We study the boundary value problem $-{\mathrm{d}iv\left(\right(\left|\nabla u\right|}^{p_1\left(x\right)-2}+{\left|\nabla u\right|}^{p_2\left(x\right)-2}\left)\nabla u\right)=f(x,u)$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in ${ℝ}^N$. Our attention is focused on two cases when $f(x,u)=\pm (-\lambda |u{|}^{m\left(x\right)-2}u+\left|u{|}^{q\left(x\right)-2}u\right)$, where $m\left(x\right)=max\{p_1\left(x\right),p_2\left(x\right)\}$ for any $x\in \overline{\Omega }$ or $m\left(x\right)<q\left(x\right)<\frac{N·m\left(x\right)}{(N-m(x\left)\right)}$ for any $x\in \overline{\Omega }$. In the former case we show the existence of infinitely many weak solutions for any $\lambda >0$. In the latter we prove that if $\lambda$ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ${\mathbb{Z}}_2$-symmetric version for even functionals...

In the paper an interface is proposed that combines flexible (fuzzy) querying and data mining functionality. The point of departure is the fuzzy querying interface designed and implemented previously by the present authors. It makes it possible to formulate and execute, against a traditional (crisp) database, queries containing imprecisely specified conditions. Here we discuss possibilities to extend it with some data mining features. More specifically, linguistic summarization of data (databases),...

It was shown that there is a statistical learning problem – a version of the expectation maximization (EMX) problem – whose consistency in a domain of cardinality continuum under the family of purely atomic probability measures and with finite hypotheses is equivalent to a version of the continuum hypothesis, and thus independent of ZFC. K. P. Hart had subsequently proved that no solution to the EMX problem can be Borel measurable with regard to an uncountable standard Borel structure on $X$, and...

This paper verifies a result of [9] concerning graphoidal structure of Shenoy's notion of independence for Dempster-Shafer theory belief functions. Shenoy proved that his notion of independence has graphoidal properties for positive normal valuations. The requirement of strict positive normal valuations as prerequisite for application of graphoidal properties excludes a wide class of DS belief functions. It excludes especially so-called probabilistic belief functions. It is demonstrated that the...

The knowledge of causal relations provides a possibility to perform predictions and helps to decide about the most reasonable actions aiming at the desired objectives. Although the causal reasoning appears to be natural for the human thinking, most of the traditional statistical methods fail to address this issue. One of the well-known methodologies correctly representing the relations of cause and effect is Pearl's causality approach. The paper brings an alternative, purely algebraic methodology...

The class of contrast intensification operators is formally defined and it's lattice structure studied. The effect of these operators in the referential classifications derived from special kinds of fuzzy relations is also determined. Results and examples are presented providing contrast intensification operators which keep quasi-uniformity structures generated by fuzzy relations while diminishing the fuzziness or the entropy of the relations.

Multiple-valued logics are useful for dealing with uncertainty and imprecision in Knowledge-Based Systems. Different problems can require different logics. Then we need mechanisms to translate the information exchanged between two problems with different logics. In this paper, we introduce the logical foundations of such logics and the communication mechanisms that preserve some deductive properties. We also describe a tool to assist users in the declaration of logics and their communication mechanisms....

Four notions of factorizability over arbitrary directed graphs are examined. For acyclic graphs they coincide and are identical with the usual factorization of probability distributions in Markov models. Relations between the factorizations over circuits are described in detail including nontrivial counterexamples. Restrictions on the cardinality of state spaces cause that a factorizability with respect to some special cyclic graphs implies the factorizability with respect to their, more simple,...

Temporal Constraint Networks are a well-defined, natural and efficient formalism for representing temporal knowledge based on metric temporal constraints. They support the representation of both metric and some qualitative temporal relations and are provided with efficient algorithms based on CSP techniques. Recently, a generalization based on fuzzy sets has been proposed in order to cope with vagueness in temporal relations. In this paper we generalize some earlier definitions for Fuzzy Temporal...