Reliable Computing:
Special = Issue on the=20 Linkages Between Interval
Mathematics and Fuzzy Set Theory
Guest = editor:=20 Weldon A. Lodwick

Reliable Computing will devote a = special=20 issue to
papers that address the interrelationship between interval=20 mathematics
and fuzzy set theory. The connection between interval = mathematics=20 and
fuzzy set theory is evident in the extension principle,=20 arithmetic,
logic, and in the mathematics of uncertainty. Much of the = research to
date has been in the use of interval mathematics in fuzzy = set=20 theory, in
particular fuzzy arithmetic and fuzzy interval analysis. = This may=20 be
because intervals can be considered as a particular type of fuzzy=20 set.
The impact of fuzzy set theory on interval mathematics is not = quite=20 as
evident. For example, it is clear that fuzzy logic, fuzzy control, = fuzzy
neural networks, and fuzzy cluster analysis, are four important = areas=20 of
fuzzy set theory. The impact of interval analysis on these four = areas=20 is
not as apparent. Can the development in these areas of fuzzy set=20 theory
inform research in interval mathematics?

There are areas of interval = mathematics and=20 fuzzy set theory that have
developed in parallel with little or no=20 interchange of ideas. In
particular the extension principle of Zadeh = and the=20 united extension of
R.E. Moore as well as subsequent research in this = area=20 has largely been
developed independently. Both are related to = set-valued=20 functions. Is
there a useful underlying unifying mathematics? = Secondly,=20 dependencies
and their effect on the resulting arithmetic has more = recently=20 been a
part of the fuzzy set theory literature and approaches=20 independently
developed from what has been known in the interval = analysis=20 community
almost since the beginnings of interval analysis research. = Are=20 there
other areas of interval analysis research that would be useful = for=20 the
fuzzy set theory community to know about?

One of the paths of interval = mathematics=20 research has led to validation
analysis. Is there a useful comparable = counterpart for fuzzy set theory?
Interval analysis is the way to = model the=20 uncertainty arising from
computer computations. Thus, interval = analysis=20 shares mathematical
uncertainty modeling with the field of fuzzy set = theory.=20 So,
fundamentally, what are the common points between interval = analysis=20 and
fuzzy set theory? In interval analysis, convergence of algorithms = has
been an area of research. Are there extensions of these = approaches=20 to
fuzzy algorithms? In the area of interval analysis, much work has=20 been
done in validation methods for differential equations. A few=20 research
papers have appeared in this area in the fuzzy set theory = setting.=20 Are
there areas of cross-fertilization? There are many research = papers=20 in
the area of optimization in both interval analysis and fuzzy set=20 theory.
What is the interrelationship between interval and fuzzy=20 optimization?
Is there a fundamental mathematical foundation out of = which=20 both arise?

The following lists a few areas = of=20 interest.  It is indicative and not
exhaustive.

- Fuzzy and interval mathematical = analysis
- Comparative analysis of the interval and fuzzy logics
- = Upper=20 and lower dependency bounds in interval and fuzzy
= mathematics
-=20 Dependency analysis in interval and fuzzy computations
- Fuzzy and = interval=20 methods in classification (cluster) analysis
- The use of fuzzy set = theory=20 and interval analysis methods in
neural networks
- Interval = and=20 fuzzy ordering methods
- The use of interval analysis and fuzzy set = theory in=20 neural
networks
- The use of interval analysis and fuzzy = set theory=20 in surface
modeling, interpolation and approximation
- The=20 application of interval analysis to fuzzy algorithms and vice
=20 versa
- The methods and relationship between interval and = fuzzy
=20 optimization
- Fuzzy and interval logic controllers
- Computer = systems in=20 support of fuzzy number data types and
associated numerical = algorithms=20 akin to such interval analysis computer
systems as that of = S.Rump,=20 "INTLAB---Interval Laboratory" at:
http://www.= ti3.tu-harburg.de/~rump/intlab/index.html
-=20 Interval and fuzzy methods for differential equations
- Convergence = and=20 complexity analysis of interval and fuzzy algorithms
- One of the = uses of=20 interval analysis is in the validation of
solutions under=20 computational and data errors. Is there a comparable use
of = fuzzy set=20 and possibility theory in the validation of solutions under
=20 uncertainty?

In addition to new results in = theoretical=20 analysis, innovative
applications, and computer implementations, we = invite=20 insightful
surveys. Please send a copy of your manuscript (in = electronic=20 form
preferably---LATEX, including the style files required, = postscript=20 or
pdf format) to:

Professor Weldon A. = Lodwick
Department of=20 Mathematics---Campus Box 170
P.O. = Box=20 173364
weldon.lodwick [at] cudenver [dot] edu
Telephone:=20 +1 303 556-8462

Schedule:
June 15, 2002: = Deadline for=20 submission of papers to the special issue.
March 15, 2003: Revisions to accepted = papers=20 due.

Manuscripts will be subjected to = the usual=20 reviewing process
and should conform to the standards and formats as=20 indicated in the
"Information for Authors" section inside the back = cover=20 of
Reliable Computing. Contributions should not exceed 32 = pages.

&nbs= p;    =20 Reliable Computing
=      =20 Volume 8, issue 1, 2002

= Mathematical=20 Research

Two Ways to Extend the Cholesky = Decomposition=20 to Block Matrices  with
Interval Entries
Uwe Schaefer=20
1-20

Approximate Quantified Constraint = Solving=20 by  Cylindrical Box
Decomposition
Stefan Ratschan=20
21-42

Exact Minkowski Products of N = Complex Disks=20
Rida  T.  Farouki, Helmut Pottmann
43-66

Verification of Invertibility of = Complicated=20 Functions
over Large Domains
Jens Hoefkens, Martin=20 Berz
67-82

A New Subdivision Strategy for = Range=20 Computations
Paluri S. V. Nataraj, Suresh Mandir Sheela=20
83-92

&nbs= p;  =20 Information

Special Issue on the Linkages = Between=20 Interval
Mathematics and Fuzzy Set=20 Theory
93-95