Duality in process of noncommutative deformation and topological nature of Cherepanov-Rice integral

In t his paper it is showed, that for the noncommutative deformation simultaneously there exist also loading deformation H, and unloading deformation \(H^v\). The real deformation is a combination of these types of deformations. The criterion of destruction J reflects topological character of medium, i.e. it defines properties of symmetry of medium at destruction. It is possible to tell, that during destruction the energy is released not continuously but and discretely. This situation is reflected through topological number Q or number of unloading, connected to him.

PLETHORA OF q-OSCILLATORS POSSESSING PAIRWISE ENERGY LEVEL DEGENERACY

Using the q, p-deformed oscillators as basic generating system, we obtain diverse classes (which form distinct sectors of functional continua) of novel versions of q-deformed oscillators, all of which share the property of "accidental" degeneracy within a fixed pair of energy levels Em = Em+1, m ≥ 1, occurring at the real deformation parameter fixed by an appropriate value q(m) that depends on m and on particular model. Likewise, the degeneracy E0 = Ek (where k ≥ 2) takes place, for properly fixed q = q(k), in most of those models. The formerly studied model of q-oscillator known as the Tamm–Dancoff cutoff deformed oscillator is contained in the continua as isolated special case.

REPRESENTATIVE BRAIDS FOR LINKS ASSOCIATED TO PLANE IMMERSED CURVES

In [ AC 2], A'Campo associates a link in S3 to any proper generic immersion of a disjoint union of arcs into a 2-disc. We give a sample algorithmic way to produce, from the immersion, a representative braid for such links. As a by-product we get a minimal representative braid for any algebraic link, from a divide associated to a real deformation of the polynomial defining the link.