Geometrical and physical interpretation of evolution governed by general complex algebra

AbstractIn this paper we explore a geometrical and physical matter of the evolution governed by the generator of General Complex Algebra, GC2. The generator of this algebra obeys a quadratic polynomial equation. It is shown that the geometrical image of the GC2-number is given by a straight line fixed by two given points on Euclidean plane. In this representation the straight line possesses the norm and the argument. The motion of the straight line conserving the norm of the line is described by evolution equation governed by the generator of the GC2-algebra. This evolution is depicted on the Euclidean plane as rotational motion of the straight line around the semicircle to which this line is tangent. Physical interpretation is found within the framework of the relativistic dynamics where the quadratic polynomial is formed by mass-shell equation. In this way we come to a new representation for the momenta of the relativistic particle

Hyperbolic Cosines and Sines Theorems for the Triangle Formed by Arcs of Intersecting Semicircles on Euclidean Plane

The hyperbolic cosines and sines theorems for the curvilinear triangle bounded by circular arcs of three intersecting circles are formulated and proved by using the general complex calculus. The method is based on a key formula establishing a relationship between exponential function and the cross-ratio. The proofs are carried out on Euclidean plane

On the Problem of Definability of the Computably Enumerable Degrees in the Difference Hierarchy

© 2018, Pleiades Publishing, Ltd. Questions of definability of computably enumerable degrees in the difference hierarchy (degrees of sets from finite levels of the Ershov difference hierarchy) are studied. Several approaches to the solution of this problem are outlined

Degrees of categoricity of rigid structures

© Springer International Publishing AG 2017. We prove that there exists a properly 2-c.e. Turing degree d which cannot be a degree of categoricity of a rigid structure

Turing Degrees in Refinements of the Arithmetical Hierarchy

© 2018, Springer Science+Business Media, LLC, part of Springer Nature. We investigate the problem of characterizing proper levels of the fine hierarchy (up to Turing equivalence). It is known that the fine hierarchy exhausts arithmetical sets and contains as a small fragment finite levels of Ershov hierarchies (relativized to ∅n, n < ω), which are known to be proper. Our main result is finding a least new (i.e., distinct from the levels of the relativized Ershov hierarchies) proper level. We also show that not all new levels are proper