds with Hamiltonian group actions.
Deterministic Optimization Methods and Bioinformatics
Xiang-Sun Zhang
Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing, 100080, China ([email protected]).
Research supported in part by Chinese National Natural Science Foundation through grants
10471141.
Following the tracks of statisticians and applied mathematicians in the biological research, re
searchers in OR field have involved in computational biology/Bioinformatics. In this talk some
basic problems in Bioinformatics, such as DNA sequencing, protein folding and protein align
ment, haplotyping problem of single-nucleotide polymorphisms (SNP), are introduced. Then we
give an overview on existing application of OR theory and methods, especially the deterministic
optimization techniques, in these problems and discuss the future trend of OR application in this
prospective research direction. By the word ”deterministic method”, we cover the disciplines
in OR that do not use profound knowledge of the Probability Theory and Stochastic Process.
These include Dynamic Programming (DP), Linear Programming (LP), Nonlinear Program
ming (NP)/Integer Programming (IP), Graph Theory (GT), Artificial Neural Networks (ANN),
various heuristic approach and approximation methods. As examples, recent research results of
our group are reported in this talk. Key references are given to the interested audience.
On invariant holomorphic extension
Xiangyu Zhou
Chinese Academy of Sciences, China ([email protected])
In this talk, we’ll consider some open problems arising from our solution of the extended fututre
tube conjecture, including the problem about Steiness of the complexifications of Stein domains
38
with real group actions. We’ll present a method to approach the problem which is based on
the group action version of H. Cartan’s lemma about holomorphic extension, here L2 method
could play an important role. It’s well known that Cartan’s lemma with trivial group action is
equivalent to the famous Cartan-Serre’s Theorems A and B. We’ll emphasize the background of
the problem and the method.
Mean–Risk Portfolio Selection: Continuous-Time and Single Period
Xun Yu Zhou
Department of Systems Engineering and Engineering Management,
The Chinese University of Hong Kong, Shatin, NT, Hong Kong ([email protected])
Markowitz’s mean-variance portfolio selection model becomes a mean-semivariance model if
semivariance is used to replace variance as a risk measure. In this talk we first show that the
mean–semivariance efficient strategies in a single period are ALWAYS attained irrespective of
the market condition or the security return distribution. In sharpe contrast to this, we prove
that in the continuous time setting the mean–semivariance model NEVER achieves efficiency
save for a trivial case. A more general continuous-time mean-risk model is also investigated
thoroughly. This talk is based on joint papers with Hanqing Jin, Harry Markowitz, and Jia-An
Yan.
Some aspects of unitary representations of classical groups
Chen-Bo Zhu
Department of Mathematics, National University of Singapore,
2 Science Drive 2, Singapore 117543 ([email protected])
A fundamental problem in representation theory is to determine the unitary dual Ĝ of a given
(real) reductive Lie group G, namely the collection of equivalent classes of all irreducible uni
tary representations of G. For a compact Lie group, irreducible unitary representations were
parameterized long time ago through the work of Cartan and Weyl (Cartan’s Theorem of High
est Weight and Weyl’s Character Formula). Unfortunately if the reductive Lie group is not
compact, its unitary representations are either 1-dimensional or can only be found in infinitedimensional spaces. Thus a major task is to invent ways of constructing (new) representations
and determine when the representations produced are unitary. In this talk, we will first review
three important and well-known constructions: parabolic induction, cohomological induction
and dual pair correspondence. We will also discuss a comparison technique called transfer of
unitary representations between real forms.
As mentioned, the most (and the only) obvious unitary representations of any reductive Lie
group G are its unitary characters. Starting from this collection, other unitary representations
may then be constructed through application of general procedures, and so on. Indeed, the
39
spherical unitary principal series are obtained by normalized parabolic induction from unitary
characters of a minimal parabolic subgroup and are naturally associated with certain hyperbolic
coadjoint orbits; Unitary representations associated to elliptic coadjoint orbits are obtained by
cohomological induction from unitary characters of the centralizers of elliptic elements (the
Aq (λ)’s), among them there are the discrete series representations; It is therefore reasonable to
expect that unitary representations obtained by dual pair correspondence of unitary characters
should be special as well. The second purpose of this talk is to examine this distinguished class
of unitary representations (for classical groups) and discuss their connections to the various
other general constructions and to the philosophy of unipotent representations, as expounded
by Vogan.
Arakelov inequalities and the uniformization of certain rigid Shimura varieties
Kang Zuo
Mainz Univeristy, Germany ([email protected])
Joint work with Eckart Viehweg.
Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a
rational variation of Hodge structures of weight one on Y with Higgs bundle E(1,0) + E(0,1),
coming from a family of Abelian varieties. If Y is a curve the Arakelov inequality says that the
difference of the slope of E(1,0) and the one of E(0,1) is is smaller than or equal to the degree of
the canonical sheaf. We prove a similar inequality in the higher dimensional case. If the latter
is an equality, as well as the Bogomolov inequality for E(1,0) or for E(0,1), one hopes that Y
is a Shimura variety, and V a uniformizing variation of Hodge structures. This is verified, in
case the universal covering of Y does not contain factors of rank ¿1. Part of the results extend
to variations of Hodge structures over quasi-projective manifold Arakelov inequalities and the
uniformization of certain rigid Shimura varieties.
40
The twisted semigroups crossed products and its applications to the twisted
toeplitz algebras
Sriwulan Adji
School of Mathematical Science, University Science Malaysia,
Malaysia ([email protected])
Joint work with Rizky Rosjanuardi.
We study the twisted crossed product A×α,σ Γ+ of a C ∗ -algebra A by action α of the positive cone
Γ+ of totally ordered abelian group Γ as endomorphisms on A, and a cocycle σ in Γ. We then
apply our results to prove that the C ∗ -algebras generated by two isometric σ-representations of
Γ+ are canonically isomorphic. Hence we obtain the generalization theorem of Ji.
On two functionals connected to the Laplacian in a class of doubly connected
domains in space-forms
A. R. Aithal
Department of Mathematics, University of Mumbai,
Mumbai-98, India ([email protected])
Joint work with Anisa M.H.C. (University of Mumbai).
In S n (Hn ), let B1 be a ball of radius r1 , and let B0 be a smaller ball of radius r0 such that
B0 ⊂ B1 . For S n we consider r1 < π. Consider Ω = B1 \B0 . Let u = u(Ω) be the solution of
the Dirichlet problem −Δu = 1 in Ω vanishing on the boundary. Denote the Energy E(u) by
E(Ω). Consider the family F = {B1 \B0 } of domains in S n (Hn ).
Put Ω0 = B(p, r1 )\B(p, r0 ) for any fixed p ∈ S n (Hn ). We prove the following theorems;
Theorem 1: The Energy functional E(Ω) on F assumes maximum at Ω if and only if Ω = Ω0 ,
i.e., when the balls are concentric.
Theorem 2: The first eigenvalue λ1 (Ω) on F assumes maximum at Ω if and only if Ω = Ω0 .
The above theorems are proved after developping the ‘shape calculus’ for Riemannian manifolds
for the stationary problem, and the eigenvalue problem. This work was inspired by the following
paper: Kesavan S, “On the two functionals connected to the Laplacian in a class of doubly
connected domains”
Proceedings of the Royal Society of Edinburgh 133A (2003) 617-624.
42
Subordination and superordination of analytic functions associated with linear
operators
Rosihan M. Ali
School of Mathematical Sciences, Universiti Sains Malaysia,
11800 USM Penang, Malaysia, ([email protected])
�
Let A(p) denote the class of normalized analytic p-valent functions f (z) = z p + n=p+1 an z n
defined on the open unit disk Δ. Recently Dziok and Srivastava introduced the linear operator
Hp (α1 , . . . , αl ; β1 , . . . , βm ) : A(p) → A(p) defined by
Hp (α1 , . . . , αl ; β1 , . . . , βm )f (z) = z p +
∞
�
(α1 )n−p . . . (αl )n−p an z n
,
(β1 )n−p . . . (βm )n−p (n − p)!
n=p+1
where (λ)n = λ(λ + 1)(λ + 2) · · · (λ + n − 1). This Dziok-Srivastava operator
Hp (α1 , . . . , αl ; β1 , . . . , βm ) is the Hadamard product or convolution of the function f with a
function associated to the generalized hypergeometric function. The Dziok-Srivastava opera
tor includes various other linear operators, such as the Hohlov linear operator, the CarlsonShaffer linear operator, the Ruscheweyh derivative operator, and the generalized BernardiLibera-Livingston integral operator.
Another linear operator studied recently is the multiplier transform defined by
�
∞ �
�
k+λ r
p
Ip (r, λ)f (z) := z +
ak z k (λ ≥ 0).
p+λ
k=p+1
In this talk, we consider the class of normalized analytic functions associated with either
the Dziok-Srivastava operator Hp (α1 , . . . , αl ; β1 , . . . , βm ) or the multiplier transform operator
Ip (r, λ). In particular, we give some applications of differential subordination and superordina
tion to obtain sufficient conditions for these functions to be subordinated as well as superor
dinated to given univalent functions. Relevant connections of our results with those in several
earlier investigations are indicated.
CR-submanifolds of generalized Sasakian space form
Falleh R. M. Al-Solamy
Department of Mathematics, King AbdulAziz University,
P. O. Box 80015, Jeddah 21589, Saudi Arabia ([email protected])
In 1978, A. Bejancu introduced and studied CR-submanifold of Kaehler manifold [2,3] which
generalises invariant submanifold and anti-invariant submanifold. Since then many papers ap
peared on this topic with ambient manifold as Sasakian space form [4], cosymplectic space form
43
[6], Kenmotsu space form [5] etc. Recently Algre, Blair and Carriago [1] introduced generalized
Sasakian space form which generalizes above mentioned space forms.
The aim of this talk is to announce some of our recent results of CR-submanifolds of generalized
Sasakian space form. We have studied the sectional curvature, the Ricci tensor and scalar
curvature of a CR-submanifold.
[1] Alegre, P., Blair, D. E. and Carriazo, A., “Generalized Sasakian space form” Israel J. Math.
141, (2004).
[2] Bejancu, A., “CR-submanifold of a Kaehler manifold I” Proc. Amer. Math. Soc. 69,
135-142, (1978).
[3] Bejancu, A., “CR-submanifold of a Kaehler manifold II” Trans. Amer. Math. Soc. 250,
333-345, (1979).
[4] Kobayashi, M., “CR-submanifold of a Sasakian manifold” Tensor N. S. 35, 297-307, (1981).
[5] Shahid, M. H., “On CR-submanifolds of certain class of almost contact manifold” Rend.
Dell. Istituto 24, 147-159, (1992).
[6] Shoeb, M., Shahid, M. H. and Sharfuddian, A., “On submanifolds of cosymplectic manifold”
Soochoo J. Math. 27(2), 161-174, (2001).
On the Group Structure of the Jacobian of Some Hyperelliptic Curve over Several
Fp
Christine Abegail M. Antonio
Sophia University, Tokyo, Japan (tingarcia [email protected])
One of the most important problems in computational algebraic geometry is to find an efficient
algorithm to find the number of points of abelian varieties over finite fields. A special class of
these abelian varieties are called hyperelliptic curves. These type of curves exist for every genus,
g ≥ 1. A hyperelliptic curve of genus 1 is called an elliptic curve. Originally, these type of curves
are studied mainly for aesthetic reasons. However, recently, they have been studied extensively
for cryptographic purposes.
The main concept of this paper is to find the group structure of the jacobian of the hyperelliptic
curves. This can be done by doing two things. First, we count the number of elements of the
jacobian using Mumfords representation and then we solve for the order of an element by using
an explicit algorithm on performing group operations on this abelian group. To test the efficiency
of our method, we apply it to the hyperelliptic curve of genus 2, v 2 + uv = v 5 + 5u4 + 6u2 + u + 3
over several finite prime fields, Fp .
44
Existence and Uniqueness of the Solution to a Volevic System of Linear Equations
with General Singularity
Carlene P. Arceo
Department of Mathematics, University of the Philippines, Diliman
Joint work with Jose Ernie C. Lope and Jose Maria L. Escaner IV.
We establish the existence and uniqueness of the solution to a Volevic system of linear PDEs. Our
results are generalizations of those obtained by Elscher, Lope, Tahara and Baouendi-Goulavic.
Creative Mathematical Modeling and Visualization via Spreadsheets
Deane E. Arganbright
Korea Advanced Institute of Science and Technology,
South Korea ([email protected])
The computer spreadsheet is firmly established as the principal mathematical tool of the work
place. In addition, it is increasingly recognized as an excellent tool to use creatively in all areas
of the mathematical sciences and their applications, including geometry, calculus, operations
research, statistics, the natural and social sciences, engineering, and art. Many of the most
exciting developments for mathematical applications of spreadsheets involve dynamic systems
and modeling. This paper highlights the usage of spreadsheets for mathematical modeling and
mathematical visualization.
First, we demonstrate how to use the spreadsheet creation process itself to develop mathematical
concepts, reversing the standard approach of developing the mathematics traditionally before
creating a computer implementation. Second, we show how to employ standard spreadsheet
tools to produce exceptional interactive graphic displays that include animation, and how to
use them effectively in the visualization of mathematical concepts and algorithms as well as
for classroom and professional demonstrations. This highly visual presentation features live,
interactive computer projections employing the most widely used spreadsheet, Microsoft Excel.
Illustrations come from diverse areas of mathematics, including the areas mentioned above.
Actions of pointed Hopf algebras on quantum torus
Viatcheslav A. Artamonov
Department of Algebra, Faculty of Mechanics and Mathematics, Moscow State University
([email protected])
Let Oq the associative algebra with a unit element over a field k generated by elements
X1±1 , · · · , Xr±1 , Xr+1 , · · · , Xn subject to defining relations Xi Xi−1 = Xi−1 Xi = 1, 1 � i � r
45
and Xi Xj = qij Xj Xi , 1 � i, j � n. Here qij are element of k such that qij = qij qji = 1 for all
i, j. The algebra Oq is an algebra of quantum polynomials. It is a generic algebra of quantum
polynomials if all multiparameters qij with 1 � i < j � n; are independent in the multiplicative
abelian group k ∗ of the field k. The algebra Oq can be considered as a coordinate algebra of
product of a quantum torus and a quantum plane [BrG, M].
In non-commutative algebraic geometry an action of a “finite quantum group” on a quantum
space means an action of some finite dimensional Hopf algebra H on Oq . In my talk I shall
consider the case when H is a pointed Hopf algebra. It is shown that there exists a class of
standard cocommutative pointed finite dimensional Hopf algebra acting on Oq . An action of H
is a composition of Hopf algebra homomorphism from H onto some standard algebra and an
action of the standard one on Oq . It follows that an action of H on Oq is reduced to action
of some automorphism group and some skew derivations of Oq . Moreover the subalgebra of
invariants of this action is left and right Noetherian and Oq is finitely generated left and right
module over the subalgebra of invariants.
In the case when the number n of variables is at least three was considered in [A]. It is interesting
to mention that in the case n = r = 2 a classification of automorphism group of Oq is similar to
a classification of two-dimensional crystallographic groups.
[A] Artamonov V. A., Pointed Hopf algebras acting on quantum polynomials, J. Algebra 259(2003),
N 2, 323-352.
[BrG] Brown K. A., Goodearl K. R., Lecture on algebraic quantum groups. Birkhauser, Basel,
Boston, 2002.
[M] Montgomery,S.: Hopf Algebras and Their Actions on Rings, Regional Conf. Ser. Math.
Amer. Math. Soc., Providence RI, 1993.
√
Actions of G(3, 3) =< u, v : u3 = v 3 = 1 > on Q∗ ( −n)
Muhammad Ashiq
Department of Basic Sciences & Humanities, College Of E & ME,
National University of Sciences & Technology, Rawalpindi, Pakistan ([email protected])
√
The imaginary
quadratic fields are defined by the set {a + b −n : a, b ∈ Q} and denoted
√
positive integer. In this paper we have proved that if
by Q( √−n), where n is a square-free
√
√
2
a+ −n
a+ −n
∗
∈ Q ( −n) = { c
: a, a c+n , c ∈ Z, c �= 0} then n √
does not change its value
α =
c
in the orbit αG(3.3). Also we show that the number of orbits of Q∗ ( −n) under the action of
G(3.3) are 2[d(n) + 2d(n + 1) − 4] and 2[d(n) + 2d(n + 1) − 6] according to n is even or√odd,
except for n = 3 for which there are exactly eight orbits. Also, the action of G(3.3) on Q∗ ( −n)
is always intransitive.
46
S-extremal additive codes over GF(4)
Evangeline P. Bautista
Ateneo de Manila University,
Philippines ([email protected])
Joint work with Philippe Gaborit (University of Limoges)
This paper presents a bound for the minimum distance of a code and its shadow and defines
s-extremal additive codes over GF(4). It also presents a bound for the length of such a code
and classifies these codes for d = 1,2,3.
Predictive-model of numerical ability test (NAT) and achievement test (AT) in
college algebra as basis for construction of module for bridging program
Marleonie M. Bauyot
San Pedro College, Visiting Faculty, Ateneo de Davao University,
Philippines (marleonie [email protected])
Result of the study entitled: “Predictive-Model of Numerical Ability Test (NAT) and Achieve
ment Test (AT) in College Algebra as Basis for the Construction of the Module for Bridging
Program” showed that using scale level of interpretation as adapted by Opeña and Magno (2003),
the proficiency level of students in both NAT and AT in College Algebra is disadvantaged. Em
ploying Pearson r of two-tailed test at .01 level of significance, a coefficient of correlation of 0.42
was obtained which means that a moderate correlation exists between the variables NAT and AT.
Using least squares method, a mathematical model defined by an equation ŷ =38.788+0.234x
was obtained and a score of 17 out of 40 in the NAT is considered a cut-off score that will
deny admission of students in col! lege algebra course and place them under bridging program.
Based on the findings, the researcher developed a bridging instrument which can be used as the
module for a bridging program. Such module was checked by five (5) experts in mathematics
and obtained 99.45 % evaluation in terms of validity and readability. Coverage of the module
includes topics involved in the NAT and basic algebra. The module, as recommended in the
study, will prepare students performing poorly in mathematics to enter college algebra course.
On the solutions of “Ghost” manifolds problem
S. Borok
Department of Mathematics, Ben-Gurion University of the Negev,
Israel, P.O.Box 653, Be’er-Sheva 84105 ([email protected])
Joint work with I. Goldfarb and V. Goldshtein.
47
It is known that processes which take place in complex chemical kinetics and combustion systems
have very different time scales. It is often desirable to decouple such hierarchical systems into
slow and fast processes for reduction of their complexity. One of such reduction methods is
based on Intrinsic Low-Dimensional Manifolds (ILDM). This method successfully locates slow
manifolds of considered system but also has a problem: it produces additional objects (“ghost”
manifolds) that do not have any connection to the dynamics of the system. The problem
of “ghost” manifolds identication/discrimination is addressed in this paper. We analyse twodimensional singularly perturbed systems of ordinary differential equations and present two
numerical approaches to determine “ghost” manifolds and distinguish them from the true ones.
The approaches are based on special properties of invariant manifolds and singularly perturbed
systems.
Semi-parametric inference for non-parametric parameter measures
B. M. Brown
Department of Statistics and Applied Probability,
National University of Singapore, Singapore ([email protected])
Non-parametric parameters in Statistics are usually simple and interpretable, for example a
difference in location parameters, a concordance coefficient, or a probability that treatment A is
better than treatment B. The corresponding sample versions usually have the striking property
that at a neutral or null parameter value, the distribution is known exactly from combinatorial
properties, irrespective of the distribution which may generate the data. This distribution-free
property, which is central to the appeal of non-parametric methods, usually does not hold for
non-null parameter values, and thus obstructs the carrying-out of statistical inference for the
parameter.
A general method is described which, by semi-parametrically assuming the data comes from a
distribution family which is related to a standard family such as the normal in the minimal pos
sible way, allows inference procedures such as testing and confidence intervals to be formulated
for non-null values. The case of a concordance measure, Kendall’s tau, is used as an illustration.
Resolution on n-order functionally-differential equations with operator coefficients
and delayed variables in Hilbert space
Chan Roath
Royal Academy of Cambodia ([email protected])
We consider the following n-order functionally-differential equations with operator coefficients
and delayed variables in Hilbert space:
Lnp U (t) = f (t),
Lnpo U (t) = f (t)
48
where
Lnp
≡
Dtn
−
n−1
m
��
Akj Shkj Dtk ,
Dtk =
k=0 j=0
Lnpo ≡ Dtn −
n
−1 �
m
�
dk
ik dtk
[Akj + Akj (t)] Shkj +hkj (t) Dtk .
k=0 j=0
Here Akj and Akj (t) are abstract constant and variation operator coefficients, respectively;
hkj and hkj (t) are constant and variation delayed variables, respectively; Shkj +hkj (t) U (t) �
U (t − hkj − hkj (t)) is a translation operator. For t ∈ R and λ ∈ C, let
Rp (λ) ≡ (λn E −
n−1
m
��
Akj λk e−iλhkj )−1
k=0 j=0
⎛
Rpo (λ, t) ≡
⎝λn E −
n−1
m
��
k=0 j=0
⎞−1
[Akj + Akj (t)] λk e−iλ(hkj +hkj (t)) ⎠
.
0,α
Let X n,α
t0 and Y t0 be the Hilbert spaces respectively containing functions with norms:
R+
R+
2
�U (t)� =
�
∞
t0
2αt
e
�n−1
�
�U
(k)
(t)�2x
+ �U
(n)
(t)�2y
�
dt,
�U (t)�2 =
�
∞
e2αt �U (t)�2y dt,
t0
k=0
where t0 ≥ −∞, α ∈ R. We determine conditions that must be satisfied by Rp (λ), Rpo (λ, t),
0,α
Akj , Akj (t), hkj , hkj (t) for the operators Lnp , Lnpo : X n,α
t0 →
Y
t0 to be continuous and invertible.
R+
R+
Asymptotic distribution of Cramér-von Mises statistics for ARCH processes
Subhash Ajay Chandra
University of the South Pacific, Fiji Islands (chandra [email protected])
In my talk, the limiting Gaussian distribution of a class of Cramér-von Mises statistics {TN }
for two-sample problem pertaining to empirical processes of the squared residuals from two
classes of ARCH processes is elucidated. A distinctive feature is that, unlike the residuals of
ARMA processes, the asymptotics of {TN } depend on those of ARCH volatility estimators.
Based on the asymptotics of {TN }, the reliability of the confidence interval, relative asymptotic
efficiency, effect of the ARCH specification and robustness measures for some ARCH residual
distributions. In contrast with the independent, identically distributed or ARMA settings, these
studies illuminate some interesting features of ARCH residuals.
49
On the occupation time large deviations of symmetric simple exclusion process
Chih-Chung Chang
Department of Mathematics, National Taiwan University ([email protected])
We first review an occupation time large deviations result [1], which is a joint work with Clau
dio Landim and Tzong-Yow Lee. This result concerns the large deviations principle for the
occupation time of one site in the two dimensional symmetric simple exclusion process. It
is shown that the decay probability rate is of order t/logt and the rate function is given by
Υα (β) = (π/2){sin−1 (2β − 1) − sin−1 (2α − 1)}2 . The study of the large deviations of occupation
time difference of several sites will also be reported.
[1] Chang, C.C., Landim, C. and Lee, T.Y. (2004). Occupation time large deviations of two
dimensional simple exclusion process. Ann. Probab. 32, No. 1B, 661-691.
Curvature, fundamental groups and Hausdorff distance
Wen-Haw Chen
Department of Mathematics, Tung-Hai University,
Taichung 407, Taiwan ([email protected])
A fundamental question in Riemannian geometry is to investigate the relation between curvature
and topology. To study it, the concept of Hausdorff distance (or Gromov-Hausdorff distance) is
an very important tool. In this talk, I will report two new results about this topic developed by
the author. First we consider a simply connected Riemannian manifold with a lower sectional
curvature bound, and the discrete subgroups in the group of isometries of this manifold. By
developing a notion of Hausdorff distance between these groups and then proving a rigidity
result, we conclude that close geometric structures will imply close group structures. Secondly,
we consider the fundamental groups and diameters of compact Riemannian manifolds with
positive Ricci curvature. Sharp estimates on the first betti number of these manifolds and the
ratio of diameters between the universal covering of such a manifold and the manifold itself
will be given. Moreover, we also show a weak Margulis’s lemma for positively Ricci curved
manifolds. Our main method is the equivariant Hausdorff distance theory developed by Fukaya
and Yamaguchi. Some interesting examples about these two results will be discussed.
Characterization of Sporadic Simple Groups
Guiyun Chen
Department of Mathematics, Southwest China Normal University,
400715, Chongqing, P. R. China
50
Joint work with Zhangjia Han.
It is well-known fact that properties of subgroups of a finite group influence its structure. Here
we concern how the orders of maximal Abelian subgroups influence the structure of group and
come to that sporadic simple groups are uniquely determined by the set of their maximal abelian
subgroups orders.
It was first time researched how the orders of maximal Abelian subgroups influence the structure
of the group in [1]. And it is proved the following simple groups are uniquely determined by the
set of their maximal abelian subgroups orders: P SL(2, 2n ), Sz(22m+1 ), An (n ≤ 10), K3 -groups,
Mathieu groups and Janko groups, .
The topic is interesting for maximal abelian subgroups are between minimal subgroups and
maximal subgroups. Here we shall prove all sporadic simple groups are are uniquely determined
by the set of their maximal abelian subgroups orders, and give some subsets of the set of their
maximal abelian subgroups orders, which can also determine the groups.
[1] Wang Linhong, A characterization of some classes of finite simple groups.
Singular value assignment with low rank matrices
Chu Delin
Department of Mathematics, National University of Singapore, Singapore
([email protected])
Analogous to the pole assignment problem where eigenvalues of a square matrix are relocated,in
this talk we consider the problem of reassigning singular values of a rectangular matrix by
additive low rank matrices. Precise and easy-to-check necessary and sufficient conditions under
which the problem is solvable are completely characterized. The constructive proof makes it
possible to compute such a solution numerically.
Perturbations of differential expressions with logarithmically decaying coefficients
preserving the nullities
Juancho A. Collera
Department of Mathematics and Computer Science,
University of the Philippines, Baguio City, Philippines
Joint work with Marian P. Roque (University of the Philippines)
The essential spectrum and nullities of differential expressions of the form:
µ0 =
r
�
bk (log t)βk tρk Dtρk
k=0
51
have been classified. A class of relatively compact perturbations of the above expression which
do not alter these properties has also been defined. Recently, admissible perturbations of this
expression which preserve the essential spectrum have been classified. In this paper, we define
a class of admissible perturbations for the above form of differential expressions which preserves
the nullities of these expressions.
Homogenous factorisations of the Johnson graphs
Maria Cristeta N. Cuaresma
University of the Philippines, Los Baños, Philippines ([email protected])
Let Γ be an undirected graph with vertex set V Γ and edge set EΓ. A homogeneous factorisation
(M, G, Γ, E) of index k for Γ is a partition of into E of EΓ into k parts such that the groups
M, G ≤ Aut(Γ) are vertex-transitive with M fixing the parts of the partition setwise and G acting
transitively on the parts of the partition. In this paper, obtain all homogeneous factorisations
for the Johnson graphs J(n, r). The Johnson graph J(n, r) has for its vertex-set the r-subsets
of a set of size n where two vertices are joined by an edge if they intersect at r − 1 points.
We show that no homogeneous factorisation exists for J(n, r) when r ≥ 4. For r = 3, there are
nine non-isomorphic homogeneous factorisations of index three for J(8, 3) when M = AGL(1, 8)
and G = AΓL(1, 8). The only other homogeneous factorisation for r = 3 exists for J(q + 1, 3)
with P GL(2, q) ≤ M � G ≤ P ΓL(2, q) and q = 2n where n is a power of an odd prime p and
this factorisation has index p.
The homogeneous factorisations for J(n, 2) are also completely determined except for two cases
both occurring when G is affine. The first unresolved case occurs when G is one-dimensional,
though a homogeneous factorisation is found to exist for J(q, 2) when AGL(1, q) ≤ M � G ≤
AΓL(1, q) with q = 2n where n is a power of an odd prime p and this factorisation has index p.
However we have not been able to show that this is the only homogeneous factorization occurring
in this class. The other unresolved case occurs when M0 (the stabilizer of the zero vector in
M ) contains the commutator subgroup of G2 p , for even p. On the other hand, the only other
homogeneous factorization for J(n, 2) exists when n − 1 = q ≡ 1(mod 4) with P SL(2, q) ≤ M ≤
P ΓL(2, q), where G is a 3-transitive subgroup of P ΓL(2, q)and this factorisation is of index two.
A free boundary problem arising from the pricing of strike reset options
Min Dai
Department of Mathematics, National University of Singapore,
2 Science Drive 2, Singapore 117543 ([email protected])
Joint work with Yue-Kuen Kwok, Lixin Wu (Hong Kong University of Science and Technology),
Fahuai Yi, Yang Zhou (South China Normal University)
52
A strike reset option allows its holder to reset the strike price to the prevailing underlying price
at a moment voluntarily chosen by the holder. In this talk, we formulate the pricing problem
of the option as a free boundary problem where the free boundary corresponds to the optimal
strike reset policy. Some interesting properties of the free boundary are exploited and strictly
proved with the help of PDE arguments. Existence and uniqueness of the free boundary problem
is addressed. We also provide an efficient numerical approach to computing the free boundary.
The non-Archimedean metric diophantine approximations
Eveyth Deligero
Keio University, Japan ([email protected]; [email protected])
The invariance principle for non-Archimedean Diophantine approximations is a joint works of the
author, Michael Fuchs (National Chiao Tung University) and Hitoshi Nakada (Keio University)
In the field of formal Laurent series L
� with Haar measure m and non-Archimedean norm | · |
satisfying |f | < 1, we are interested the following Diophantine inequality:
�
�
�
�
P
�
f − �
< 1
,
�
Q
� q n+ln
deg(Q) = n,
(P, Q) = 1,
(1)
where ln is a given sequence of positive integers.
We give the recent development on the conditions on ln so that the following results hold: the
0- 1 law, the strong law of large numbers, the central limit theorem and the invariance principle.
law of
The invariance principle, which implies functional central limit theorem and functional
�
−1n = ∞
q
iterated logarithm, assumes that ln is a sequence of positive integers satisfying ∞
n=0
and either
�
(C1) lim ln = l < ∞ or (C2) lim ln = ∞, lim
q −lj exists.
n→∞
n→∞
i→∞
i<j≤i+li
Furthermore, we discuss one of the tools in obtaining the invariance principle which is the
construction of a 2-dependent process from the sets of f satisfying inequality (1). To have a
better understanding, we start with the construction of a 1-dependent process from the sets of f
satisfing the above inequality, which is also the main tool in obtaining the authors result (with
H.Nakada), on the central limit theorem for non-Archimedean Diophantine approximations.
Asymptotic analysis for 1D lattice Boltzmann method
Irwan Ary Dharmawan
Department of Physics, Universitas Padjadjaran,
Jl. Raya Bandung-Sumedang km.21 Jatinangor Sumedang 45363, Indonesia
(dharmawan@fisika-unpad.org)
53
Joint work with Michael Junk (Universitaet Konstanz).
In this seminar, we will introduce a method to analyze the lattice Boltzmann method algorithms.
Mathematically, the kinetic equation is asymptotically connected to the Navier-Stokes equation
by a singular limit. Here we proposed a method which is based on asymptotic analysis in
connection with standard truncation error considerations. Compared to the Chapman Enskog
analysis which is usually taken as basis for the analysis of lattice Boltzmann methods, our method
is based on the so called diffusive scalling which has first been considered by Sone. Morever, in
order to present idea of basic ideas of the analysis, we will consider simple 1D lattice Boltzmann
with D1Q3 stencil. Furthermore, we will apply this analysis to regular and irregular grid cases.
Intertwining operators, H-type groups and transference
A. H. Dooley
School of Mathematics, University of New South Wales
Let G = KAN be the Iwasawa decomposition of a rank one semi-simple Lie group. It is wellknown that the group N is a group of Heisenberg type. This fact allows us to describe the
structure of the intertwining operators of G, and in particular, to show how the identity can
be approached by uniformly bounded representations. Combining these remarks with a recent
theorem of Astengo, Cowling and di Blasio gives an approach to the Baum-Connes conjecture
for Sp(n, 1).
Chaos Behaviour in a Brusselator Reaction Mechanism
Jose Maria L. Escaner IV
Department of Mathematics, University of the Philippines, Diliman
Joint work with Polly W. Sy and Carlene P. Arceo.
The Brusselator Reaction Mechanism obtained from the Euler method is investigated and a
chaotic phenomenon in the sense of Marotto’s definition of chaos is established.
A new quantum code of length 20
Frederic Ezerman
Mathematics Department, Ateneo de Manila University ([email protected])
Finding good Stabilizer Quantum Codes is equivalent to finding Self-Dual Additive Codes over
GF(4) under the Trace Inner Product. There exist complete classifications for such codes up
to length 14, and a partial classification for length 18. Researchers have been trying to come
54
up with good codes of higher lengths. Yet, the size of the search space grows prohibitive as
the length of the code increases. Going beyond length 16 is a challenge, and not many good
codes have been constructed for such lengths. In this paper, an algorithm designed to find
good Stabilizer Quantum Codes via Additive Codes over GF(4) is presented. Making use of the
recently discovered circulant-based codes constructed by Gulliver and Lark-Kim and modifying
the general rule for lengthening a code proposed by Gaborit et al., an algorithm is constructed
for the said purpose. Based on the algorithm, a program to get good codes of higher lengths
and higher minimum distance is written and run. This paper presents a new extremal quantum
code of length 20, showing that the algorithm, applied appropriately and given the necessary
time, is an effective tool in extending the search for good quantum codes to lengths ≥20.
WP or IP issues for nonlinear wave equation
Daoyuan Fang
Department of Mathematics, Zhejiang University, Hangzhou 310027, China ([email protected])
In this talk we will present some new results (joint with Chengbo Wang) about the well posedness
or ill posedness of the Cauchy problem for nonlinear wave equation. The goal of that is to try
to understand the impact of the interplay among the regularities of the data, decay estimates
and nonlinearity on the well posedness. Part of the results is listed as follows:
Let |β | = l, k + l > 1, k, l ∈ N , set α :=
l−2
k+l−1 ,
sc (k, l) :=
n
2
+ α , we consider
�u = ±uk (∂u)β
(1)
Theorem 1 (LWP Result) The equation (1) is s-LWP in H s for
� n+5
4 ∨ sc (0, l), l ≥ 2 and n ≥ 2
s>
n−1
2 ∨ sc (k, 1) l = 1 and n ≥ 3
For ILP results, we concern on the “focusing” type equation
�u = |u|k |∂t u|l−1 ∂t u
with k + l > 1, k, l ≥ 0 and k, l ∈ R and set s̃ :=
interpreted as �u = |u|k−1 u .
n+1
4
+
(2)
l−1
k+l−1
Theorem 2 (Supercritical ILP)
Let n ≥ 1, l �= 2 or l = 2 with k = 0, then the equation (2) is
• s-ILP in Ḣ s for s ∈ (1 − n/2, sc )
• s-ILP in H s for s ∈ (1 − n/2, sc ) if sc > 1
55
. For l = 0, this equation is
• w-ILP in Ḣ s for s ∈ (1 − n/2, sc ] and l �= 2
• w-ILP in H s for s ∈ (1 − n/2, sc ] and l �= 2 with sc ≥ 1.
2ns̃
Set E = {s ∈ (sc , s̃) | s ∈ (− n2 , n+1
∧ n2 ) if l < 2}, F = {s ∈ (− n2 , 0] ∩ (sc , s̃) or 0 < s ∈ E}
Theorem 3 (Subcritical ILP)
Let n ≥ 3, l �= 2 and s˜ > sc , then we have s-ILP in H˙ s for (2) with s ∈ E. Furthermore, we
have s-ILP in H s for s ∈ F if there exist m, m̃ ∈ N ∩ [0, n2 ) such that
s˜ > (
n−1
n−1
n+1
m) ∨ (
m̃ + 1) >
+α .
2n
2n
4
On subseries of divergent series
D. K. Ganguly
Department of Pure Mathematics, University of Calcutta, 35,
Ballygunge Circular Road, Kolkata-700019, India ([email protected])
The paper presents a study on subseries of divergent series. Namely, given a divergent series of
non-negative real numbers we consider the set of sequences of zeros and ones placing in front of
each of these numbers to create a subseries. Choosing standard metrics such as the Frechet or
Baire metrics we determine the ‘size’ (in the sense of Baire category and porosity) of the set of
sequences of zeros and ones for which the resulting subseries converges.
How to incorporate non-invariant boundary conditions in invariant solutions
Joanna Goard
University of Wollongong, Australia ([email protected])
It is generally believed that in order to solve initial and boundary value problems using Lie sym
metry methods, the boundary and initial conditions need to be left invariant by the infinitesimal
symmetry generator which admits the invariant solution. This talk will demonstrate how only
less restrictive conditions need be imposed on the initial and boundary values in order that they
be recoverable with a particular symmetry generator.
56
A chorin-type formula for solutions to a closure model for the von
Karman-Howarth equation
V. N. Grebenev
Institute of Computational Technologies SD RAS,
Novosibirsk, 630090, RUSSIA ([email protected])
Joint work with M. Oberlack (Technische Universitat Darmstadt).
The article is devoted to studying the Millionshtchikov closure model (a particular case of a
model introduced by Oberlack) for isotropic turbulence dynamics which appears in the context
of the theory of the von Karman-Howarth equation. We write the model in an abstract form that
enables us to apply the theory of contractive semigroups and then to present a solution to present
a solution to the initial-boundary value problem by Chorin-type formula. This formula was
discovered by Chorin in an attempt to find solutions of the Navier-Stokes equations. The formula
obtained is of a special interest in view of its application to find the so-called infinitesimally close
asymptotic solutions to the original model. The solution obtained may be used as the initial
approximation to a solution of Oberlack’s model by using the so-called method of prolongation
with respect to a parameter of the model.
The research was supported by the Russian Foundation of Basic Research (04-00-00209) and by
DFG Foundation.
Applying the Structures of Standard Gray Codes
in Constructing Certain Types of Snakes
L. Haryanto
Faculty of Electrical Engineering, Delft University of Technology,
Delft, 2628 CD, Netherlands ([email protected])
A minimum-weight ordered basis B = (b1 , b2 , . . . , bk ) that generates a linear [n, k, 4]-code
C = (c0 = 0, c1 , . . . , c2k −1 )
is derived from a basis of�
a Reed-Muller code R(m − 2, m), which is a linear [nm , km , 4]-code
m
with nm = 2m and km = m−2
i=0 (i ). For every i ∈ {1, 2, . . . , k}, the basis vector bi corresponds
to an ordered set Bi = (i1 , i2 , i3 , i4 ) called block and the order of the four integers i1 , i2 , i3 , i4 ∈
{0, 1, 2, . . . , n − 1} in Bi can be chosen such that every integer i ∈ {0, 1, . . . , n − 1} occurring in
any block is always at the same position of the four possible positions (fixed position property ).
For this basis B, let B = (B1 , B2 , . . . , Bk ) be the corresponding list of blocks. By applying the
recursive formula T1� = (1), Ti� = (Ti�−1 , i, Ti�−1 ), 1 < i ≤ k, of the standard Gray code G(k) to
the indices of blocks in B and by considering that every block is a subsequence of four integers,
we construct a transition sequence Tk [B] = (Tk� [B], Bk ) that generates a code
S = (w0 , w1 , w2 , . . . , w4·2k −1 ),
57
in where all the 2k codewords cx = w4x of C are evenly distributed.
If the basis B is ordered and chosen properly, then S is a snake-in-the-box code (briefly, a snake ).
In order to test if the generated code S is indeed a snake, an algorithm, which is based on some
conditions of ordering blocks in B (cf. [Lukito, A., Bounds for the Length of Certain Types
of Distance Preserving Codes, Ph.D. Thesis, Delft University of Technology, 2000]) and on the
critical weight-2 words generated by a subsequence of the form (i3 , i4 , B1 , Ba , . . . , Bb , B1 , j1 , j2 ),
is developed.
There are covers of the hypercube Qn , 8 < n ≤ 16, by eight disjoint translations of a snake
S and the corresponding eight translation vectors always constitute a group. In this range of
values of lengths n, with respect to the number of disjoint snakes and to the algebraic structures
of these translation vectors, this result is stronger than the predicted cover of hypercubes by
sixteen disjoint snakes presented by Wojciechowski in [Combinatorica, 14(4), 1994, 491-496].
Resolution of the system of fuzzy integer inequalities
Hu Cheng Feng
I-Shou University, Taiwan ([email protected])
This work considers the resolution of the system of fuzzy integer inequalities. The concept of
level sets is adopted to convert this problem into a crisp(traditional) nonlinear integer program.
It is shown that a system of fuzzy integer inequalities with concave membership functions can be
reduced to a regular convex integer programming problem. The p-th power Lagrangian method
is introduced to deal with the resulting convex integer programming problem as a sequence of
linearly constrained convex integer programming problems. Moreover, due to the polyhedral
nature of the constraint set of the linearly constrained convex integer program, a branch-and
bound procedure is then applied for solving the sequence of linearly constrained convex integer
programming problems. Some computational results are included to illustrate the theory and
solution procedure.
Dominated convergence theorems involving small riemann sums
Ch. Rini Indrati
Department of Mathematics, Gadjah Mada University, Yogyakarta, Indonesia
([email protected] or [email protected])
Joint work with Lee Peng Yee (National Institute of Education, Singapore)
The Henstock integral is well-known. Lee (1989) in Lanzhou Lectures on Henstock Integration
gave a relationship between the Henstock integral and the LSRS (Locally Small Riemann Sum)
58
property. Darmawijaya and others gave the FSRS (functionally-small-Riemann-sum) property
for the Henstock integral on the real line. Convergence theorems for the integral have been
proved using conditions involving small Riemann sums. Gong used the FSRS property for the
Henstock integral on the real line to give some convergence theorems. He used the dominated
convergence theorem to deduce his theorems. Indrati extends Gong’s convergence theorem to the
n-dimensional space and establish its connection with the equi-Henstock integrability theorem.
In this paper, we define two interval functions Uδ and Vδ using Riemann sums of Henstock inte
grable functions, as major and minor functions. Then we formulate two dominated convergence
theorems for the Henstock integral in the n-dimensional space.
Long time asymptotics of the heat semigroup on nilpotent covering manifolds
S. Ishiwata
Institute of Mathematics, Tsukuba University,
Tsukuba-shi Ibaraki, 305-8571, JAPAN ([email protected])
In 1993, E. B. Davies proved that a long time asymptotics of the heat kernel of a secondorder differential operator on R with periodic coefficients. After that, C. Batty, O. Bratteli, P.
Jørgensen and D. Robinson obtained the asymptotics on several nilpotent Lie groups. On the
other hand, a local central limit theorem on abelian covering of compact Riemannian manifolds
is obtained by M. Kotani and T. Sunada in 2000.
� is approximated by the semigroup
In this talk, we explain that the heat semigroup on M
generated by Albanese sub-Laplacian on the nilpotent Lie group by using an approximation of
� by a sequence of uniformly distributed nets.
M
An integral equation for a particular solution of a non-uniquely solvable interior
Riemann problem on a region with corners
Munira Ismail
Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia,
81310 UTM Skudai, Johor Darul Tazim, Malaysia ([email protected])
Joint work with Ali Hassan Mohamed Murid and Bahrom Sanugi.
In a previous work we have obtained a new integral equation related to the Riemann problem
for a simply connected region with corners and is applicable only for the uniquely solvable
interior Riemann problem. We have obtained its numerical scheme using Picard iteration method
that eliminates all singularities successfully. In this paper we will obtain a particular solution
for the non-uniquely solvable interior Riemann problem on a region with corners requiring it
59
satisfy additional constraints at the origin of the coordinate system. The integral equation
is modified to accommodate the additional constraints giving a new integral equation that is
uniquely solvable. Thus a particular solution of the non-uniquely solvable Riemann problem
on a region with corners can be obtained. Our numerical scheme will be adjusted to suit the
new integral equation. Our presentation will begin with an introduction followed by a review
of the definition and the solvability of the Riemann problems and the integral equation for the
uniquely solvable interior Riemann problem for a simply connected region with corners. The
discussion will focus on how we obtained the new integral equation related to the non-uniquely
solvable interior Riemann problem for a simply connected region with corners and its additional
constraints and lastly numerical examples will be presented.
Gas-kinetic BGK schemes on moving and adaptive meshes
Jin Changqiu
Mathematics Department, Hong Kong University of Science and Technology ([email protected])
Joint work with Kun Xu.
The moving mesh methods are classified in two main groups with respect to the grid movement
relative to the overall solution algorithm, namely static and dynamic ones. Generally, the
motion of the grid can be chosen arbitrarily in flow simulations. But, different mesh movement
has significant impact on the numerical solutions. This paper is to study two types of moving
mesh methods incorporated with gas-kinetic BGK schemes for the Navier-Stokes equations.
For multi-material flows, the Lagrangian grid is considered to be the best choice due to its shock
capturing of material interfaces. However, the purely Lagrangian calculation produces very
distorted grids, which reduces the accuracy of the numerical solution, and eventually stops the
computation. Meanwhile, the mesh generation method based on the elliptic solver constructs
smooth grid and keeps the computation running, but with over-diffused at material interface.
Therefore, a method is suggested to ensure the continuing geometric quality of the computational
grid, while keeping the rezoned grid at each time step as close as possible to the Lagrangian
grid.
In this paper, firstly we are going to develop a gas-kinetic BGK scheme in the Lagrangian
framework following a unified transformation. At the same time, in order to avoid the mesh
distortion and resolve the rapid flow variation region, an adaptive and moving mesh method
is implemented in the scheme as well. For the unsteady flow simulation, the mesh basically
follows the fluid velocity and automatically is refined in the interfaces with large gradients. In
order to guarantee the geometric quality of the computational grid, the coupling between mesh
moving and adaptation has to be carefully designed. In the current study, the mesh adaptation
is based on the variational principle, and the method recently proposed by Tang et al has been
successfully implemented in the gas-kinetic BGK scheme. Many 1D and 2D test cases will be
presented.
60
Graphing calculator as a teaching and learning aid for secondary school students
and teachers
Hailiza Kamarulhaili
School Of Mathematical Sciences, Universiti Sains Malaysia, Penang ([email protected])
Joint work with Lee Boon Sim.
This paper discussed the possibility and suggestion of the implementation of graphic calculator
in teaching and learning Mathematics in the Malaysian’s upper secondary school in order to
meet the challenge of today’s accelerated technology. This project has been carried out at the
fourth and the fifth former level, where at this level it is mandatory for the students as well as
the teachers to be introduced to technology in assisting the students and the teachers to learn
and to teach Math in a more meaningful way. Therefore we hope the pilot study that we have
conducted would be able to help them in making Mathematics fun to learn as well as fun to teach.
A survey has been conducted on students and teachers at three secondary schools in Northern
area of Malaysia. They are in Kedah, Penang and Perak. All these three schools were not been
exposed to Graphic calculator before. The survey was to seek students and teachers point of
views on Mathematics before and after the demonstration of graphic calculator. Analysis was
made from the survey and brought to the suggestions of a new curriculum change, teachers’
attitudes and students’ preparation towards the new technology.
Simulation study for partially linear model with random covariates
Sri Haryatmi Kartiko
Faculty of Mathematics and Natural Sciences,
Gadjah Mada Unversity, Yogyakarta, Indonesia ([email protected])
Partially linear or semiparametric regression model is of the form
Yi = Xi� β + g(Ti ) + �i ,
i = 1, · · · , n
where Xi� β is the parametric component, g(Ti ) is the nonparametric component. g and β =
(β1 · · · , βp )� are the unknown parameters. In the paper by Gao (1995a), Xi = (Xi1 , ..., Xip )� and
Ti are fixed design points, �i are random errors are of mean zero and fnite varance, independent
of X. This model is usefull in situation where the responses Yi and predictors Xi are linearly
dependent but Yi are nonlinearly related to the independent variable Ti .
In this paper, a similar model with random covarates is considered. Instead of observing Xi , we
observe Wi = Xi + Vi , i = 1, · · · , n, where the measurement
errors Vi are i.i.d. independent of
�
(Yi , Xi , Ti ) with mean zero and covariance matrix uu .
61
The suggested estimator for β is β̂n = (W̃ � W̃ − n
likelihood the form of the estmator of β provides
ĝn (t) =
n
�
�
−1
�
uu ) W̃ Ỹ
. Maximizing the weighted
wnj (t)(Yj − Wj� β̂n )
j=1
as the estimator of g(t).
Simulation study is perform in this study to clarify the asymptotic result for the estimator of β
which is
�
�
−
→
−1
−1
n1/2 (β̂n − β) N (0,
Γ
)
�2
where Γ = E[(� − U � β)X − E(X|T )]
and Splus.
+ E[(U U � −
�
�2
uu )β]
+ E(U U � �2 ), by using Xplore
Conditionally positive definite (Fourier) hyperfunctions
Dohan Kim
Department of Mathematics, Seoul National University, Seoul 151-747, Korea
Making use of the heat kernel method we define a conditionally positive definite (Fourier) hy
perfunction and obtain a Bochner-Schwartz type theorem for hyperfunctions which generalizes
both the Bochner-Schwartz type theorem for distributions of Gelfand-Vilenkin in [GV] and our
version of Bochner-Schwartz theorem for (Fourier) hyperfunctions in [CCK2, CK1].
A geometric viewpoint of geophysical fluid dynamics in the tropics?
Tieh-Yong Koh
School of Physical & Mathematical Sciences,
Nanyang Technological University, Singapore ([email protected])
Recent advances in Geophysical fluid dynamics (GFD) theory came from the classical view of
fluid dynamics under Lagrangian and Hamiltonian formulations (Goldstein, 1980) and received
further elaboration in recent GFD textbooks (Salmon, 1998). The basic idea in the Hamiltonian
approach is that the description of fluid dynamics is independent of the particular coordinate
system used and depends only on the geometric character of the phase space of the system
in question. In general, geophysical fluids have a tendency to evolve quickly towards a slow
manifold in its phase space and in this ”adjustment” process, transient waves are emitted.
Such abstract conceptualization is clearly illustrated in the quasi-geostrophic theory (Charney,
1948), an elementary theory for extra-tropical GFD. A bit more advanced is the semi-geostrophic
theory, which can be rendered in mathematically rigorous form using the Hamiltonian approach.
62
Therefore, a challenge arises: can this kind of approach be applied to GFD in the tropics? If so,
there will be a big step forward in our understanding of the tropical atmosphere and oceans.
The ramifications of a proper GFD theory for the tropics are really wider than hinted above.
The last half-century has witnessed the advent of computers and the realization of computa
tional prediction of atmospheric states from known observations and guesses of initial conditions.
Numerical weather prediction (NWP) has reached unprecedented skill in that state-of-the-art
predictions provide reliable 3-day local forecasts and reasonable 1-week regional weather outlooks
in the mid-latitudes. However, forecasts of tropical weather are hardly reliable even within the
same day. The reasons are many-fold: (1) the dearth of atmospheric soundings (observations)
in tropical regions; (2) the inherent shorter spatio-temporal scales of convective weather; and
(3) the lack of basic understanding of tropical GFD to guide the development of NWP schemes
for the tropics.
For the last three years, a small focused effort has begun in Singapore to research into im
proving NWP for Southeast Asia, then under the auspices of Temasek Laboratories at National
University of Singapore (NUS), and now carried on in Temasek Laboratories at Nanyang Tech
nological University. A mesoscale NWP model that has demonstrated capability in mid-latitude
weather forecasts has been implemented as a research tool. However, the challenges of predict
ing mesoscale weather in Southeast Asia are many, even with or perhaps especially with such
a sophisticated computational model. Current numerical schemes and parameterizations of un
resolved processes for computational prediction were developed originally to accurately capture
mid-latitude weather. E.g. sound waves are often filtered out or slowed down in mesoscale sim
ulations with dubitable impact on atmospheric convection (Tijm and van Delden, 1999), which
is fundamentally important in the tropics. An improved understanding of the fundamentals of
tropical GFD would bear directly upon issues like the reliability of NWP in the tropics and
perhaps suggest improvements to parameterizations and numerical schemes for NWP.
This presentation is aimed to be an introduction to the use of the geometric approach in formu
lating geophysical fluid dynamics in the tropics and a call to regional mathematicians to engage
in further research in this topic.
Variable Selection for the Single-index Model
Efang Kong,
National University of Singapore, Singapore ([email protected])
Joint work with Yingcun Xia.
Single-index model (SIM) is one of the most popular semi-parametric models in econometrics
and statistics due to its dimension reduction capability. Many methods have been developed for
its parameter estimation. It is well known that the estimation precision and prediction accuracy
suffer from irrelevant variables in the model. However, little work has been done on variable
selection for SIM. We show in this paper the delete-m-out cross- validation method (CV(m)) can
63
be used for variable selection in SIM. However, the range of m for the method to be consistent in
SIM is different from those in linear regression models and in nonparametric models. Due to its
heavy computational burden, CV(m) is in fact not very applicable in SIM. A new method, called
the nonparameterized cross-validation (NCV) method, is proposed and proved to be consistent.
Simulations suggest that not only the NCV method is much less time consuming, but also has
better performance than CV(m) and the method based asymptotic normality of the estimator
of the index parameter. Applying the method to the Swiss banknotes data, a SIM with selected
variables indeed has much better prediction ability than a SIM with all the variables. Other
real data sets are also investigated.
Krylov solvers for grain growth simulation using phase-field modeling
Bipin Kumar
Department of Mathematics, NUS, Singapore ([email protected])
Joint work with Prasad Patnaik and Lin Ping.
The evolution of microstructures is fundamental to many fields including, biology, hydrodynam
ics, chemical reactions, material science etc. Understanding the kinetics of micro structural
evolution is the central dogma in developing novel materials with desired properties. Simula
tion of such processes through mesoscopic means is fast gaining popularity as a complementary
and cost effective tool to labor intensive experimentation. In the literature, a wide variety of
modeling attempts for the simulation of grain growth kinetics exist, such as, vortex methods,
variational approaches, Monte Carlo, phase-field modeling etc. . The phase-field modeling has
the distinct advantage of diffuse interfaces, where the micro structure is described by the well
known Cahn-Hilliard and Allen-Cahn equations [1,2]. However, all the existing approaches have
used only the explicit forward Euler method in time and finite difference for spatial discretiza
tion. The fine grid sizes (512x512, 1024x1024) and the large number of phase-field variables (at
least 36 to avoid grain coalescence) makes this approach both memory and CPU intensive. To
this end, no implicit or semi-implicit versions have been attempted in the literature, despite the
well known advantages of the latter schemes (higher time step sizes, stability and accuracy). In
the present study, we have solved the Allen-Cahn (more popularly known as time dependent
Ginzbug-Landau) equation by using a semi-implicit scheme, with an explicit treatment for the
non-linear terms. Krylov subspace methods have been employed to solve the resulting algebraic
equations for all the phase field variables. Since, the matrix is highly sparse, an efficient CSR
format has been adopted. The sparse version of GMRES and CGM solvers (with/ without preconditioners) is developed and their performance is compared. The temporal evolution of grain
growth and the mean grain size distributions have been visualized and compared. It should be
mentioned that, by choosing a semi-implicit route, we have made the scheme even more memory
and CPU intensive. We believe, this disadvantage can be easily circumvented by the possible
gains through higher time step sizes. Furthermore, we plan to develop an MPI (parallel) version
of the CGM and GMRES solvers. We also intend to extend these simulations to 3-d to study
more realistic cases.
[1] L.Q.Chen, Anuunal Rev. Mater. Res, 32, 113-140 (2002).
64
[2] C.E.Krill and L.Q.Chen, Acta Materialia, 50, 3057-3073 (2002).
A class of matrix operators and their properties
Miyeon Kwon
University of Wisconsin-Platteville ([email protected])
For a fixed sequence ϕ = {ϕn }, define the matrix operator Mϕ on �2 by
Mϕ (x)j =
∞
�
ϕj∨k xk .
k=0
Here j ∨ k = max{j, k}. The matrix representation of Mϕ under the standard basis of �2 is
⎛
⎞
ϕ0 ϕ1 ϕ2 . . .
⎜ ϕ1 ϕ1 ϕ2 . . . ⎟
⎜
⎟
Mϕ = ⎜ ϕ ϕ ϕ . . . ⎟ .
2
2
2
⎝
⎠
..
..
.. . .
.
.
.
.
In this talk, we present the necessary and sufficient condition for Mϕ to be bounded or compact
and characterize its membership in Schatten-von Neumann class Sp , p > 1 as well. We also give
some other interesting properties.
On a class of solvable lie algebras and groups of dimension 5
Anh Vu Le
Ho Chi Minh University of Pedagogy,
Vietnam ([email protected]; [email protected])
The paper considers a subclass of the class of MD5-algebras and MD5-groups, i.e., five dimen
sional solvable Lie algebras and Lie groups such that their orbits in the co-adjoint represntation
(K-orbit) are orbit of zero or maximal dimension. The investigation is devoted to the desciption
of the geometry of K-orbits of given MD5-groups. The foliations associated to these groups are
also presented in the paper.
Extension of functions with small oscillation
Denny H. Leung
Department of Mathematics, National University of Singapore,
Singapore ([email protected])
65
Joint work with Wee Kee Tang.
A function that is the pointwise limit of a sequence of continuous functions is said to be of
Baire Class One. A classical theorem of Kuratowski says that every Baire one function on a Gδ
subspace of a Polish (= completely metrizable) space X can be extended to a Baire one function
on X. In this talk, I will describe an ordinal index for Baire one functions introduced by Kechris
and Louveau and use it to prove a refinenement of Kuratowski’s Theorem.
Qualitative properties for solutions on a fourth order degenerate parabolic
equation in higher space dimensions
Junjie Li
Department of Mathematicas, Yuquan Campus, Zhejiang University,
Hangzhou 310027, Peoples Republic of China ([email protected]).
We are concerned with existence, positivity and long-time behavior of solutions to the following
Newmann problem of a fourth order degenerate parabolic equation in higher space dimensions
ut + div(|u|n �Δu) = 0 in Ω × (0, T )
∂Δu
∂u
=
= 0 on ∂Ω × (0, T )
∂ν
∂ν
(3)
u(x, 0) = u0 (x) in Ω
(4)
where Ω ⊂ RN (N ≥ 2) is an open and bounded domain, ν denotes the unit outer normal vector
to ∂Ω. This problem appears in the lubrication theory for thin viscous films and the function u
is the height of the film. Our main results are as follows:
(I)(Existence) Assume that Ω ∈ RN (N ≥ 2) is an open and bounded domain with the boundary
−2 N +4
of class C 1,1 (or C 0,1 if Ω is convex), and assume that 18 < n < min{ 2N
N −2 , N −2 }. Assume further
�
+2
that the initial value 0 ≤ u0 ∈ W 1,2 (Ω) satises Ω uα+1
< ∞ for α ∈ ( 12 − n, min{2, N
0
N −2 } − n)
being subject to some conditions. Then problem (3) has a nonnegative weak solution u(x, t) ∈
∗
W 1,2 (0, T ; (W 1,q (Ω))� ) for some q ∗ > 1.
(II)(Asymptotic behavior) Let n, α and u0 as in (I) and let u be the weak solution of (3) obtained
−6
in (I) for T = ∞. Assume further that α + n > N
N −2
. Then
�
1
limt→∞ u(x, t) =
u0 (x) ∈ W 1,2 (Ω).
|Ω| Ω
Resampling the generalized estimating functions for analysis of longitudinal data
Yue Li
Department of Statistics and Applied Probability,
National University of Singapore ([email protected])
66
Joint work with You-Gan Wang.
Sandwich estimator is commonly used as the variance estimator of parameters in GEE model.
However, sandwich has been found to underestimate the true variance and may not be consistent
some cases. Parzen, Wei and Ying (1994, Biometrika) proposed a pivotal method to perturb
the estimating equations for the variance estimation, while they require the estimating functions
must be pivotal or at least asymptotically pivotal. We propose resampling methods to perturb
the estimating equations without pivotal assumptions. We investigated the consistency of the
resulting variance estimators and confidence interval coverage probabilities. Simulation results
indicate that the variance estimators by our proposed methods can not only correct the bias
of sandwich but also improve the confidence interval coverage. Real data sets are also used for
illustration.
Various modelings of fluid/structure interactions
Christian Licht
Laboratoire de Mecanique et Genie Civil,
CASE COURRIER 048, Universite Montpellier 2,
Place Eugene Bataillon 34095 Montpellier Cedex 05 FRANCE ([email protected])
We examine various linear or non-linear initial boundary value problems stemming from model
ings of fluid/structure interactions. In all cases the structure is a linear elastic body surrounded
by fluids, whereas the fluids may fill up bounded or unbounded domains, may be perfect or
viscous of non-Newtonian type, and gravity or surface tension effects on the free-surfaces of the
fluids may be taken into account. We confine to small motions.
A common feature of our mathematical treatment of these problems is a suitable description of
the mechanical state of the coupled system in terms of an element of an Hilbert space where
the mechanical energy induces a hilbertian norm. Then, a (non-)linear evolution equation may
describe the transient motions, and the theory of semi-groups of (non-)linear operators sup
plies existence and uniqueness results. This same formalism also provides spectral properties
and convergence results of finite element methods using appropriate truncations in the case of
unbounded fluid domains.
Modelling volcanic ashfall using partial differential equations
Leng Leng Lim
Institute of Information and Mathematical Sciences, Massey University,
Auckland, New Zealand ([email protected])
Joint work with Robert McKibbin and Winston L. Sweatman.
67
This short communication discusses a mathematical model to calculate the concentration of
volcanic ash within the atmosphere following an eruption and how it is eventually distributed
on the ground. The model is derived from conservation equations for a continuum, and the
particle concentration is described by the well-known advection-dispersion equation. The model
is also suitable for other similar particulate releases.
There have been similar modelling attempts, but most have concentrated on a single-layered
atmospheric structure. Because of the complexity of atmospheric flow velocities and turbulence,
we model the atmosphere as a horizontally-layered system. Within each layer, which is assumed
to be moving steadily and parallel to the ground, the velocity components, particle terminal
speed and the dispersion coefficients are assumed uniform, but may be different for each layer.
This allows for elevation-dependent wind and turbulence profiles, as well as the agglomeration
of particles which alters their terminal speeds. The resulting mathematical problem involves a
set of coupled linear partial differential equations subject to boundary conditions found from
physical considerations.
Some characteristics classes in number theory
Lim Meng Fai
National University of Singapore, Singapore 117543 (g0306132 @nus.edu.sg)
Let k be a commutative ring with identity and A a commutative associative k-algebra with
unity. We introduce a Dennis trace map mod n, from K1 (A; Z/n) to Ω1dR (A)/(n), the module
of differentials mod n. In particular, we will be interested in the case when A is a number ring
of a number field F . Through the Dennis trace map, we give a sufficient condition for the class
group of A to contain an element of order n. We shall apply these to the cases of quadratic
√
number fields and Q( 3 m) , where m �≡ ±1 mod 9 and squarefree. If time permits, we shall also
give a method of constructing intertwining matrices, using the results developed in the talk.
The material covered is part of the speakers Master thesis which consists mainly on doing a
survey on [2]. The part on intertwining matrices comes from [1].
[1] Berrick A J, Intertwiners and the K-theory of commutative rings, J. reine angew. Math., 569
(2004), 55-101.
[2] Max Karoubi and Thierry Lambre, Quelques Classes Caracteristiques en Theorie des Nom
bres, J. reine angew. Math. 543 (2002), 169-186.
Unfolding complete graphs, paths, and cycles
Yvette F. Lim
Department of Mathematics De La Salle University,
2401 Taft Avenue, 1004 Manila ([email protected])
68
If two non-adjacent vertices of a graph G having a common neighbor are identified to obtain
the graph H, we say that H is a 1-f old of G. We also say that G is a 1-unfold of H. For
convenience, any graph G is regarded as a 0-fold of G. If k is any positive integer, we say that
H is a k-fold of G if H can be obtained from G by folding G iteratively k times. Thus, a k-fold
of G is a 1-fold of a (k − 1) − fold of G. If H is a k-fold of G, we call G a k-unfold of H. The
symbol F −1 (G) denotes the set of all k-unfolds of G for k ≥ 0. In this paper, we investigate the
sets F −1 (G), where G is a complete graph, a path or a cycle.
Lengths of pairs of complex 5 × 5 matrices
W. E. Longstaff
School of Mathematics & Statistics, The University of Western Australia,
35 Stirling Highway, Crawley, WA 6009, Australia
Let {A, B} be a pair of n × n matrices over a field IF . By definition, the length l(A, B) of this
pair is the smallest positive integer l such that every matrix in the unital algebra generated
by A and B is a linear combination of words in A and B of length at most l. It has been
conjectured that l(A, B) ≤ 2n − 2. A. Paz has shown this to be the case for n ≤ 4. We establish
the conjecture for complex 5 × 5 matrices. The technique of proof easily establishes the smaller
size cases. Also, for any field IF and any positive integer m it is shown that there exist m × m
matrices C, D over IF such that the length of the pair {C, D} is 2m − 2.
On the Construction of Semiperfect Colorings of Symmetrical Patterns
Manuel Joseph C. Loquias
Department of Mathematics, University of the Philippines, Diliman
Joint work with Rene P. Felix.
If G is the symmetry group of an uncolored symmetrical pattern then a perfect coloring of the
pattern is a coloring in which all symmetries in G permute the colors. Otherwise, the coloring
is said to be nonperfect. Senechal, in [2], suggested that a systematic study of nonperfect
colorings might become useful and interesting. In particular, we look at nonperfect colorings
in which the symmetries permuting the colors form a subgroup of index 2 in G. Rigby referred
to such colorings as semiperfect colorings in [1]. We develop a method for the construction of
semiperfect colorings. However, the method may give rise to perfect colorings and we address
this problem by determining equivalent conditions to know whether the coloring that will be
obtained is perfect or semiperfect.
[1] J. F. Rigby. Symmetry in geometry: A personal view. Symmetry: Culture and Science,
1:6376, 1990.
[2] M. Senechal. Color symmetry. Comput. Math Applic., 16 No. 5-8:545553, 1988.
69
Applications of the compensated compactness method on hyperbolic conservation
systems
Yun-Guang Lu
Department of Mathematics, University of Science and Technology of China,
Hefei, China
(yglu [email protected])
An important class of the equations arising in applications are the nonlinear system of conser
vation laws. The basic question in this area which has been adressed by the most important
contributors in this field (Riemann, Friedrichs, Lax, Glimm, DiPerna and Lions) is the existence
of solutions to these equations. This helps to answer the question if the modelling of the natural
phenomena at hand has been doen correctly, if the problem is well posed. In the past years,
we studied the existence of weak solutions for some nonlinear hyperbolic equations arisen in the
compressible fluid and gas mechanics, elasticity mechanics and the theory of combustion. The
method we used in the study is the theory of the compensated compactness. Through a care
ful construction of exact solutions of the classical Fuchsian equations, we obtained the explicit
entropy-entropy flux pairs of Lax type for some physical models. The necessary estimates for
the major terms in these entropies are obtained by the use of singular perturbation theory of
the ordinary differential equation. The special entropies provide a convergence theorem in the
strong topology for the artificial viscosity method, Friedrichs- Lax scheme method when applied
to these equations and used together with the theory of the compensated compactness.
Self-dual Codes over Z2 x Z2 and Groups of 2 x 2 Binary Matrices
Bernardo R. Marquez
Department of Computer Science, College of Computer Studies
Ateneo de Naga University, Philippines [email protected]
This talk will present some additional results on a unifying approach to the study of self-dual
codes over Z2 × Z2 . By defining a self-dual code with respect to a 2 × 2 binary matrix and also
with respect to a set of such matrices, different families of self-dual codes over Z2 × Z2 can be
characterized by groups of 2 × 2 binary matrices. These families of codes include the Euclidean
self-dual codes over the commutative rings on Z2 × Z2 and the self-dual codes over F4 with
the Hermitian inner product, with tZhe Euclidean trace inner product, and with the Hermitian
trace inner product. Preliminary results of this study were presented in the 2002 Conference on
Algebra, which was held at the Ateneo de Manila University in December 2002.
Wavelet Solutions of Certain Ill-posed Problems
Vinod Mishra
Department of Mathematics, Sant Longowal Institute of Engineering & Technology,
Longowal 148 106 (Punjab) India
70
An ill-posed problem is one for which a small perturbation on the boundary specification (g)
can amount to a big alteration on its solution, if it exists. That is, if the solution exists, it does
not depend continuously on data (g). Considering Meyer multiresolution analysis, the problem
is regularized to a well-posed problem in scaling spaces. The Fourier transform of g has a rapid
decay at high frequencies and are cut by the Meyer wavelet, which possesses compact support
in frequency domain, amounting to exact solution when data error tends to zero. The following
problems are analyzed:
I.
k(x)uxx (x, t) = ut (x, t),
t ≥ 0, 0 < x < 1,
u(0, t) = gn (t), ux (0, t) = 0,
where
gn (t) =
II.
�
n−2 cos 2n2 t if 0 ≥ t ≤ t0
0
if t > t0 .
k(x)uxx (x, t) = ut (x, t) + ux (x, t),
t ≥ 0, 0 < x < 1,
u(0, t) = g(t), ux (0, t) = 0.
Teaching Applications of Planar Graphs with Maple
E. J. Moore
Department of Mathematics, Faculty of Applied Science,
King Mongut’s Institute of Technology North Bangkok
Joint work with S. Charoenwed, S. Boonpisuthi, C. Janyaoudom, and U. Phalavonk.
Graph theory is now extensively used to develop mathematical models for many real-world prob
lems arising, e.g., in engineering, operations research, industrial mathematics, and in electrical,
computer, transport networks etc. The Maple Mathematical Software package contains a set of
procedures which can quickly solve many graph theory problems and which are very useful for
teaching students of graph theory to solve real-world problems. In this paper an example is given
of an application of the theory of planar graphs to the modeling of an electrical circuit board
and of the use of Maple to solve the graph theory model. In electrical circuit boards, connectors
may not cross since this would lead to undesirable electrical contact at crossing points. When
necessary, insulated wires may be used where connectors cross, but this can complicate the man
ufacture of the circuit boards and increase their cost. An alternative approach is to decompose
the electrical circuit into the minimum number of layers such that in each layer connectors do
not cross and insulating wires are not required. In a graph theory model a layer corresponds to
a planar graph. The graph theory problem is therefore to find the minimum number of planar
subgraphs contained in the graph that represents the electrical circuit. A short summary of the
theory of planar graphs will be given and an algorithm developed to find a lower bound on the
71
number of planar subgraphs contained in any given graph. A Maple 9.5 procedure based on this
graph theory algorithm will be described which can find a lower bound on the number of layers
required for any given electrical circuit.
On Factorial Numbers
Felix P. Muga II
Mathematics Department, Ateneo de Manila University
A representation of numbers using factorial numbers as bases is proposed. The existing base
number systems like the binary and the decimal systems cannot represent all the rational num
bers as a finite sequence of digits. For example, there is no binary number that is equal to
the rational number 1/5. In this paper, we shall show that a base number representation with
base b can represent a rational number p/q exactly if q has no factor that is relatively prime to
b. Otherwise, the base number representation is non-terminating and repeating. We shall also
prove that a rational number can be expressed exactly a factorial number with a finite number
of digits.
Probability and stochastic processes using a Riemann-type integration
P. Muldowney
University of Ulster
Basic probability theory and its many offshoots were put on a firm mathematical footing by
A.N. Kolmogorov and others in the twentieth century, by basing the notion of expectation on
the Lebesgue integral of the random variable with respect to a probability measure on the
sample space. This approach places a heavy burden of mathematical structure on the primitive
notion of probability or likelihood. An extension of the Riemann integral, which possesses the
desirable properties of the Lebesgue integral, was discovered independently by Ralph Henstock
and Jaroslav Kurzweil in the 1950’s. This approach is not based on measure theory, and it gives
rise to a different formulation of probability and stochastic processes.
Subgroups P GL(2, q) as homomorphic images of group
Qaiser Mushtaq
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
Joint work with Saima Riaz.
The extended modular group, P GL(2, Z) is the group of linear fractional transformations z →
az+b
l
cz+d with integral coefficients satisfying ad − bc = ±1. Triangle groups Δ(l, m, k) =< x, y : x =
72
y m = (xy)k = 1 > are evolved when P GL(2, Z) =< x, y : x2 = y 3 = t2 = (xt)2 = (yt)2 = 1 >
acts on the projective line over a finite field Fq , namely P L(Fq ), where q is a prime power.
We focus our attention on obtaining triangle groups with k = 7, l = 2, m = 3 and q to be a
prime p. An action of the modular group can be depicted by a coset diagram. In this paper we
create a non-simple fragment of a coset diagram which is composed of at least two circuits such
that more than two points of the fragment are fixed by the element (xy)7 and at least one of the
vertices is a common vertex of the two circuits. Let w1 and w2 be the words (elements) induced
by the paths of these circuits . We apply a process established by Q. Mushtaq (QJM, 2(39),
1988) on the elements (words) w1 , w2 and w1 w2 of P GL (2, Z) where each word corresponds
to the respective path and obtain an equation f (θ) = 0 which yields solutions θ � s in a suitable
finite field Fp , (p ≡ ±1(mod 7)). Thus corresponding to each θ ∈ Fp and suitable p, there exists
a coset diagram D(θ, p) which will contain the fragment and which will depict the permutation
representations of the triangle group Δ(2, 3, 7).
Non-associative algebras with right identities
Ki-Bong Nam
Department of Mathematics, Univ. of Jeonju,
Chon-ju 560-759, Korea; and
Department of Mathematics and Computer science,
University of -Whitewater, Whitewater, WI 53190 ([email protected])
Joint work with Seul Hee Choi (University of Jeonju).
The Lie admissible non-associative algebra Nn,n,0 over a feild of characteristic zero is spanned
by {ea1 x1 · · · ean xn xj11 · · · xjnn ∂u |a1 , · · · , an , j1 , · · · , jn ∈ Z, 1 ≤ u ≤ n} with the usual evaluation
mapping product multiplication is defined in the papers [1], [3-7]. We find the non-associative
algebra automorphism groups Autnon (Nn,n,0 ) and Autnon (Nn,0,n ) of the non-associative algebras
Nn,n,0 and Nn,0,n respectively by using auto-invariant [2] and by introducing a generalized Kro
necker delta. We prove Theorem 3 of the paper [5] differently in this talk. If time permits, we
will discuss on the relations of Lie-admissible non-associative, Lie, and Weyl algebras and some
unsolved problems of them.
[1] Mohammad H. Ahmadi, Ki-Bong Nam, and Jonathan Pakinathan, Lie admissible nonassociative algebras, Algebra Colloquium, Vol. 12, No. 1, World Scientific, March, 2005, 113-120.
[2] I. N. Herstein, Topics in Ring Theory, The University of Chicago Press, 1965, 1-10.
[3] V. G. Kac, Description of Filtered Lie Algebra with which Graded Lie algebras of Cartan type
are Associated, Izv. Akad. Nauk SSSR, Ser. Mat. Tom, 38, 1974, 832-834.
[4] Ki-Bong Nam, Generalized W and H Type Lie Algebras, Algebra Colloquium 6:3, (1999),
329-340.
[5] Ki-Bong Nam, On Some Non-Associative Algebras Using Additive Groups, Southeast Asian
Bulletin of Mathematics, Vol. 27, Springer Verlag, 2003, 493-500.
73
[6] Ki-Bong Nam and Seul Hee Choi, Automorphism group of non-associative algebras W N2,0,0 1 ,
Vol. 1 (2005), No.1, JOCMO, SAS publishers, 35-44.
[7] R. D. Schafer, Introduction to nonassociative algebras, Dover, 1995, 128-138.
Linear Optimization Under Uncertainty: Persistency and Its Applications
Karthik Natarajan
Department of Mathematics, National University of Singapore,
Singapore 117543 ([email protected])
Joint work with Teo Chung Piaw and Song Miao.
Given a linear optimization problem, where the objective function c̃ is randomly distributed,
we would like to evaluate the probability distribution of the optimal decision variables. We
call this the persistency problem for linear optimization under uncertainty. We propose a conic
programming framework that solves the persistency problem for the class of linear programs
under a limited set of known marginal moments for the objective coefficients. The simulation
results indicated that the results obtained under our approach are promising and robust.
On the class number and Sylow 2-group of the ideal class group of quadratic fields
Fidel R. Nemenzo
University of the Philippines (fi[email protected])
In this talk we investigate the Sylow 2-group of the ideal class group of certain quadratic fields
and determine the exact power of 2 dividing the class number. We use the theory of ideals and
a classic theorem by Legendre on the solvability of the Diophantine equation ax2 + by 2 = z 2 to
give elementary proofs to known results on the class number of quadratic fields.
Some BQ-rings of Linear Transformations
S. Nenthein
Department of Mathematics, Faculty of Science, Chulalongkorn University,
Bangkok 10330, Thailand ([email protected])
Joint work with Y. Kemprasit. By a BQ-ring we mean a ring whose bi-ideals and quasi-ideals
coincide. It is known that every (Von Neumann) regular ring is a BQ-ring. If V is a vector space
over a division ring, let L(V ) denote the ring of all linear transformations α : V → V under
usual addition and composition. Since L(V ) is regular, L(V ) is a BQ-ring. For a subspace W
of V , let L(V, W ) be the subring of L(V ) consiting of all α ∈ L(V ) with Im α ⊂ W . It is shown
in this paper that L(V, W ) is regular only the case that V = W or W = {0} but L(V, W ) is
always a BQ-ring.
74
A class of boundary value problem For high order differential equations
Anh Tuan Nguyen
Ho Chi Minh City University of Pedagogy
New sufficient conditions of the existence end uniqueness of the solutions of the boundary for a
functional differential equations of n-th order with certain functional boundary conditions are
constructed by a method of a priori of estimates
Optimization issues in the construction of multidimensional wavelets
Hyungju Park
Korea Institute for Advanced Study
The wavelet construction from a multiresolution generated by a finite number of compactly
supported scaling functions in any dimension can be reduced to the problem of extending a
matrix with Laurent polynomial entries. As the extended matrix is not unique, one can consider
the set of all possible extensions which produces a design space or parametrization for wavelet
construction. This paper aims to clarify the process of obtaining such a design space and subse
quently optimizing the wavelet construction with respect to certain design goals (e.g. frequency
response, regularity or linear phase etc). The method relies on Grobner basis computation to
solve the algebraic relations produced during the process. A conjecture is proposed regarding
the feasibility of paraunitary matrix completion.
Generalized Spectrum of Grassmann Codes
Arunkumar R. Patil
Department of Mathematics, Indian Institute of Technology Bombay,
Powai, Mumbai 400076, India ([email protected])
Let C be a linear [n, k]q -code, that is, a k-dimensional subspace of the n-dimensional vector
space F nq over the finite field F q of q elements. The spectrum of C is the sequence Ai (C),
0 ≤ i ≤ n, of positive integers defined by Ai (C) := #{c ∈ C : �c� = i}, where the norm
�c� of a codeword c = (c1 , . . . , cn ) ∈ C is the number of nonzero coordinates in c, that is,
�c� := #{i : ci �= 0}. More generally, one can associate the norm to a subspace D of C by
defining �D� := #{i : there exists c ∈ D with ci �= 0}. The double sequence Ari (C), 0 ≤ i ≤ n,
0 ≤ r ≤ k, where
Ari (C) := #{D : D subspace of C with dim D = r and �D� = i},
is called the generalized spectrum of C. The generalized spectrum of a code gives a more refined
information than the classical spectrum and the higher weights. It was shown by Kløve (1992)
that the generalized spectra satisfy an analogue of the MacWilliams identities.
75
We begin by considering the codes C(�, m) obtained from the Grassmannian G�,m of �-dimensional
subspaces of F m
q , with its Plücker embedding. These codes have been studied by Ryan (1987),
Nogin (1996) and by Ghorpade and Lachaud (2000). In particular, the minimum distance and
some of the higher weights of C(�, m) are known. The spectrum of C(�, m) is known only in
the special case of � = 2 and is given by Nogin (1996) using the classification of bilinear skewsymmetric forms of a given rank. Recently, Vincenti and Montanucci have partially succeeded
in determining the generalized spectrum of C(2, 4) in connection with a characterization of the
Klein quadric that goes back to Tallini (1955). We show that their result about the generalized
spectra can be obtained by alternative means and also complete their calculations.
We also consider the problem of in the determination of spectra associated to the so called
Schubert codes, which were introduced by Ghorpade and Lachaud (2000), and have been studied
by Hao Chen (2000), Guerra and Vincenti (2004), and by Ghorpade and Tsfasman (2005). We
make a small beginning here by determining the complete weight hierarchy for the Schubert
codes Cα (2, 4) corresponding to all the Schubert subvarieties Ωα of the Klein quadric G2,4 .
Divisibility of some hypergroups defined from groups
Sajee Pianskool
Department of Mathematics, Faculty of Science, Chulalongkorn University,
Bangkok 10330, Thailand ([email protected])
Joint work with S. Chaopraknoi (Chulalongkorn University). A hypergroup (H, ◦) is said to
be divisible if for every x ∈ H and every positive integer n, there exists y ∈ H such that
x ∈ (y, ◦)n where (y, ◦)n denotes the set y ◦ y ◦ · · · ◦ y (n copies). The following hypergroups
defined from groups are known. If G is an abelian group and ρ is an equivalence of G defined
by xρy ⇐⇒ x = y or x = y −1 , then (G/ρ, ◦) is a hypergroup where xρ ◦ yρ = {(xy)ρ, (xy −1 )ρ}.
Also, if G� is any group and N � G� , then (G� , �) is a hypergroup where x � y = N xy. The
purpose of this paper is to show that for a finite G, (G/ρ, ◦) is divisible if and only if G is of
odd order. In addition, for a finite G� , that N = G� is necessary and sufficient for (G� , �) to be
divisible.
The Effect of Desalination Plants on the Salinity of the Arabian Gulf Waters
Anton Purnama
Department of Mathematics and Statistics, College of Science Sultan Qaboos University,
PO Box 36, Al-Khod 123 Muscat, Sultanate of Oman ([email protected])
Joint work with H.H. Al-Barwani (Sultan Qaboos University) and Ronald Smith (Loughborough
University).
Seawater desalination in the Arabian Gulf is the reliable source of water supply to the population
of Kuwait, Saudi Arabia, Bahrain, Qatar and the United Arab Emirates. It is a common practice
that once a seawater desalination plant is built, its daily water production capacity is then
76
gradually increased to make up deficiencies in supplies of water as drought conditions worsen
and water demands increase due to rapid industrial development and sharp population growth.
The physical features of the Arabian Gulf are its shallow semi-enclosed nature and arid climate.
The Gulf is connected to the Gulf of Oman through the narrow Strait of Hormuz. Due to the
arid nature of its bordering lands, water in the Gulf can have an extremely high evaporation rate
leading to hypersaline conditions. The Arabian Gulf is environmentally very fragile, and any
further loss of water by desalination plants operated along its coast would deteriously change the
salinity. A mathematical model is developed to reveal multiplicative dependence of the salinity
on factors associated with incoming river flow, evaporation and desalination plants. Using a
semi-enclosed sea of simple geometry the effect of seawater desalination at the northern coasts
of the Arabian Gulf is found to be more severe.
Simulations of multi-phase flow using Runge-Kutta discontinuous Galerkin
methods with conservative approach of interfaces
Jianxian Qiu
Department of Mathematics, Nanjing University,
Nanjing, Jiangsu, P.R. of China 210093 ([email protected])
Joint work with Tiegang Liu (IHPC, Singapore) and Boo Cheong Khoo (National University of
Singapore).
In the presentation we will describe our recent work on using Runge-Kutta discontinuous
Galerkin (RKDG) finite element methods for multi-phase flow simulations in one dimension
where the moving material interfaces by is effected by a conservative method. In generally,
the method for solving multi-phase compressible flow comprises two parts: one is the technique
for solving in the single-phase and another is treatment of material interfaces. A relatively
dominant difficulty for simulation of compressible multi-phase flow is the treatment of moving
material interfaces and their vicinity. In the literature there are some methods to overcome this
difficulty. The ghost fluid method (GFM) developed by Fedkiw et al.[3] has provided a flexible
way to treat the two-medium flow. The main appealing features of the GFM are its simplicity,
easy extension to multi-dimensions and maintenance of a sharp interface without smearing. The
GFM makes the interface ”invisible” during computations and the computations are carried
out as for a single-medium manner such that its extension to multi-dimensions becomes fairly
straightforward. In [4], Liu et al. proposed a modification to the original GFM via solving the
shock-shock relation of the Riemann problem near the interface and employing the obtained
solution for the specification of the ghost cell status on both sides of the interface. This has
led to even greater robustness of the GFM-related method for application of strong shock. Al
though the GFM is simple and effective, it is still not a conservative method. In our proposed
conservative method, we use both the exact Riemann solvers as fluxes for multi-phase flow at the
interfaces and adaptive interface cell; that is every cell contains only one medium and the size of
neighboring cell next to the interfaces is modified according to the movement of interface. For
solving of single-phase flow, we adopt the RKDG method rather than finite difference method
used in [3,4]. RKDG method is a very popular method in solving single-phase flow; it is a high
77
order finite element method suitable for hyperbolic conservation laws while encompassing the
useful features from high resolution finite volume schemes, such as the exact or approximate
Riemann solvers, TVD Runge-Kutta time discretizations, and limiters. Numerical results for
both gas-gas and gas-water flow in one dimension are provided to illustrate the validity of the
RKDG with conservative treatment of the interfaces.
[1] B. Cockburn, G. Karniadakis, C.-W. Shu, The development of discontinuous Galerkin meth
ods, part 1: overview, Lecture Notes in Computational Science and Engineering, Springer, Vol.
11, (2000), pp. 3-50.
[2] B. Cockburn, C.-W. Shu, Runge-Kutta discontinuous Galerkin method for convectiondomi
nated problems, J. of Sci. Comput., 16, (2001), 173-261.
[3] R.P. Fedkiw, T. Aslam, B. Merriman, S. Osher, A non-oscillatory Eulerian approach to
interfaces in multimaterial flows (the Ghost Fluid Method) , J. Comp. Phys., 152 (1999),457
492.
[4] T.G. Liu, B.C. Khoo, K.S. Yeo, Ghost fluid method for strong shock impacting on material
interface, J. Comp. Phys., 190 (2003), 651-681.
On the Edge Reduction Number of Graphs
Blessilda P. Raposa
Mathematics Department, De La Salle University, Manila
A unit graph is a graph drawn such that adjacent vertices are one unit apart. The edge-reduction
number of a graph G, denoted by ed(G), is the minimum integer k, such that there exist k edges
of G whose deletion yields a unit graph. We show that the edge-reduction number of unit graphs
and unicyclic graphs is equal to zero. We also find some bounds for the edge-reduction number
of graphs.
Identification of moving average process with infinite variance
Dedi Rosadi
Department of Mathematics, Gadjah Mada University,
Sekip Utara, Yogyakarta, Indonesia ([email protected])
In the traditional Box-Jenkins modelling procedure, we use the sample autocorrelation function
as a tool for identifying the plausible models for empirical data. In this paper, we consider the
sample normalized codifference as a new tool for the preliminary order identification of moving
average process with infinite variance. From simulation studies, we find that the proposed
method may perform as well as the Rosenfelds method (Rosenfeld, 1976).
[1] Rosadi, D. and Deistler, M. (2004) Estimating the codifference function of linear time series
models with infinite variance. Preprint. Institute for Mathematical Methods in Economics, Re
78
search Unit Econometrics and System Theory, Vienna University of Technology. Available on
http://dedirosadi.staff.ugm.ac.id/paper/codifference.pdf
[2] Rosadi, D. (2005) Order Identification for Gaussian Moving Average Using the Codifference
Function, Journal Of the Statistical Computation and Simulation, In Press.
[3] Rosenfeld (1976) Identification of time series with infinite variance. Applied Statistics 25,
147153.
Subdivision Number: Powers of Paths and Cycles
Leonor A. Ruivivar
Mathematics Department, De La Salle University,
2401 Taft Avenue, Manila, Philppines
Joint work with Severino V. Gervacio, Isagani B. Jos, and Yvette F. Lim.
A graph G is called a unit-distance graph (in the plane) if its vertices can be represented by
distinct points in the plane such that any two points representing adjacent vertices are 1 unitdistance apart. To subdivide an edge ab of G means to replace the edge ab with two new edges
ac and cb, where c is new vertex not in G. If every edge of a graph of a graph is subdivided, the
resulting graph is a unit-distance graph. The smallest integer k such that there exist k edges
of G which when subdivided yields a unit distance graph is called the subdivision number of G,
denoted by sd(G).
Exact values of sd(G) are derived here when G is the square of a cycle or the cube of a path.
Upper bounds for the subdivision numbers of higher powers of the path and the cycle are also
obtained.
Wave Front Solutions of Continuous Neural Networks
Sittipong Ruktamatakul
Department of Mathematics, Faculty of Science, Mahidol University,
Rama VI Rd, Bangkok 10400, Thailand
Joint work with Yongwimon Lenburya (Mahidol University) and Jonathan Bell (University of
Maryland Baltimore County). In this work, we first examine the existence, uniqueness and
stability of the traveling wave front solutions of the integral differential equations arising from
modeling a cortical layer of nerve cells. For this purpose, we formulate a single cell layer neural
network model with lateral inhibition connectivity. We then discuss briefly a self-excitatory cell
layer coupled to a second layer of non-self-excitatory cells that provide inhibitory feedback to
the first layer while getting excitatory stimuli from the first layer.
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On the absence of arbitrage opportunity for the fractional black-scholes model
Rattikan Saelimy
Department of Mathematics and Computer Science, Prince of Songkla University,
Pattani, Thailand, 94000([email protected])
Joint work with Tran Hung Thao (Academy of Science and Technology, Vietnam)
After introducing an approximate approach to a fractional Black-Scholes Model, we discuss the
principle of Absence of Arbitrage Opportunity (principle of AAO) for this model: We prove that
in spite of the fact that, in general, there exists an arbitrage opportunity for a fractional model,
there will be no arbitrage for our approximate model while the approximation can be made with
any small enough exactitude. This is an advantage of our approximate approach.
On the Cardinality of the Set of Solutions to Congruence Equation Associated
with Seventh Degree Form
Siti Hasana Sapar
Laboratory of Theoretical Mathematics, Institute for Mathematical Research,
Universiti Putra Malaysia ([email protected])
Joint work with Kamel Ariffin Mohd Atan and Muhamad Rushdan Md Said.
Let x = (x1 , . . . , xn be a vector in the space Z n with Z ring of integers and q be a positive
integer, f a polynomial in x with coefficients in Z. The exponential sum associated with f is
defined as
�
S(f ; q) =
exp(2πif (x)/q)
where the sum is taken over a complete set of residues modulo q.
The value of S(f ; q) has been shown to depend on the estimate of the cardinality |V |, the number
of elements contained in the set V = {x mod q|fx ≡ 0 mod q} where fx is the partial derivative
of f with respect to x.
This paper will give an explicit estimate of |V | for polynomial f (x, y) in Zp [x, y] of degree seven.
Earlier authors have investigated similar polynomials of lower degrees. The polynomial that we
consider in this paper is as follows:
f (x, y) = ax7 + bx6 y + cx5 y 2 + sx + ty + k.
The approach is by using p-adic Newton Polyhedron technique associated with this polynomial.
80
Irreducible Representations of Some Two-Groups
Nor Haniza Sarmin
Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia,
81310 UTM Skudai, Johor ([email protected])
Joint work with Azhana Ahmad and Fong Wan Heng.
Group theory is one of the mathematical tools most commonly used in quantum chemistry and
spectroscopy. Most applications of group theory to physical problems are, more particularly,
applications of representation theory. One reason is that representation theory reduces the
abstract properties of groups to numbers. The main part of studying representation theory is
to look at its irreducible representations. Irreducible representations are important to chemist
since they provide directly a great deal of information about the nature of vibrational and
electronic wavefunctions. Irreducible representation for an atom also provides the ways for
labelling orbitals, determining molecular orbitals formation and determining vibrational motions
for a molecule. In this research, the irreducible representation for some two-groups are obtained.
Cyclic projective planes
Bernhard Schmidt
Nanyang Technological University, Singapore ([email protected])
A projective plane with a point regular cyclic automorphism group is called a cyclic projec
tive plane. In 1938, J. Singer constructed cyclic projective planes for all prime power orders.
Singer’s planes are called Desarguesian projective planes. Singer conjectured that every pro
jective plane of prime power order is necessarily Desarguesian, i.e. arises from his construc
tion using finite fields. Hall (1956) and Bruck (1960) verified Singer’s conjecture for orders
2,3,4,5,7,8,9,11,13,16,25,27,32,49,64,81. Apparently, no further results have been obtained on
Singer’s conjecture since then. Remarkably, Bruck (1960) conjectured that Singer’s conjecture
is false and suggested to search for counterexamples. I have followed Bruck’s suggestion and will
present the results of my search in this talk.
Application of wavelet transform to problems involving integral kernels
Alexander Shapeev
Department of Mathematics, National University of Singapore,
Singapore ([email protected])
Many problems of mathematical physics can be expressed in terms of integral equations. But
computation of integral operators is time-consuming and hence this approach is not popular.
However, application of wavelet transforms can make this approach viable.
81
The approach proposed is based on the observation that one can approximately represent the
matrix of an integral operator as a product of wavelet transform matrices and a sparse matrix.
This representation leads to an efficient procedure for computing the integral kernels which gives
a significant speed-up in computations thus making integral operator approach competative with
other approaches.
The proposed approach is applied to problems of contruction of harmonic and conformal map
pings. For harmonic mapping problem, there is a formula for solution representation which can
be efficiently applied with the help of wavelet transforms. The problem of conformal domain
mapping can be formulated in terms of integral equation which is solved iteratively using the
fast algorithm of computing the integral operators.
Implicit finite-difference scheme with approximation error O(τ 4 , h8 ) for the heat
conduction equation
V. P. Shapeev
TAM SB RAS, Novosibirsk, Russia ([email protected])
Implicit two-layer difference scheme DS(4,8) with residual term R(τ ; h) ∼ O(τ 4 , h8 ), where τ and
h are grid steps in time and space, is constructed for one-dimensional heat conduction equation.
The scheme stencil consists of 10 points: 5 points on the bottom time layer and 5 points on
the upper layer. The scheme formulas were found by the method of indefinite coefficients under
condition that its residual term R(τ, h) on solution of the differential problem had the highest
possible order of smallness with respect to τ and h[1, 2]. In so doing, all differential consequences
of heat equation were taken into account.
In solving Dirichlets problem, besides the system of difference equations of scheme DS(4,8)
written for the interior grid points one needs two more equations, because within this scheme one
can not obtain independent equations in points near the left and right boundaries. Therefore,
in the present research in points near the left and right boundaries the scheme DS(4,8) was
completed by well-known scheme DS(2,4) on six-point stencil with R(τ, h) ∼ O(τ 2 , h4 ). As the
result, the system of difference equations of scheme DS(4,8) was supplemented with two equations
of scheme DS(2,4) and became determined.
Convergence of the difference solution was tested on problems with known exact solutions. In
numerical experiments on a succession of grids at τ = ch2 it was found out that difference
solution of Dirichlets problem for scheme DS(4,8) combined with DS(2,4) converges with at least
sixth order. The developed implicit finite-difference scheme is free of stiff limits on choosing τ
and h ratio, which is typical for explicit schemes, and converges with third order on grid sequence
at τ = ch (c = const). In order to realize Dirichlet boundary conditions, the scheme DS(4,8)
was completed by scheme DS(3,6) with R(τ, h) ∼ O(τ 3 , h6 ). The scheme DS(4,8) combined with
DS(3,6) converges with eighth order at τ = ch2 (c = const) for some h < h0 .
It is established that numerical calculations with a given high accuracy on compound schemes
DS(4,8) ∪ DS(2,4) and DS(4,8) ∪ DS(3,6) demand much less grid nodes and computational time
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than on scheme DS(2,4) alone. The higher accuracy we need, the more advantageous are the
first schemes.
[1] Shapeev A.V., Shapeev V.P. High order approximation schemes. Proceedings of the 3rd
European Conference on Numerical Mathematics and Advanced Applications, Finland, July
26-30, 1999.
[2] Shapeev A.V., Shapeev V.P. Difference schemes of increased order of accuracy for solving
elliptical equations in domain with curvilinear boundary. Computational Mathematics and
Mathematical Physics, Vol 40, No. 2, pp. 213-221.
Controlled steering, obstacle avoidance and posture stabilisation of car-like mobile
robots via a Lyapunov-based approach
Bibhya Sharma
University of the South Pacific, Fiji Islands (sharma [email protected])
Joint work with Jito Vanualailai.
In this paper we propose an efficient algorithm that considers the multi-task of control and
motion planning of a car-like mobile robot and its posture stabilisation within a constrained
and obstacle-ridden workspace. We have developed a new scheme for posture stabilisation of
the final configurations. We essentially treat the painted boundaries of a predefined parking
bay as static obstacles, which have to be avoided for collision-free parking maneuvers. For this,
we fix circular obstacles at equal intervals on these boundaries and attach to them sufficient
repulsive potentials for the desired avoidance. As a result, we have trajectories with faster
convergence rates and near-optimal postures. This, together with the other kinematic and the
dynamicss constraints have been treated simultaneously, for the first time, via the Lyapunov
based approach. In addition, the approach intrinsically guarantees stability of the kinodynamic
system. We demonstrate the efficiency of the control algorithm with results through simulations
from different scenarios.
On p.p. rings which are reduced
K. P. Shum
The Chinese University of Hong Kong, Hong Kong, SAR ([email protected])
The concept of locally reduced rings will be introduced. Some characterization theorems for
reduced p.p.-rings will be given. Our results strengthen and extend the results of Fraer and
Nicholson and others on p.p. rings in the literature.
83
A linear time knowledge based sorting algorithm
Paul M E Shutler
National Institute of Education, Nanyang Technological University,
Singapore ([email protected])
Well known sorting algorithms such as Quicksort, Heapsort and Shellsort work on the basis
of minimal knowledge about the list being sorted (i.e. they can only tell if x < y or y < x
is true) and as a consequence have O(N logN ) complexity where N is the number of elements
being sorted. Modern sorting needs (e.g. processing data for web-based search engines) demand
the rapid sorting of very large lists, and so the limitations of minimal knowledge are no longer
sustainable. This short communication describes the analysis of the simplest possible knowledge
based sorting algorithm i.e. when the final sorted position of each element can be estimated
with reasonable accuracy. Although this means that the list can be partially presorted, random
variations in the local density of elements in the final sorted list mean that the presorted list must
be expanded by some factor R in order to avoid balking excess elements to the end of the list and
incurring a heavy penalty in the final sort. We analyse the computation time as a function of R,
identify the optimal expansion factor, and show that the predicted O(N ) complexity is in good
agreement with the CPU time taken to sort randomly generated lists of length N = 103 − 105
on a PC.
On the structure of some matrix ring related to injectivity and its generalisations
Chilada Somsup
Department of Mathematics, Mahidol University,
Bangkok 10400, Thailand ([email protected])
Joint work with Nguyen Van Sanh.
In this report, we construct some matrix rings over skew-fields and find wome important prop
erties by using recent results related to injectivity and its generalizations. We also use some new
techniques to illustrate them.
Presentations of trivial reals and Kolmogorov complexity
Frank Stephan
Departments of Computer Science and Mathematics, National University of Singapore,
3 Science Drive 2, Singapore 117543 ([email protected])
Joint work with Guohua Wu (Nanyang Technological University).
84
A real number α between 0 and 1 is identified with the sequence of its binary digits after a
dot, so it is a member of {0, 1}∞ . Such a number is called left-r.e. iff the finite strings left of
it, that is, the representatives of smaller dyadic numbers, are recursively enumerable. A pre
sentation
α ∈ {0, 1}∞ is a prefix-free and recursively enumerable subset of {0, 1}∗ such that
� of −|σ|
. Note that α has a presentation iff α is a left-r.e. real.
α = σ∈V 2
Clearly if α is computable then every presentation is computable but most incomputable reals
have also incomputable presentations. For example, an r.e. and incomputable α has an incom
putable presentation, namely V = {0n 1 : α(n) = 1}. But Downey and LaForte proved that
some incomputable α has only computable presentations. For the further investigations, let A
be the set of all left-r.e. and incomputable α which have only computable presentations. The
following three results are obtained.
First, the members of A are characterized: Let α be left-r.e. and incomputable. Then α ∈
/ A
iff there is a recursive approximation α0 , α1 , . . . such that (1) α0 < α1 < . . . < α, (2) there is a
recursive function h such that all digits of αn beyond h(n) are 0 and (3) for every computable
function g and every m there exist n ≥ m and s ≥ g(n) with αs+1 ≥ αs + 2−n .
Second, it is shown that every member α of A is strongly Kurtz-random. Here α is strongly
Kurtz-random iff there is no recursive function f such that the prefix-free Kolmogorov complex
ity of α(0) . . . α(f (n) − 1) is less than f (n) − n for all n.
Third, a corollary of the second result is that A is disjoint to the Turing ideal of trivial reals.
Here a real α is trivial if there is a constant c such that the prefix-free Kolmogorov complexity
of α(0) . . . α(n − 1) is bounded by c plus the complexity of n itself.
Downey and LaForte showed already that A does not intersect the Turing filter of the reals with
promptly simple Turing degree. Since Nies established the existence of a promptly simple and
trivial real α, the members of A are all Turing incomparable to this α.
Feedforward neural networks model for forecasting trend and seasonal time serioes
Subanar
Mathematics Department, Gadjah Mada University,
Yogyakarta, Indonesia ([email protected])
Joint work with Suhartono.
Many business and economic time series are non-stationary time series that contain trend and
seasonal variations. Seasonality is a periodic and recurrent pattern caused by factors such as
weather, holidays, or repeating promotions. A stochastic trend is often accompanied with the
seasonal variations and can have a significant impact on various forecasting methods. In this
paper, we will investigate the issue of how to use the feed forward neural networks (FFNN)
for modeling effectively time series with both trend and seasonal patterns. Limited empirical
studies on seasonal time series forecasting with neural networks show that some find neural
85
networks are able to model seasonality directly and prior deseasonalization is not necessary, and
others conclude just the opposite. In this research, we study particularly on the effectiveness
of data preprocessing, including detrending and deseasonalization on FFNN modeling and fore
casting performance. We use two kinds of data, simulation and real data. Simulation data are
examined on both additive and multiplicative of trend and seasonality patterns. The results
are compared to those obtained from the classical time series model such as the Box-Jenkins
seasonal autoregressive integrated moving average (ARIMA) models. Our result shows that a
combination of detrending and deseasonalization is the effective data preprocessing on the use
of FFNN for forecasting trend and seasonal time series.
Keywords: Feed forward neural networks, trend, seasonality, time series, forecasting.
A Construction of Balanced Maximum Counting Sequences
I Nengah Suparta
Department of Mathematics, Faculty of Electrical Engineering and Computer Science,
TU Delft, P.O. Box 5031, 2600 GA Delft, Netherlands.
The author is on leave from Dept. of Math. Education, IKIP Singaraja,
Bali, Indonesia ([email protected])
A counting sequence of length n is a (closed) list of all 2n binary n-bit codewords such that each
codeword appears exactly once. The Hamming distance between two n-bit codewords is the
number of bit positions where they differ. The average distance of a closed counting sequence of
length n is the average Hamming distance between the 2n pairs of successive codewords in the
sequence. A counting sequence of length n which has average distance equal to n − 12 is called
a maximum counting sequence. The number of bit changes in bit position i, 1 ≤ i ≤ n, in a
counting sequence of length n, is called the transition count of the bit position i. A balanced
maximum counting sequence is one in which the transition counts of any two bit positions in the
list differ by at most 2. The existence of balanced maximum counting sequences is a longstanding
conjecture of Robinson and Cohn (IEEE Transaction on Computers, C-30, No. 1, (1981) 17
23). In this talk we shall present a construction of balanced maximum counting sequences, and
therefore prove the conjecture of Robinson and Cohn.
Coalgebra of Generalized Power Series Rings
Budi Surodjo
FMIPA Universitas Gadjah Mada Sekip Utara, Yogyakarta, Indonesia (surodjo [email protected])
Joint work with Setiadji Sri Wahyuni and Jurusan Matematika.
Let S be a commutative strictly ordered monoid and R be a commutative ring with an iden
tity. A construction of multiplication between scalar and vector of RS via S-action defines a
subbimodule of RS over its polinomial ring. Generaly over this operation the generalized power
series ring is not a subbimodule of RS . If S is cancellative over its order; we know R[[S]] is a
86
subbimodule of RS . Base on the finiteness conditions of the generalized power series, we first
observe the structure of R[[S]]f and find out coalgebras of R[[S]], over an Artinian monoid and a
Notherian ring. Since R[[S]] is an algebra then we give a sufficient condition, such that (R[[S]]∗ )f
is a coalgebra over R.
Sheaves of algebras over locally Boolean spaces
U. M. Swamy
Dept of mathematics, Andhra University,
Visakhapatnam-530 007, INDIA ([email protected])
A triple (S,P,X) is called a sheaf over X if S and X are topological spaces and P is a local
homeomorphism of S onto X. Sheaves over locally Boolean spaces (i.e, locally compact, totally
disconnected and Hausdorff spaces) are of special nature, in view of the duality between these
and a class of algebraic structures. This duality is analogous to the stone duality between
locally Boolean spaces and Boolean rings. An algebra R=(R,+,0) of type (2,2,0) is called an
almost Boolean ring (ABR) if it satisfies all the identities of a Boolean ring, except possibly
the associativity of +. The concept of an ideal in an ABR is introduced, analogous to that of
a ring, and proved that the set of maximal ideals together with the hull-kernel topology is a
locally Boolean space. Further , if (S,P,X) is a sheaf over a locally Boolean space X, then the
set of continuous local sections on compact open subsets of X has the structure of an almost
Boolean ring, whose space of maximal ideals is homeomorphic to X. On the other hand , if R is
any almost Boolean ring , then a sheaf(S,P,X) is constructed, where X is the space of maximal
ideals of R, and proved that the ABR of local sections of this sheaf is isomorphic to the given
ABR. These two correspondences between ABR’s and sheaves over locally Boolean spaces are
proved to be inverses to each other.
A sheaf (S,P,X) is called a sheaf of algebras of the given type T if, for each p in X, the stalk Sp is
an algebra of type T and the fundamental operations are continuous . An ABR which is also an
algebra of type of T satisfying certain compatibility conditions is called a T-ABR. For a given
type T of algebras, a duality between T-ABR’s and sheaves of T-algebras over locally Boolean
spaces is obtained . The structure theory of ABR’s is discussed in detail and, in particular, the
subdirectly irreducible ABR’s are determined.
On Galois extensions with automorphism group as Galois group
George Szeto
Department of Mathematics, Bradley University,
Peoria, Illinois 61625 ([email protected])
Let B be a ring Galois extension of B G with Galois group G such that B G is a projective
separable C G -algebra where C is the center of B. Then it is shown that G = AutB G (B) and
87
K = �1� where K = {g ∈ G | g(c) = c for all c ∈ C} if and only if either B is an indecomposable
DeMeyer-Kanzaki Galois extension of B G or B = B G e⊕B G (1−e) where e and 1−e are minimal
central idempotents in B. This is a generalization of the case for Galois algebras. Moreover, the
class of indecomposable Galois extentions are also studied.
Some Properties of the Fisher Index of Discrimination
Choon Peng Tan
Faculty of Information and Communication Technology, Universiti Tunku Abdul Rahman,
13, Jalan 13/6, 46200 Petaling Jaya Selangor, Malaysia ([email protected])
We consider tests of two simple hypotheses where the probability of the Type I error α is fixed
and the probabilities of the Type II errors do not exceed a fixed β and the Fisher index of
discrimination is used as a measure of separation of the two hypotheses. In Tan and Tang
(2004), tests on the parameters of a density function belonging to the exponential class family
using the log likelihood-ratio statistic were discussed. It was shown that the Fisher index as a
function of the minimum test sample size converges to a constant depending only on α and β as
the sample size approaches infinity. In this paper, we shall extend the results of Tan and Tang
(2004) to the case where the sampling density function does not belong to the exponential class
and to a general test statistic different from the log likelihood-ratio statistic. Applications to
estimating the minimum sample size of such tests will be discussed. The relationship between
the Fisher index and the relative entropies of the density functions specified under the two
hypotheses will be established.
[1] C.P. Tan and S.F. Tang, ”A characterization of some tests of simple hypotheses using the
Fisher index of discrimination,” Proceedings of the International Sri Lankan Statistical Confer
ence: Visions of Futuristic Methodologies, Basil M. de Silva and Nitis Mukhopadhyay (Editors),
pp. 507 - 514, 28 - 30 December 2004.
Some convexity properties of Orlicz-direct-sum of Banach spaces
Sornsak Thianwan
Department of Mathematics, Faculty of Science, Chiangmai University,
Chiangmai 50200, Thailand ([email protected])
Joint work with Suthep Suantai.
In this paper, we define direct sum of Banach spaces by using Orlicz functions, and consider it
equipped with both Luxemburg and Amemiya norms. Criteria for strict convexity,
locally
�and
n �
uniform convexity in Orlicz-direct-sum space are presented and it is shown that ( i=1 Xi )φ
has presented property (H) if and only if each Xi has the property (H).
88
Classes of identities involving the Dedekind eta function and eisenstein series
Pee Choon Toh
National University of Singapore, Singapore ([email protected])
Joint work with Heng Huat Chan and Shaun Cooper.
Let q = e2πτ where Im(τ ) > 0. The Dedekind eta function, η(τ ), is defined by
η = η(τ ) = q
∞
�
1
24
(1 − q n ).
n=1
In 1955, Newman established remarkable arithmetic properties for the coefficients of η d when
d ∈ D = {2, 4, 6, 8, 10, 14, 26}. Furthermore, Serre (1985) proved that for even d, η d is lacunary
if and only if d belongs to the same set D.
In this work, we obtain partial generalizations of Newman and Serres result. We will use the
theory of elliptic functions and some results of Ramanujan to construct new identities involving
Jacobis Theta functions. These identities when specialized, will yield explicit representations for
η d F where d ∈ D and F is some linear combination of Eisenstein series. Of particular interest is
the representation for η 26 . In 1972, Dyson stated that there exists representations for η d when
ever d = 3, 8, 10, 14, 15, 21, 24, 26, 28, 35, 36 · · · . This list of numbers with the exception of 26
corresponds to the dimensions of finite dimensional simple Lie algebras, and this correspondence
was independently established by MacDonald. The representation of η 26 was known to Atkin
(before 1968) but has remained unpublished.
On Henstock’s version of multiple stochastic integral
Tin-Lam Toh
National Institute of Education, Nanyang Technological University,
Singapore ([email protected])
Joint work with Chew Tuan Seng (National University of Singapore).
It is well known and often emphasized in texts that it is impossible to define stochastic integrals
using the Riemann approach, since the integrators have paths of unbounded variation, and
the integrands are highly oscillatory. The deficiency of the Riemann approach is due to the
uniform meshes used in the Riemann sums. Uniform mesh is unable to handle highly oscillatory
integrands and integrators. A way out of this apparent impasse of the Riemann approach
was introduced by J. Kurzweil and R. Henstock independently in 1950s. They used non-uniform
meshes (meshes that vary from point to point) in the definition of the Riemann-Stieltjes integral.
This technically minor but conceptually important modification of the classical definition of
89
Riemann leads to the integrals which are more general than the Riemann-Stieltjes integral and
the Lebesgue-Stieltjes integral.
It is not surprising that this approach using non-uniform meshes has been successfully used to
give alternative definitions to the stochastic integral with respect to Brownian motion, and even
with respect to semimartingales, see [1, 2, 3].
The theory of Multiple Stochastic Integral was first studied by N. Wiener in 1938. The integrand
in this case consists of real-valued functions whose domain lie on Rn . This study was followed
up in greater details by K. Ito in the early 1950s. Similar to his study of the stochastic integral
in one-dimension, he gave a non-explicit L2 -procedure in defining what we call the Multiple
Ito-Wiener integral.
In this presentation we shall discuss how the generalized Riemann approach can be used to give
an alternative definition to the Multiple Stochastic Integral. The advantage of the generalized
Riemann approach lies in its explicit L2 - procedure and its more intuitive approach in nature.
In constructing the integral on Rn , attention will be paid on the diagonal and the non-diagonal
part. Discussion will also be made on the characterization of the characterization of the class of
all integrable functions in terms of the primitive processes.
[1] Toh T.L., Chew T.S., A Variational Approach to Ito’s Integral, Proceedings of the SAP’s 98,
Taiwan, P291-2999, World Scientific, Singapore, 1999.
[2] Toh T.L., Chew T.S., Tay J.Y., The Non-Uniform Riemann Approach to Ito’s Integral, Real
Analysis Exchange, 2000.
[3] Toh T.L., Chew T.S., The Riemann Approach to Stochastic Integration using Non-Uniform
Meshes, Journal of Mathematical Analysis & Applications, 2000.
On the subnormaliser condition for subgroups
Phi Hung Tong Viet
University of Natural Sciences,
Vietnam ([email protected]; [email protected])
Joint work with Bui Xuan Hai.
A subgroup H of G is said to satisfy the subnormaliser condition in G if for every subgroup K of
G such that H� K, it follows that NG (K) ≤ NG (H). In this paper, we study the properties of
groups with all subgroups satisfying the subnormaliser condition. We establish the connection
between such groups and the so called T -groups, i.e., the groups, in which the normality is a
transitive relation.
90
Stability and consistency of kinetic upwinding for advection-diffusion-equations
Manuel Torrilhon
Department of Mathematics, Hong Kong University of Science and Technology
([email protected])
Numerical methods based on kinetic models of fluid flows are increasingly popular to solve vis
cous and inviscid fluid equations in a finite volume approach. These methods rely on the tools
of kinetic gas theory as a physical modeling framework for computational algorithms for con
tinuum partial differential equations. This physical guidance leads to very robust and accurate
methods, like the so-called BGK-scheme.
The gas-kinetic BGK method approximately solves the BGK model of Boltzmann’s equation at
each cell interface and obtains a numerical flux from integration of the distribution function. It
is extended to viscous fluid flow by use of the Chapman-Enskog distribution function.
This talk introduces the algorithm of the BGK-scheme and investigates stability and consistency
of the numerical method applied to a linear advection diffusion equation. We will discuss several
limiting cases originating from the actual scheme in order to explain the mechanism of kinetic
upwinding.
We show the error of the viscous BGK scheme to be generally first order in time and space.
However, it is proven that a third order “super-convergence” is present in regimes where the
grid Reynolds number is large, i.e., the flow is under-resolved. The stability results consider the
explicit time marching and demonstrate the upwinding ability of the kinetic method. Further
more, the stabily range in the under-resolved case goes beyond that of common finite difference
methods, since the BGK-scheme is shown to allow the time step to be controlled from the
advection only.
These new results for consistency and stability of kinetic schemes demonstrate the ability of
physically constructed numerical methods to produce minimized error constants and improved
stability conditions. The talk will also discuss what can be learnt for standard methods for
viscous flow from the present results.
On the relations between the classes of QF3-rings and hereditary rings
Maliwan Tunapan Department of Mathematics Mahidol University,
Bangkok 10400, Thailand ([email protected])
Joint work with guyen Van Sanh.
Menabu Harada (1965, 1978,1980) has studied some generalizations of QF-rings by introducing
two following conditions:
(i) Every non-small right R-module contains a non-zero injective submodule;
(ii) Every non-cosmall right R-module contains a non-zero projective direct summand.
91
(iii) Every indecomposable injective module E is hollow, i.e., every proper submodule is small
in E.
The main results in our report are the following 2 Theorems:
Theorem 1. Let R be a right perfect right QF-3 ring. The following conditions are equivalent:
(1) R is right hereditary;
(2) ei R/ei J t is injective for each primitive idempotent ei and for any integer t;
(3) R is Morita equivalent to a direct sum of ring of upper-triangular matrices over a division
rings
Theorem 2. Let R be a right perfect ring. If J 2 (R) = 0 or R is right hereditary, then
thefollowing conditions are equivalent:
(1) The condition (i) holds;
(2) The condition (ii) holds and each ei R contains a unique minnimal submodule;
(3) R is right QF-3.
Irregular Sampling
Kim Tuan Vu
Department of Mathematics, University of West Georgia,
Carrollton, GA30118 ([email protected])
Joint work with A. Boumenir (University of West Georgia)
Let f (t) ∈ L2 (R) and fˆ(λ) be its Fourier transform. In signal processing f is called a (finite
energy) signal, and fˆ is the frequency content of the signal f . Signal f is bandlimited with
the bandwidth T if f has no frequency higher than T . Most of signals like human voices, are
bandlimited.
The fundamental of digital signal processing is the Shannon sampling formula
f (t) =
∞
�
n=−∞
f
� nπ � sin(T t − nπ)
T
T t − nπ
,
that allows to recover a bandlimited signal with bandwidth T from its samples at equidistant
points spaced Tπ apart. The sampling rate Tπ per second is called the Nyquist sampling rate.
This is the minimum rate at which the signal needs to be sampled in order to reconstruct it
92
exactly. Denote by N (R) the number of sampled points on the interval [−R, R], the Nyquist
sampling rate tells us that in order to recover a signal with bandwidth T exactly, one must have
limR→∞
N (R)
T
≥ .
2R
π
All available sampling formulae, such as Shannon, Paley-Wiener, and Kramer, need a priori the
information of the bandwidth, in order to set the sampling rate.
Now suppose that our receiver gets a signal from a unknown source. So no information about
the bandwidth of the signal is known. What conditions should we put on a set of sampling
points so that we can recover a bandlimited signal f with unknown bandwidth, and how do we
recover the signal?
A solution of this problem is based on the Gelfand-Levitan theory for inverse spectral problems
and Kramer’s theorem. The connection between the two is that given a sequence of points
satisfying certain conditions, we can construct a self-adjoint Sturm-Liouville problem such that
these points are precisely its eigenvalues. Using eigenfunctions of this Sturm-Liouville problem,
Kramer’s theorem then allows us to employ the spectral theorem as a sampling formula.
Measure entropy of discrete dynamical systems
Widodo
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Gadjah Mada
University, Yogyakarta, Indonesia, (widodo [email protected])
Let(X, β, µ) be a measure space with µ(X) = 1, and let f : X → X be a function from
X into itself. The iterations of f provide discrete dynamical systems whose behavior can be
very complicated. In discrete dynamical systems, it is investigated the behavior of an iteration
�x, f 1 (x), f 2 (x), f 3 (x), · · · , f n (x), · · · �of every point x ∈ X, where f 0 (x) =: x and f n (x) =:
(f ◦ f ◦ f ◦ · · · f )(x), for n = 1, 2, 3, · · · , i.e. f n is the n-fold composition of f with itself. For
A ⊂ X, we define f i (A) =: {f i (x) : x ∈ A} and f −i (A) =: {x ∈ X : f i (x) ∈ A}, i = 0, 1, 2, 3, · · · .
There are three indicators for the complexity of discrete dynamical systems, i.e. measure entropy,
topological entropy and Liapunov Exponent. This paper is intended to review the concept of
measure entropy.
For every finite sub -algebra β of , we define f −1 (α) =: {f −1 (A) : A ∈ α}. Let α and β
be two finite sub σ-algebras of β . The join of α and β is defined by α ∨ β =: {A ∩ B :
A ∈ αandB ∈ β}. In this case β is said to be a refinement of α, in symbols, α < β, if
for every B ∈ β there exists A ∈ α such that B ⊂ A The function f is called µ-invariant
if µ(f −1 (B)) = µ(B), for all B ∈ B. If α = {A1 .A2 , · · · , Ak } is a finite sub σ-algebra of B,
k
�
we define H(α) =: − µ(Ai )lnµ(Ai ). Measure entropy of f with respect to α is defined by
i=1
1
H(α
∨
f −1 (α) ∨ f −2 (α) ∨ · · · ∨ f −(n−1) (α)).
n→∞ n
h(f, α) =: lim
93
The measure entropy of f is defined
by h(f)=: suph(f, ): is a finite sub -algebra of . In this paper it is also studied the important
properties of h(f).
Following these properties, it is studied the behaviour of successive iterations of piecewise linear
transformations (plt) on the unit interval [0,1], which are allowed to have finitely many of
discontinuity points. First, it will be discussed the basic properties of plt and its measure
entropy. It is well known that for continuous case, there exists a simple way for calculating the
measure entropy of plt, called Rohlin Formula. But it is still a problem either this formula is
true for discontinuous case or not. In this study the problem above is investigated. The methods
used in this paper are plus and minus symbol sequences, and plus and minus orbits. By using
these methods, the invariant measure can be constructed explicitly and finally it is proved that
Rohlin Formula is true for discontinuous case.
The memory effect in fast impregnation processes
Jagath K Wijerathna
Department of Mathematics, University of Colombo,
Colombo 03, Sri Lanka ([email protected])
In literature, impregnation processes have been modelled widely using classical darcys law. These
models give us reasonable approximations of real situation in most of the industrial molding
processes but they are far away from the real process when volume forces of the flow are varying
with time. In our model we consider a correction of classical darcys law which is called darcys law
with memory[2]. Darcys law with memory is an integrodifferential equation with a parameter
dependent integral which is not easy to solve using direct standard discretization. We solve this
equation with an alternative approach using semigroup theory on linear evolution operators and
results of numerical experiments shows that when the volume forces of the flow is asymptotically
steady darcys law with memory recovers the classical darcys law otherwise the darcys law with
memory gives a better approximation to the reality rather than classical darcys law.
Keywords: Porous media, Homogenization, time dependent permeability
Totally disconnected locally compact groups and their automorphisms
George Willis
University of Newcastle, Australia ([email protected])
Let G be a totally disconnected locally compact group and let α be an automorphism of G.
Then the scale of α is the positive integer
s(α) := min{[α(U ) : α(U ) ∩ U ] : U ≤ Gis compact and open}
94
and any compact open subgroup U at which the scale is attained is said to be tidy for α. These
ideas form the basis of a structure theory for totally disconnected locally compact groups.
For comparison with the connected case, recall that the solution of Hilbert’s Fifth Problem tells
us that connected locally compact groups may be approximated by Lie groups and that Lie
group theory is based ultimately on linear algebra. The scale and tidy subgroups fill the role
of linear algebra in the study of totally disconnected groups: subgroups tidy for α correspond
to the Jordan canonical form of a linear transformation and the scale of α corresponds to its
eigenvalues.
These correspondences will be explained in more detail and illustrated with examples in the
talk.
Benders’ partitioning algorithm for an integrated one-dimensional cutting stock
and transportation problem
Sirirat Wongprakornkul
Department of Statistics, Faculty of Science, Khon Kaen University, Thailand
([email protected])
Large-scale problems, where the number of variables is large, typically cannot be solved because
of time and memory storage limitations. One way to solve large-scale problem was to partition
the problem into smaller size sub problems. First of all, the integration of the one-dimensional
cutting stock problem with multiple cutting facilities and the transportation problem (1D-CSP
&TP) was the focus for applications with large-scale problems. In this subject, Benders De
composition approach was applied to partition the particular problem into multi-sub problems.
For partitioning the problem, two methods were presented, called B1 and B2. In B1, there
were two groups of variables: X’s are CSP variables and Y’s are TP variables. After the relaxed
mathematical model of 1D-CSP&TP was formed; B1 would partition the problem into multi-sub
problems of one-dimensional cutting stock problems and a master problem. In B2, there were
three groups of variables: X’s are CSP variables, Y’s are TP variables, and Z’s are balanced
variables. After the relaxed mathematical model of 1D-CSP&TP was formed; B2 would parti
tion the problem into multi-sub problems of one-dimensional cutting stock problems, multi-sub
problems of transportation problems, and a master problem. Advantages and disadvantages
were discussed to compare with both methods.
Intervals containing exactly one c.e. degree
Guohua Wu
School of Mathematics, Statistics and Computer Science, Victoria University of Wellington,
Wellington, New Zealand ([email protected])
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It is known that there are intervals of d.c.e. degrees containing no c.e. degrees, and that if
an interval of d.c.e. degrees containing two or more c.e. degrees, then this interval contains
infinitely many c.e. degrees. We proved that the bi-isolating degrees are dense in the high c.e.
degrees. After this, Arslanov asked whether the bi-isolating degrees occur in every jump class.
In this paper, we prove that there are low bi-isolating degrees, providing a partial solution to
Arslanov’s question.
A Feasible Direction Algorithm Without Line Search for Solving Max-Bisection
Problems
Fengmin Xu
Faculty of Science, Xi’an Jiaotong University, Xi’an, 710049, P.R.China
([email protected])
Joint work with Xu Chengxian and Xue Honggang.
This paper concerns the solution of the NP-hard max-bisection Problems. NCP functions are
employed to convert max-bisection problems into continuous non-linear programming problems.
Solving the resulting continuous nonlinear programming problem generates a solution that gives
an upper bound on the optimal value of the max-bisection problem. From the solution, the
greedy strategy is used to generate a satisfactory approximate solution of the max-bisection
problem. A feasible direction method without line searches is proposed to solve the resulting
continuous nonlinear programming, and the convergence of the algorithm to KKT point of
the resulting problem is proved. Numerical experiments and comparisons on well-known test
problems, and on randomly generated test problems show that the proposed method is robust,
and very effcient.
On Characterizations of a commutator Galois extension
Lianyong Xue
Department of Mathematics, Bradley University
Peoria, Illinois 61625, USA ([email protected])
Let B be a ring with 1, C the center of B , G a finite automorphism group of B, B G the set
of elements in B fixed under each element in G, and VB (B G ) the commutator subring of B G in
B. Then B is called a commutator Galois extension with Galois group G if VB (B G ) is a Galois
extension with Galois group G|VB (B G ) ∼
= G. In this paper, it is shown that B is a commutator
Galois extension with Galois group G is equivalent to each of the following statements:
(1) VB (B G ) ∗ G is an Azumaya C G -algebra.
(2) VB (B G ) ∗ G is a Hirata separable extension of VB (B G ).
(3) B ∗ G is a Hirata separable extension of B G and contains B G as a direct summand as a
96
B G -bimodule.
(4) B is a Galois extension with Galois group G such that B ∼
= B G ⊗C G VB (B G ) by the multiG
plication map and VB (B ) is finitely generated projective over C G .
Moreover, a relationship is studied between the following classes of Galois extensions: (1) com
mutator Galois extensions and Azumaya Galois extensions (that is, B is a Galois extension
with Galois group G such that B G is an Azumaya C G -algebra) , and (2) commutator Galois
extensions and centrally projective Galois extensions (that is, B is a Galois extension of B G and
centrally projective over B G ).
Some results on the irreducible modules for the algebraic groups and
corresponding lie algebras of type A
Jiachen Ye
Department of Applied Mathematics, Tongji University,
Shanghai 200092, Peoples Republic of China ([email protected] )
A monomial basis and a filtration of subalgebras for the universal enveloping algebra U(g) of a
complex simple Lie algebra gl of type Al is given in this note. In particular, a new multiplicity
formula for the Weyl module V (λ) of U(gl ) is obtained in this note. Moreover, we describe the
weight set of irreducible modules for the algebraic groups of type A, and prove that the weight
set of an irreducible module L(λ) is the same as that of the Weyl module V (λ) when λ ∈ Λ1 is
a restricted weight.
Global stability of elementary waves for certain dissipative hyperbolic
conservation laws
Huijiang Zhao
Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences
([email protected])
This talk is concerned with some recent results on the global stability of certain elementary waves
for a class of hyperbolic systems of conservation laws with dissipative terms. The contents are
based on recent works joint with Professors Feimin Huang, Kenji Nishihara, Tong Yang, and
Yinchuan Zhao.
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