Triangular norms in the study of probabilistic metric spaces as a special kind of associative functions defined on the unit interval. These functions have found applications in many areas since then. In this study, we present new methods for constructing triangular norms and triangular conorms on an arbitrary bounded lattice under some constraints. Also, we give some illustrative examples for t...

We are concerned with the limit distribution of lt-norms RN (t) = ‖XN‖t (of order t) of samples XN = (X1, . . . , XN ) of i.i.d. positive random variables, as N → ∞, t → ∞. The problem was first considered by Schlather(10), but the case where {Xi} belong to the domain of attraction of Gumbel’s double exponential law (in the sense of extreme value theory) has largely remained open (even for an e...

Recently, in [4], we have discussed the following distributive equation of implications I(x, T1(y, z)) = T2(I(x, y), I(x, z)) over t-representable t-norms, generated from strict t-norms, in interval-valued fuzzy sets theory. In this work we continue these investigations, but with the assumption that T1 is generated from nilpotent t-norms, while T2 is generated from strict t-norms. As a byproduc...

Let fn denote a kernel density estimator of a bounded continuous density f in the real line. Let Ψ(t) be a positive continuous function such that ‖Ψf‖∞<∞. Under natural smoothness conditions, necessary and sufficient conditions for the sequence √ nhn 2 log h−1 n supt∈R ∣∣Ψ(t)(fn(t)−Efn(t))∣∣ (properly centered and normalized) to converge in distribution to the double exponential law are obtaine...

We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of L∞(0, T ; L2...

We present a procedure for constructing multi-valued tnorms and t-conorms. Our construction uses a pair of single-valued t-norms and the pair of dual t-conorms to construct interval-valued t-norms u and t-conorms t. In this manner we can combine desirable characteristics of different t-norms and t-conorms; furthermore if we use the t-norm ∧ and t-conorm ∨, then (X,u,t) is a superlattice, i.e. t...