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Assembled here is a collection of articles presented at a NATO
ADVANCED STU DY INSTITUTE held at Puerto de la Cruz, Tenerife,
Spain during the period of July 10th to 21st, 1989. In addition to
the editors of these proceedings Professor Larry L. Schumaker from
Vanderbilt University, Nashville, Tennessee, served as a member of
the international organizing committee. The contents of the
contribu tions fall within the heading of COMPUTATION OF CURVES AND
SURFACES and therefore address mathematical and computational
issues pertaining to the dis play, modeling, interrogation and
representation of complex geometrical objects in various scientific
and technical environments. As is the intent of the NATO ASI
program the meeting was two weeks in length and the body of the
scientific activities was organized around prominent experts. Each
of them presented lectures on his current research activity. We
were fortunate to have sixteen distinguished invited speakers
representing nine NATO countries: W. Bohm (Federal Republic of
Germany), C. de Boor (USA), C.K. Chui (USA), W. Dahmen (Federal
Republic of Germany), F. Fontanella (Italy), M. Gasca (Spain), R.
Goldman (Canada), T.N.T. Goodman (UK), J.A. Gregory (UK), C.
Hoffman (USA), J. Hoschek (Federal Republic of Germany), A. Le
Mehaute (France), T. Lyche (Norway), C.A. Micchelli (USA), 1.1.
Schumaker (USA), C. Traas (The Netherlands). The audience consisted
of both young researchers as well as established scientists from
twelve NATO countries and several non-NATO countries.
This volume contains both invited lectures and contributed talks
presented at the meeting on Total Positivity and its Applications
held at the guest house of the University of Zaragoza in Jaca,
Spain, during the week of September 26-30, 1994. There were present
at the meeting almost fifty researchers from fourteen countries.
Their interest in thesubject of Total Positivity made for a
stimulating and fruitful exchange of scientific information.
Interest to participate in the meeting exceeded our expectations.
Regrettably, budgetary constraints forced us to restriet the number
of attendees. Professor S. Karlin, of Stanford University, who
planned to attend the meeting had to cancel his participation at
the last moment. Nonetheless, his almost universal spiritual
presence energized and inspired all of us in Jaca. More than
anyone, he influenced the content, style and quality of the
presentations given at the meeting. Every article in these
Proceedings (except some by Karlin hirnself) references his
influential treatise Total Positivity, Volume I, Stanford
University Press, 1968. Since its appearance, this book has
intrigued and inspired the minds of many researchers (one of us, in
his formative years, read the galley proofs and the other of us
first doubted its value but then later became its totally committed
disciple). All of us present at the meeting encourage Professor
Karlin to return to the task of completing the anxiously awaited
Volume 11 of Total Positivity.
This volume contains both invited lectures and contributed talks
presented at the meeting on Total Positivity and its Applications
held at the guest house of the University of Zaragoza in Jaca,
Spain, during the week of September 26-30, 1994. There were present
at the meeting almost fifty researchers from fourteen countries.
Their interest in thesubject of Total Positivity made for a
stimulating and fruitful exchange of scientific information.
Interest to participate in the meeting exceeded our expectations.
Regrettably, budgetary constraints forced us to restriet the number
of attendees. Professor S. Karlin, of Stanford University, who
planned to attend the meeting had to cancel his participation at
the last moment. Nonetheless, his almost universal spiritual
presence energized and inspired all of us in Jaca. More than
anyone, he influenced the content, style and quality of the
presentations given at the meeting. Every article in these
Proceedings (except some by Karlin hirnself) references his
influential treatise Total Positivity, Volume I, Stanford
University Press, 1968. Since its appearance, this book has
intrigued and inspired the minds of many researchers (one of us, in
his formative years, read the galley proofs and the other of us
first doubted its value but then later became its totally committed
disciple). All of us present at the meeting encourage Professor
Karlin to return to the task of completing the anxiously awaited
Volume 11 of Total Positivity.
Assembled here is a collection of articles presented at a NATO
ADVANCED STU DY INSTITUTE held at Puerto de la Cruz, Tenerife,
Spain during the period of July 10th to 21st, 1989. In addition to
the editors of these proceedings Professor Larry L. Schumaker from
Vanderbilt University, Nashville, Tennessee, served as a member of
the international organizing committee. The contents of the
contribu tions fall within the heading of COMPUTATION OF CURVES AND
SURFACES and therefore address mathematical and computational
issues pertaining to the dis play, modeling, interrogation and
representation of complex geometrical objects in various scientific
and technical environments. As is the intent of the NATO ASI
program the meeting was two weeks in length and the body of the
scientific activities was organized around prominent experts. Each
of them presented lectures on his current research activity. We
were fortunate to have sixteen distinguished invited speakers
representing nine NATO countries: W. Bohm (Federal Republic of
Germany), C. de Boor (USA), C.K. Chui (USA), W. Dahmen (Federal
Republic of Germany), F. Fontanella (Italy), M. Gasca (Spain), R.
Goldman (Canada), T.N.T. Goodman (UK), J.A. Gregory (UK), C.
Hoffman (USA), J. Hoschek (Federal Republic of Germany), A. Le
Mehaute (France), T. Lyche (Norway), C.A. Micchelli (USA), 1.1.
Schumaker (USA), C. Traas (The Netherlands). The audience consisted
of both young researchers as well as established scientists from
twelve NATO countries and several non-NATO countries."
In recent years, considerable progress has been made in the
understanding of problems of learning and generalization. In this
context, intelligence basically means the ability to perform well
on new data after learning a model on the basis of given data. Such
problems arise in many different areas and are becoming
increasingly important and crucial towards many applications such
as in bioinformatics, multimedia, computer vision and signal
processing, internet search and information retrieval, datamining
and textmining, finance, fraud detection, measurement systems,
process control and several others. Currently, the development of
new technologies enables to generate massive amounts of data
containing a wealth of information that remains to become explored.
Often the dimensionality of the input spaces in these novel
applications is huge. This can be seen in the analysis of
micro-array data, for example, where expression levels of thousands
of genes need to be analyzed given only a limited number of
experiments. Without performing dimensionality reduction, the
classical statistical paradigms show fundamental shortcomings at
this point. Facing these new challenges, there is a need for new
mathematical foundations and models in a way that the data can
become processed in a reliable way. The subjects in this
publication are very interdisciplinary and relate to problems
studied in neural networks, machine learning, mathematics and
statistics.
The recent appearance of wavelets as a new computational tool in
applied mathematics has given a new impetus to the field of
numerical analysis of Fredholm integral equations. This book gives
an account of the state of the art in the study of fast multiscale
methods for solving these equations based on wavelets. The authors
begin by introducing essential concepts and describing conventional
numerical methods. They then develop fast algorithms and apply
these to solving linear, nonlinear Fredholm integral equations of
the second kind, ill-posed integral equations of the first kind and
eigen-problems of compact integral operators. Theorems of
functional analysis used throughout the book are summarised in the
appendix. The book is an essential reference for practitioners
wishing to use the new techniques. It may also be used as a text,
with the first five chapters forming the basis of a one-semester
course for advanced undergraduates or beginning graduates.
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