The main bottleneck of the research in metric space searching is the so-called curse of dimensionality, which makes the task of searching some metric spaces intrinsically diicult, whatever algorithm is used. A recent trend to break this bottleneck resorts to probabilistic algorithms , where it has been shown that one can nd 99% of the elements at a fraction of the cost of the exact algorithm. T...

This paper summarizes recent work on the theory of epsilon entropy for probability distributions on complete separable metric spaces. The theory was conceived [3] in order to have a framework for discussing the quality of data storage and transmission systems. The concept of data source was defined in [4] as a probabilistic metric space: a complete separable metric space together with a probabi...

The notion of a bead metric space is defined as a nice generalization of the uniformly convex normed space such as $CAT(0)$ space, where the curvature is bounded from above by zero. In fact, the bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces and normed bead spaces are identical with uniformly convex spaces. In this paper, w...

In this paper, we define the concept of probabilistic like Menger (probabilistic like quasi Menger) space (briefly, PLM-space (PLqM-space)). We present some coupled fixed point and fixed point results for certain contraction type maps in partially order PLM-spaces (PLqM- spaces).

Simple heuristics often show a remarkable performance in practice for optimization problems. Worst-case analysis falls short of explaining this performance. Because this, “beyond worst-case analysis” algorithms has recently gained lot attention, including probabilistic algorithms. The instances many problems are essentially discrete metric space. Probabilistic such nevertheless mostly been cond...

We define and study a quantale-valued Wijsman structure on the hyperspace of all non-empty closed sets of a quantale-valued metric space. We show its admissibility and that the metrical coreflection coincides with the quantale-valued Hausdorff metric and that, for a metric space, the topological coreflection coincides with the classical Wijsman topology. We further define an index of compactnes...

The method for numerical study of the open quantum systems is proposed. It is based on our recent results in the field of numerical integration with respect to probability measures in complete separable metric spaces. Our approximations satisfy the condition of being exact on a class of polynomial functionals of a given degree. The method does not require preliminary discretisation of space and...

Based on previous results that study the completion of fuzzy metric spaces, we show that every intuitionistic fuzzy quasi-metric space, using the notion of fuzzy metric space in the sense of Kramosil and Michalek to obtain a generalization to the quasi-metric setting, has a bicompletion which is unique up to isometry.

In this paper, the notion of cyclic $varphi$-contraction in fuzzymetric spaces is introduced and a fixed point theorem for this typeof mapping is established. Meantime, an example is provided toillustrate this theorem. The main result shows that a self-mappingon a G-complete fuzzy metric space has a unique fixed point if itsatisfies the cyclic $varphi$-contraction. Afterwards, some results inco...