It was, and still is, a fundamental law of crystallography that fivefold rotational symmetry and three-dimensional (3D) lattice periodicity are mutually incompatible. With this in mind, the Israeli materials scientist Dan Shechtman did not trust his eyes when he inspected electron diffractograms of rapidly solidified Al-Mn. What he expected to see was either diffraction patterns with sharp Brag...

A simple argument is given as to why it is always trivial to calculate Lefschetz and Nielsen numbers for iterated function systems or dynamical systems in hyperspaces. The problem is reduced to a simple combinatorical situation on a finite set. In the papers [3], [4] the question was raised whether the Lefschetz or even the Nielsen number might be used in the hyperspace H(X) of all nonempty com...

Many ecological and evolutionary studies require to quantify the degree of adaptation subterranean species caves or other systems. In 1962, Kenneth A. Christiansen (1924–2017) coined term “troglomorphism” illustrate process suite adaptive traits organisms (“troglomorphic traits” “troglomorphies”) (Christiansen 1962). Although this was proposed in a paper published French, its derivatives (e.g. ...

We deal with a hyperspace selection problem in the setting of connected spaces. present two solutions this illustrating difference between selections for nonempty closed sets, and those at most two-point sets. In first case, we obtain characterisation compact orderable latter case --- that same property is equivalent to existence ternary relation on space, known as cyclic order, gives so called...

We investigate two closure-type properties, the Reznichenko property and the Pytkeev property, in hyperspace topologies.

Abstract: Noise-based logic, by utilizing its multidimensional logic hyperspace, has significant potential for low-power parallel operations in beyond-Moore-chips. However universal gates for Boolean logic thus far had to rely on either time averaging to distinguish signals from each other or, alternatively, on squeezed logic signals, where the logic-high was represented by a random process and...

The main issue addressed in this paper is how to help learners navigate in existing web-based learning resources. Towards this issue, we introduce a learner-centered navigation path planning. The key idea is to provide learners with a space, in which they can see through WWW pages to plan a navigation path. In this paper, we also demonstrate an assistant system, which is composed of hyperspace ...

In this paper, we propose a framework based on a subsymbolic approach for the support of exploratory activities in a hyperspace. By using it, it is possible to express the semantic features of the whole hyperspace and the states of exploratory activities in topological order. This approach is applied to generate the navigation information for the exploratory activity. The space explored is chan...

Using the isomorphism from [3] between the category Pries of Priestley spaces and the category PStone of pairwise Stone spaces we construct a Vietoris hyperspace functor on the category PStone related to the Vietoris hyperspace functor on the category Pries [4, 18, 24]. We show that the coalgebras for this functor determine a semantics for the positive fragment of modal logic. The sole novelty ...