8 research outputs found
Interpretation of percolation in terms of infinity computations
In this paper, a number of traditional models related to the percolation
theory has been considered by means of new computational methodology that does
not use Cantor's ideas and describes infinite and infinitesimal numbers in
accordance with the principle `The part is less than the whole'. It gives a
possibility to work with finite, infinite, and infinitesimal quantities
numerically by using a new kind of a computer - the Infinity Computer -
introduced recently in by Ya.D. Sergeyev in a number of patents. The new
approach does not contradict Cantor. In contrast, it can be viewed as an
evolution of his deep ideas regarding the existence of different infinite
numbers in a more applied way. Site percolation and gradient percolation have
been studied by applying the new computational tools. It has been established
that in an infinite system the phase transition point is not really a point as
with respect of traditional approach. In light of new arithmetic it appears as
a critical interval, rather than a critical point. Depending on "microscope" we
use this interval could be regarded as finite, infinite and infinitesimal short
interval. Using new approach we observed that in vicinity of percolation
threshold we have many different infinite clusters instead of one infinite
cluster that appears in traditional consideration.Comment: 22 pages, 7 figures. arXiv admin note: substantial text overlap with
arXiv:1203.4140, arXiv:1203.316
Some Applications of the Percolation Theory. Brief Review of the Century Beginning
The review is a brief description of the state of problems in percolation
theory and their numerous applications, which are analyzed on base of
interesting papers published in the last 15-20 years. At the submitted papers
are studied both the cluster system of the physical body and its impact on the
object in general, and adequate mathematical tools for description of critical
phenomena too. Of special interest are the data, first, the point of phase
transition of certain of percolation system is not really a point, but it is a
critical interval, and second, in vicinity of percolation threshold observed
many different infinite clusters instead of one infinite cluster that appears
in traditional consideration.Comment: arXiv admin note: text overlap with arXiv:1104.5376, arXiv:1205.0691
by other author
UN SEMPLICE MODO PER TRATTARE LE GRANDEZZE INFINITE ED INFINITESIME
A new computational methodology allowing one to work in a new way with infinities and infinitesimals is presented in this paper. The new approach, among other things, gives the possibility to calculate the number of elements of certain infinite sets, avoids indeterminate forms and various kinds of divergences. This methodology has been used by the author as a starting point in developing a new kind of computer – the Infinity Computer – able to execute computations and to store in its memory not only finite numbers but also infinite and infinitesimal ones