Ternary semigroups of topological transformations

A ternary semigroup is a nonempty set with a ternary operation which is associative. The purpose of the present paper is to give a characterization of open sets of finite-dimensional Euclidean spaces by ternary semigroups of pairs of homeomorphic transformations and extend to ternary semigroups certain results of L.M. Gluskin concerned with semigroups of homeomorphic transformations of finite-dimensional Euclidean spaces.

A Plethora of Non-Bending Surfaces of Revolution: Classifications and Explicit Parameterizations

As the title itself suggests here we are presenting extremely reach two/three parametric families of non-bending rotational surfaces in the three dimensional Euclidean space and provide the necessary details about their natural classifications and explicit parameterizations. Following the changes of the relevant parameters it is possible to trace out the ``evolution'' of these surfaces and even visualize them through their topological transformations. Many, and more deeper questions about their metrical properties, mechanical applications, etc. are left for future explorations.

Ontology of a metaphor: the way from apophaticism to cataphaticism in the cognitive path to God

This work reasonably substitutes the literary and aesthetic understanding of metaphor as a poetic technique of expressiveness with the epistemological understanding of metaphor as an instrument for expanding the semantic possibilities of perception. Ontological dimension of a metaphor is reconstructed in accordance with the concept of “inertia” of nature. Nature repeatedly reproduces the previously acquired in some area of the phenomenal, while appearance of a human proliferated such reproduction to the area of the noumenal. Repetition of such reproductions of the acquired testifies to the fractality of being. Metaphor turns the linear evolution of semantics into a complex nonlinear process through topological transformations. It becomes most efficient in the cataphatic operation of “composition”. Unlike apophaticism, metaphor not only takes to a new level of cognition, but also arranges all these levels in accordance with the fractal nature of being. Therefore, metaphor is the primary means for transforming fractality into an epistemological tool in the area of the noumenal. If analytical reasoning fractionize the idea, by increasing the entropy of semantics, the metaphorical assertions reduce it, focusing the thought and synthesizing new semantic patterns. Such a focus fractalizes semantics, which should ultimately result in a semantic explosion and, most likely, bring a man closer to God.