## Fuzzy approach and uncertainty processing

### Field characteristic

#### Copulas and Risk Analysis

We have studied limit properties of transformations of copulas.
Limit copulas are (if they exist) closely related to maximum
attractors copulas and thus their application in risk analysis
(e.g., in finance, hydrology, etc.) is expected. Observe that
copulas are Schur concave functions, what is not the case of
triangular norms. Thus our interest was turned also to
characterization of triangular norms which are Schur concave. Also
some other relaxed properties of triangular norms, especially of
their sections, have been studied. Application of attained results
is expected in several optimization tasks dealing with triangular
norms. Another group of our results concerns the algebraic
structure of triangular norms. Especially, we have introduced and
studied the Archimedean components of triangular norms,
construction of triangular norms with prescribed Archimedean
components, etc. An important role of triangular sub-norms in this
setting was stressed. Comparison meaningful functions are an
indispensable tool in measurement evaluation and subsequent
decision making. Till now, only continuous comparison meaningful
functions have been characterized. We have succeeded to
characterize all comparison meaningful functions. In this
characterization, a key role is played by the lattice polynomials.
Among other results, a mention of the building preference proposal
structures utilizing the tools of fuzzy set and fuzzy logic
theories was obtained.

#### Vagueness and Uncertainty

Parallel run with the continuation of the investigation of fuzzy
coalitional games, continuing from the previous periods, a new
topic was open. The formal presentation of geographical data is
realized via specific objects similar to $n$-dimensional vectors,
$n>2$, where the first two components are real numbers
(geographical coordinates) and the remaining components reflect
various phenomena connected with the coordinated location. They
are mathematically represented by real or integer numbers, other
discrete sets, or also by logical variables. Each of the above
components can be vague, what means that it can be processed by
means of fuzzy set theoretical tools. The vagueness is usually
reduced in the terrestrial geography, but it becomes significant
in the geographical representation of remote or on the spot
realized sensing of other planets or moons.
The research of this topic is in its opening period. It can use
the previous results derived for vector fuzzy quantities,
multi-criterion fuzzy ordering, and fuzzy logical operations.
The aim is to derive synthetical methods for parallel processing
of all relevant components respecting their vagueness and uncertainty.

### People

### Selected publications

- Butnariu D., Klement E. P., Mesiar R., Navara M.: Sufficient
triangular norms in many-valued logics with standard negation.
Archive of Mathematical Logic 14 (2005), 44, 829-849.
- Klement E. P., Mesiar R., Pap E.: Archimax copulas and invariance
under transformations. Comptes Rendus de l'Académie des
Sciences Paris. Mathématiques 340, (2005), 755-758.
- Klement E. P., Mesiar R., Pap E.: Triangular norms: Basic notions
and properties. In: Logical, Algebraic, Analytic, and
Probabilistic Aspects of Triangular Norms (Klement E. P., Mesiar
R., eds.). Elsevier, Amsterdam 2005, pp. 17-62.
- Mareš M., Mesiar R.: Aggregation of complex quantities. In:
Proceedings of AGOP'2005 - International Summer School on
Aggregation Operators and Their Applications (Mesiar R., Pasi G.,
Faré M. eds.). Universitá della Svizzeria Italiana, Lugano 2005, pp. 85-88.
- Mesiar R.: Discrete copulas - what they are. In: Joint
EUSFLAT-LFA 2005, Conference Proceedings. (Montseny E., Sobrevilla
P. eds.). Universitat Politecnica de Catalunya, Barcelona 2005, pp. 927-930.

### Grants and projects

**Mathematics, Informatics and Cybernetics: Tools and Applications**

Milan Mareš, key project of the Academy of Sciences
of the Czech Republic No. K 1019101.
**Semantic Part of Vague Verbal Data**

Milan Mareš, grant No. A 1075301 of
the Grant Agency of the Czech Academy of Science.
**Aggregation Principles in the Models of Mathematical Economy**

Milan Mareš, grant No. 402/04/1294
of the Czech Science Foundation.