No 2 (2024)

А series of hypercomplex numerical sets having a compositional structure is shown to arise in an abstract environment consisting of randomly oriented 1D geometric objects. We focus on the series’core set which is represented by a groupoid-type algebraic system with one binary operation, associative multiplication, admitting zero-dividers but having no unity; an original Cayley-type table for this set is given. Introduction of the operation of reversible addition extends the set to algebras of real, complex and hypercomplex numbers with units built of the initial simple elements. It is demonstrated that this fundamental mathematics is tightly linked with the origin of the basic equation of quantum physics.

This article is the second article in the series of “non-standard” formalism of quantum theory. It develops the set-theoretic paradigm and substantiates the notion of fundamental rotation, which was introduced at the intuitive level in the first article of the series. It is shown that the fundamental rotation is a carrier of an ordinal infinity. We prove a number of theorems on the relation between the carriers of ordinal and quantitative infinity in particular, we formulate the condition under which the carrier of infinity is a set. It is shown that for the set-theoretic continuum S ( N ) this condition is not fulfilled, and hence S ( N ), contrary to G. Cantor’s wish, is not a set. Due to the large amount of material, this article is the second part of the general article. It presents the key points of the set-theoretic paradigm, while the third part formulates and develops the ordinal paradigm.

The article puts forward and substantiates the following thesis. A mathematical model is not sufficient to define a scientific model. A scientific model must include a conceptual model and a mathematical model built on the basis of the first. As a conceptual model, it is proposed to use ontology’s, which in Computer Science are used as a representation of knowledge. It is shown that it is possible to construct ontology’s for models of modern physics, in particular for models of quantum mechanics.

The paper shows that space-time with certain geometrical properties can be a source of origin of characteristic properties of elementary material objects. It is demonstrated that such space-time is a homogeneous stationary space with rotation caused by the rotation of congruences of time-like world lines.

Considering the particle as a spherical resonator of “electromagnetic” de Broglie waves, it was shown that the propagation of spherical de Broglie waves along mutually opposite radii leads to the emergence of standing spherical waves, the nodes and antinodes of which can be associated with the spatial distribution of the particle’s characteristics. Thus, in the work it is examinated a steady-state oscillatory process inside a hollow resonator. Using a suitable phase difference between the converging and diverging spherical waves, it is possible to obtain the spatial distribution of characteristics inside the de Broglie wave resonator particle without a singularity at the center (r=0).

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