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The modern notion of the axiomatic method developed as a part of the conceptualization of mathematics starting in the nineteenth century. The basic idea of the method is the capture of a class of structures as the models of an axiomatic system. The mathematical study of such classes of structures is not exhausted by the derivation of theorems from the axioms but includes normally the metatheory of...
This position paper presents a lightweight approach for knowledge modeling centered around the notion of distinction. Concept models are represented using UML class diagrams. Distinctions between concepts are established by using attributes and stating properties of the attributes using the @L-is calculus on words. This calculus relies on simple laws to manipulate relationships between words. We show...
The completeness of quasi-metric space is researched in this paper. And the closed ball theorem of quasi-metric space is proved. Finally, the Ekeland theorem, Caristi fixed point theorem.
In many-valued logic theories, the decision and construction for Sheffer functions is an important problem. The decision for Sheffer functions is interrelated to the decision for completeness of functions set, and the solution can be reduced to determining the minimal coverings of precomplete. It's well known that each precomplete set is a function set, T(Gm), preserving the relation Gm, therefore,...
An operator describing overlapping is substituted/or the effort modality of Moss and Parikh's logic of knowledge and topology. This means that the spatial idea underlying that system, viz closeness, is replaced with covering. We ask what properties of frames can be expressed by means of the new language. It turns out that the language is, for example, capable of characterizing directed spaces. After...
With the adoption of UML by the OMG and industry as the lingua-franca of visual systems modelling, one begins to ponder what will come next in this field? This tutorial brings a vision for visual modelling beyond UML. We present and consolidate radical new notations, proposed in a series of research papers and with quickly increasing adoption by industry, for the specification of complex systems in...
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